{"id":"n-66d46e23b1cea6df","revision":0,"first_index":0,"fingerprint":"f6-b1029f17d1b64336","name":"shift_x_plus_one of basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["5","3"],["50","60","18"],["750","1350","810","162"],["15000","36000","32400","12960","1944"],["375000","1125000","1350000","810000","243000","29160"],["11250000","40500000","60750000","48600000","21870000","5248800","524880"],["393750000","1653750000","2976750000","2976750000","1786050000","642978000","128595600","11022480"],["15750000000","75600000000","158760000000","190512000000","142884000000","68584320000","20575296000","3527193600","264539520"],["708750000000","3827250000000","9185400000000","12859560000000","11573604000000","6944162400000","2777664960000","714256704000","107138505600","7142567040"],["35437500000000","212625000000000","574087500000000","918540000000000","964467000000000","694416240000000","347208120000000","119042784000000","26784626400000","3571283520000","214277011200"],["1949062500000000","12863812500000000","38591437500000000","69464587500000000","83357505000000000","70020304200000000","42012182520000000","18005221080000000","5401566324000000","1080313264800000","129637591776000","7071141369600"],["116943750000000000","841995000000000000","2778583500000000000","5557167000000000000","7502175450000000000","7202088432000000000","5041461902400000000","2592751835520000000","972281938320000000","259275183552000000","46669533039360000","5091221786112000","254561089305600"],["7601343750000000000","59290481250000000000","213445732500000000000","469580611500000000000","704370917250000000000","760720590630000000000","608576472504000000000","365145883502400000000","164315647576080000000","54771882525360000000","13145251806086400000","2151041204632320000","215104120463232000","9927882482918400"],["532094062500000000000","4469590125000000000000","17431401487500000000000","41835363570000000000000","69028349890500000000000","82834019868600000000000","74550617881740000000000","51120423690336000000000","26838222437426400000000","10735288974970560000000","3220586692491168000000","702673460179891200000","105401019026983680000","9729324833260032000","416971064282572800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","128","3072","98304","3932160","188743680","10569646080","676457349120","48704929136640","3896394330931200","342882701121945600","32916739307706777600","3423340888001504870400","383414179456168545484800"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0"],["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190698,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-d4c886dee13c11ec"]},{"id":190699,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-9daa35c48f863bb7","e-d4c886dee13c11ec"]},{"id":190700,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-9daa35c48f863bb7","e-d4c886dee13c11ec"]},{"id":190701,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-9daa35c48f863bb7","e-d4c886dee13c11ec"]},{"id":190702,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-9daa35c48f863bb7"]}],"region_exact":{},"incoming_edges":[{"id":"e-9daa35c48f863bb7","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-66d46e23b1cea6df","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"226cf1d0fbc112e2f05ad864478ffa3493742ff6faf1386806c65c2a61056d6a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c211a6e140527a22","role":"sequence","name":"basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-d4c886dee13c11ec","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-66d46e23b1cea6df","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"159d04605a67f9976f89a504aa01790872aab1b878e134384ce5782ba5049570","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c211a6e140527a22","role":"sequence","name":"basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0caafe52abc061f9","kind":"fingerprint_related","node_a_id":"n-09bd967d44af22a1","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-09bd967d44af22a1","other":{"id":"n-09bd967d44af22a1","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","18"],["525","630","189"]],"card_rows":[["1"],["30","18"],["525","630","189"],["7000","12600","7560","1512"]],"row_sums_preview":["1","48","1344","28672","516096","8257536","121110528","1660944384"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610","basis A007318"],"edges":["e-3fa86ece9d0bedd1","e-59db270ff1385597"],"text":"A178822 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A178822 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-0ee3b3aa0c5651ba","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-61a576bbbb666f30","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-61a576bbbb666f30","other":{"id":"n-61a576bbbb666f30","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["81","270","225"],["729","3645","6075","3375"],["6561","43740","109350","121500","50625"]],"card_rows":[["81","270","225"],["729","3645","6075","3375"],["6561","43740","109350","121500","50625"],["59049","492075","1640250","2733750","2278125","759375"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013610","reverse"],"edges":["e-a617f0e181e2ec6f","e-60d01f2c9f2eb56b"],"text":"A038329 \u2192 basis A013610 \u2192 reverse"},"headline":"A038329 