{"id":"n-d742a8e69ac54bc3","revision":0,"first_index":0,"fingerprint":"f6-4b28493176e02e0b","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","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"],["16","10"],["256","320","100"],["4096","7680","4800","1000"],["65536","163840","153600","64000","10000"],["1048576","3276800","4096000","2560000","800000","100000"],["16777216","62914560","98304000","81920000","38400000","9600000","1000000"],["268435456","1174405120","2202009600","2293760000","1433600000","537600000","112000000","10000000"],["4294967296","21474836480","46976204800","58720256000","45875200000","22937600000","7168000000","1280000000","100000000"],["68719476736","386547056640","966367641600","1409286144000","1321205760000","825753600000","344064000000","92160000000","14400000000","1000000000"],["1099511627776","6871947673600","19327352832000","32212254720000","35232153600000","26424115200000","13762560000000","4915200000000","1152000000000","160000000000","10000000000"],["17592186044416","120946279055360","377957122048000","708669603840000","885837004800000","775107379200000","484442112000000","216268800000000","67584000000000","14080000000000","1760000000000","100000000000"],["281474976710656","2111062325329920","7256776743321600","15118284881920000","21260088115200000","21260088115200000","15502147584000000","8304721920000000","3244032000000000","901120000000000","168960000000000","19200000000000","1000000000000"],["4503599627370496","36591746972385280","137219051146444800","314460325543936000","491344258662400000","552762290995200000","460635242496000000","287897026560000000","134951731200000000","46858240000000000","11714560000000000","1996800000000000","208000000000000","10000000000000"],["72057594037927936","630503947831869440","2561422288066969600","6403555720167424000","11006111394037760000","13757639242547200000","12897786789888000000","9212704849920000000","5038197964800000000","2099249152000000000","656015360000