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-13fdf5d9d8b2fafc","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-70416a49496e0b07","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-70416a49496e0b07","other":{"id":"n-70416a49496e0b07","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","18"],["900","1080","324"]],"card_rows":[["1"],["30","18"],["900","1080","324"],["27000","48600","29160","5832"]],"row_sums_preview":["1","48","2304","110592","5308416","254803968","12230590464","587068342272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-62ade03cd9f25f04","e-71b9a24e2c029225"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-17b84d727c0ad6e6","kind":"fingerprint_related","node_a_id":"n-13fb4d204a4545c3","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-13fb4d204a4545c3","other":{"id":"n-13fb4d204a4545c3","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","15"],["27","90","75"]],"card_rows":[["3"],["9","15"],["27","90","75"],["81","405","675","375"]],"row_sums_preview":["3","24","192","1536","12288","98304","786432","6291456"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-618e7f7547136f14","e-e1338bedb0bc5159"],"text":"A120910 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A120910 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-40d8199b54297ea9","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-4906612bf3a39f4d","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-4906612bf3a39f4d","other":{"id":"n-4906612bf3a39f4d","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","480","400"],["1728","8640","14400","8000"],["20736","138240","345600","384000","160000"]],"card_rows":[["144","480","400"],["1728","8640","14400","8000"],["20736","138240","345600","384000","160000"],["248832","2073600","6912000","11520000","9600000","3200000"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["x \u21a6 x/(x+1)"],"edges":["e-277c1139c8aed132"],"text":"A038334 \u2192 x \u21a6 x/(x+1)"},"headline":"A038334 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-416b3460b4266941","kind":"fingerprint_related","node_a_id":"n-3ccd0708169ccfae","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-3ccd0708169ccfae","other":{"id":"n-3ccd0708169ccfae","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","270","81"],["3375","6075","3645","729"],["50625","121500","109350","43740","6561"]],"card_rows":[["225","270","81"],["3375","6075","3645","729"],["50625","121500","109350","43740","6561"],["759375","2278125","2733750","1640250","492075","59049"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013610"],"edges":["e-a617f0e181e2ec6f"],"text":"A038329 \u2192 basis A013610"},"headline":"A038329 \u2192 basis A013610","chain":null}},{"id":"l-4b06ac69db03e00d","kind":"fingerprint_related","node_a_id":"n-bbf993cd74f140c5","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-bbf993cd74f140c5","other":{"id":"n-bbf993cd74f140c5","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","12"],["250","300","90"]],"card_rows":[["1"],["20","12"],["250","300","90"],["2500","4500","2700","540"]],"row_sums_preview":["1","32","640","10240","143360","1835008","22020096","251658240"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-24fdbe8219e7135e","e-294d8fb9d4a59a43"],"text":"A178820 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A178820 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-4b37e2e460fb2193","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-016c78ecb5900322","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-016c78ecb5900322","other":{"id":"n-016c78ecb5900322","name":"reversal of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","120","36"],["1000","1800","1080","216"],["10000","24000","21600","8640","1296"]],"card_rows":[["100","120","36"],["1000","1800","1080","216"],["10000","24000","21600","8640","1296"],["100000","300000","360000","216000","64800","7776"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609","reverse"],"edges":["e-31086658f6c15e80","e-7c209be1b5e86449"],"text":"A013612 \u2192 basis A013609 \u2192 reverse"},"headline":"A013612 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-4f698d1938e80aad","kind":"fingerprint_related","node_a_id":"n-8c7af6eedfa5a565","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-8c7af6eedfa5a565","other":{"id":"n-8c7af6eedfa5a565","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["25","15"],["375","450","135"]],"card_rows":[["1"],["25","15"],["375","450","135"],["4375","7875","4725","945"]],"row_sums_preview":["1","40","960","17920","286720","4128768","55050240","692060160"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-0c010b1c5e20b0cb","e-b9e43e8e38a60c73"],"text":"A178821 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A178821 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-4f8ccb9dbb3501f2","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-cb1e68669081ca55","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-cb1e68669081ca55","other":{"id":"n-cb1e68669081ca55","name":"shift_x_plus_one of basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","6"],["150","180","54"]],"card_rows":[["1"],["10","6"],["150","180","54"],["3000","5400","3240","648"]],"row_sums_preview":["1","16","384","12288","491520","23592960","1321205760","84557168640"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-1eb90284df2d6b2d","e-ca6047f896fef525"],"text":"A156992 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A156992 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5638e3fc69a56420","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-263d5a8e971d7391","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-263d5a8e971d7391","other":{"id":"n-263d5a8e971d7391","name":"basis_map_A013610 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1296","4320","3600"],["46656","233280","388800","216000"],["1679616","11197440","27993600","31104000","12960000"]],"card_rows":[["1296","4320","3600"],["46656","233280","388800","216000"],["1679616","11197440","27993600","31104000","12960000"],["60466176","503884800","1679616000","2799360000","2332800000","777600000"]],"row_sums_preview":["9216","884736","84934656","8153726976","782757789696","75144747810816","7213895789838336","692533995824480256"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013611","basis A013610"],"edges":["e-fd91849598d32af1","e-830f228615096df7"],"text":"A038319 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A038319 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-59a88d71ab9ad88e","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-f9c8690cbff4f27f","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-f9c8690cbff4f27f","other":{"id":"n-f9c8690cbff4f27f","name":"basis_map_A013610 of basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","30"],["324","1080","900"]],"card_rows":[["1"],["18","30"],["324","1080","900"],["5832","29160","48600","27000"]],"row_sums_preview":["1","48","2304","110592","5308416","254803968","12230590464","587068342272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609","basis A013610"],"edges":["e-c5caeaebd0a50686","e-e1bd9b07558f601f"],"text":"A013622 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A013622 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-6495e5cbcd2ee4b0","kind":"fingerprint_related","node_a_id":"n-e0e61e54d312e320","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-e0e61e54d312e320","other":{"id":"n-e0e61e54d312e320","name":"shift_x_plus_one of basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","3"],["50","60","18"]],"card_rows":[["1"],["5","3"],["50","60","18"],["625","1125","675","135"]],"row_sums_preview":["1","8","128","2560","57344","1376256","34603008","899678208"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-13e67b7b83b67a66","e-6859df88d728c4d6"],"text":"A085880 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A085880 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-66570788d6bcb52e","kind":"fingerprint_related","node_a_id":"n-6a76f32847485778","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-6a76f32847485778","other":{"id":"n-6a76f32847485778","name":"derivative of basis_map_A007318 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["30","18"]],"card_rows":[["3"],["30","18"],["225","270","81"],["1500","2700","1620","324"]],"row_sums_preview":["0","3","48","576","6144","61440","589824","5505024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A007318","D"],"edges":["e-66f5a589f0412003","e-5ec80f908a2ace99"],"text":"A013620 \u2192 basis A007318 \u2192 D"},"headline":"A013620 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-71af5ff3878fc920","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-6ff233938df68c62","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-6ff233938df68c62","other":{"id":"n-6ff233938df68c62","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","480","144"],["8000","14400","8640","1728"],["160000","384000","345600","138240","20736"]],"card_rows":[["400","480","144"],["8000","14400","8640","1728"],["160000","384000","345600","138240","20736"],["3200000","9600000","11520000","6912000","2073600","248832"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A007318","basis A013609"],"edges":["e-37fb8ca4839f71ce","e-20d7759351a4efa8"],"text":"A038284 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A038284 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-758c610240381aac","kind":"fingerprint_related","node_a_id":"n-24fd2fda67fd61e1","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-24fd2fda67fd61e1","other":{"id":"n-24fd2fda67fd61e1","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","10"],["36","120","100"]],"card_rows":[["1"],["6","10"],["36","120","100"],["216","1080","1800","1000"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-0e13ca82ccc1a33c","e-5c109959660b3c9c"],"text":"A038243 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038243 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-8b93c3e66d22a526","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-acd1496158b15db9","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-acd1496158b15db9","other":{"id":"n-acd1496158b15db9","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["120","200"],["1080","3600","3000"],["8640","43200","72000","40000"]],"card_rows":[["120","200"],["1080","3600","3000"],["8640","43200","72000","40000"],["64800","432000","1080000","1200000","500000"]],"row_sums_preview":["320","7680","163840","3276800","62914560","1174405120","21474836480","386547056640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609","D"],"edges":["e-31086658f6c15e80","e-a4296a25ffb8b9b1"],"text":"A013612 \u2192 basis A013609 \u2192 D"},"headline":"A013612 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-8c56d35d796023df","kind":"fingerprint_related","node_a_id":"n-c44f7114310405ad","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-c44f7114310405ad","other":{"id":"n-c44f7114310405ad","name":"Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","5"],["9","30","25"]],"card_rows":[["1"],["3","5"],["9","30","25"],["27","135","225","125"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013622"],"derivation":{"root":"n-c44f7114310405ad","steps":[],"edges":[],"text":"A013622"},"headline":"A013622: Triangle of coefficients in expansion of (3+5x)^n.","chain":null}},{"id":"l-b51e36cce4af9da5","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-18408e461c718031","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-18408e461c718031","other":{"id":"n-18408e461c718031","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["27","45"],["729","2430","2025"]],"card_rows":[["1"],["27","45"],["729","2430","2025"],["19683","98415","164025","91125"]],"row_sums_preview":["1","72","5184","373248","26873856","1934917632","139314069504","10030613004288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610","basis A013610"],"edges":["e-cc1debec4a411a9c","e-3da1106719a8deab"],"text":"A038271 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038271 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-bbe691933ecd170b","kind":"fingerprint_related","node_a_id":"n-d524cbe046c3619e","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-d524cbe046c3619e","other":{"id":"n-d524cbe046c3619e","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","120","100"],["216","1080","1800","1000"],["1296","8640","21600","24000","10000"]],"card_rows":[["36","120","100"],["216","1080","1800","1000"],["1296","8640","21600","24000","10000"],["7776","64800","216000","360000","300000","100000"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609"],"edges":["e-31086658f6c15e80"],"text":"A013612 \u2192 basis A013609"},"headline":"A013612 \u2192 basis A013609","chain":null}},{"id":"l-c4e5f47ea24b95bc","kind":"fingerprint_related","node_a_id":"n-c600c5b3860fc422","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-c600c5b3860fc422","other":{"id":"n-c600c5b3860fc422","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","9"],["225","270","81"]],"card_rows":[["1"],["15","9"],["225","270","81"],["3375","6075","3645","729"]],"row_sums_preview":["1","24","576","13824","331776","7962624","191102976","4586471424"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013610","basis A007318"],"edges":["e-b88e4ed4246b1dea","e-ae46eb6dafa4d24c"],"text":"A038221 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038221 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-c50fd46bf96ddafb","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-b885e5ca17d9686b","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-b885e5ca17d9686b","other":{"id":"n-b885e5ca17d9686b","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3600","4320","1296"],["216000","388800","233280","46656"],["12960000","31104000","27993600","11197440","1679616"]],"card_rows":[["3600","4320","1296"],["216000","388800","233280","46656"],["12960000","31104000","27993600","11197440","1679616"],["777600000","2332800000","2799360000","1679616000","503884800","60466176"]],"row_sums_preview":["9216","884736","84934656","8153726976","782757789696","75144747810816","7213895789838336","692533995824480256"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-85e8843cd1dca083","e-122262de9b39d5d9"],"text":"A038338 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A038338 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c7ae7829c29d29e3","kind":"fingerprint_related","node_a_id":"n-a2df8adfe5ad37e6","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-a2df8adfe5ad37e6","other":{"id":"n-a2df8adfe5ad37e6","name":"shift_x_plus_one of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["10","6"],["50","60","18"]],"card_rows":[["2"],["10","6"],["50","60","18"],["250","450","270","54"]],"row_sums_preview":["2","16","128","1024","8192","65536","524288","4194304"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-28b777b3c0b25332","e-1530d4c2d448dba5"],"text":"A028326 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A028326 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d15c59e85f78dbf5","kind":"fingerprint_related","node_a_id":"n-384c17a53af3442c","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-384c17a53af3442c","other":{"id":"n-384c17a53af3442c","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","20"],["144","480","400"]],"card_rows":[["1"],["12","20"],["144","480","400"],["1728","8640","14400","8000"]],"row_sums_preview":["1","32","1024","32768","1048576","33554432","1073741824","34359738368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611"],"edges":["e-62a1b0a86f60f435"],"text":"A038271 \u2192 basis A013611"},"headline":"A038271 \u2192 basis A013611","chain":null}},{"id":"l-d5b509b97271a177","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-a4e7ee9042e15b92","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-a4e7ee9042e15b92","other":{"id":"n-a4e7ee9042e15b92","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["20"],["480","800"]],"card_rows":[["20"],["480","800"],["8640","28800","24000"],["138240","691200","1152000","640000"]],"row_sums_preview":["0","20","1280","61440","2621440","104857600","4026531840","150323855360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","D"],"edges":["e-62a1b0a86f60f435","e-00fc065f23e775fd"],"text":"A038271 \u2192 basis A013611 \u2192 D"},"headline":"A038271 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-d7768caf215aff3c","kind":"fingerprint_related","node_a_id":"n-adf2466ee0571e22","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-adf2466ee0571e22","other":{"id":"n-adf2466ee0571e22","name":"derivative of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["30","50"]],"card_rows":[["5"],["30","50"],["135","450","375"],["540","2700","4500","2500"]],"row_sums_preview":["0","5","80","960","10240","102400","983040","9175040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["D"],"edges":["e-7a137375e7b37544"],"text":"A013622 \u2192 D"},"headline":"A013622 \u2192 D","chain":null}},{"id":"l-de7d71226159e368","kind":"fingerprint_related","node_a_id":"n-e7973e04614ef858","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-e7973e04614ef858","other":{"id":"n-e7973e04614ef858","name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","6"],["100","120","36"]],"card_rows":[["1"],["10","6"],["100","120","36"],["1000","1800","1080","216"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610","basis A013609"],"edges":["e-b9cdb58d54bdc029","e-23eee42afd6ddf98"],"text":"A038255 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038255 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-e1846d2ad72bbb48","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-46b66c368a0b724e","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-46b66c368a0b724e","other":{"id":"n-46b66c368a0b724e","name":"basis_map_A013610 of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["10","6"]],"card_rows":[["1"],["10","6"],["75","90","27"],["500","900","540","108"]],"row_sums_preview":["0","1","16","192","2048","20480","196608","1835008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","basis A013610"],"edges":["e-62aaaa59f91d42c1","e-d412e18869728fdb"],"text":"A038231 \u2192 D \u2192 basis A013610"},"headline":"A038231 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-f0f85c329999257b","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-25c780f7547ef542","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-25c780f7547ef542","other":{"id":"n-25c780f7547ef542","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","12"],["400","480","144"]],"card_rows":[["1"],["20","12"],["400","480","144"],["8000","14400","8640","1728"]],"row_sums_preview":["1","32","1024","32768","1048576","33554432","1073741824","34359738368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","basis A013610"],"edges":["e-9c08db68889738c4","e-56dc4c4a20ef3f27"],"text":"A027467 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A027467 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-fa016910d2fa3ec7","kind":"fingerprint_related","node_a_id":"n-66d46e23b1cea6df","node_b_id":"n-ef590ccf585da7ab","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-ef590ccf585da7ab","other":{"id":"n-ef590ccf585da7ab","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["270","162"],["6075","7290","2187"],["121500","218700","131220","26244"]],"card_rows":[["270","162"],["6075","7290","2187"],["121500","218700","131220","26244"],["2278125","5467500","4920750","1968300","295245"]],"row_sums_preview":["432","15552","497664","14929920","429981696","12039487488","330225942528","8916100448256"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013610","D"],"edges":["e-a617f0e181e2ec6f","e-66e39408aad9696c"],"text":"A038329 \u2192 basis A013610 \u2192 D"},"headline":"A038329 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-fa29148e2dbf6ef8","kind":"fingerprint_related","node_a_id":"n-05c39cd3844b146d","node_b_id":"n-66d46e23b1cea6df","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-05c39cd3844b146d","other":{"id":"n-05c39cd3844b146d","name":"basis_map_A007318 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","3"],["25","30","9"]],"card_rows":[["1"],["5","3"],["25","30","9"],["125","225","135","27"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A007318"],"edges":["e-66f5a589f0412003"],"text":"A013620 \u2192 basis A007318"},"headline":"A013620 \u2192 basis A007318","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:            1;\n  1:            5,             3;\n  2:           50,            60,            18;\n  3:          750,          1350,           810,            162;\n  4:        15000,         36000,         32400,          12960,           1944;\n  5:       375000,       1125000,       1350000,         810000,         243000,         29160;\n  6:     11250000,      40500000,      60750000,       48600000,       21870000,       5248800,        524880;\n  7:    393750000,    1653750000,    2976750000,     2976750000,     1786050000,     642978000,     128595600,     11022480;\n  8:  15750000000,   75600000000,  158760000000,   190512000000,   142884000000,   68584320000,   20575296000,   3527193600,    264539520;\n  9: 708750000000, 3827250000000, 9185400000000, 12859560000000, 11573604000000, 6944162400000, 2777664960000, 714256704000, 107138505600, 7142567040;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5*n+3*n*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5*n + 3*n*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5*n + 3*n*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610","basis A007318"],"edges":["e-542183e09dec6678","e-9daa35c48f863bb7"],"text":"A196347 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A196347 \u2192 basis A013610 \u2192 basis A007318","chain":null}
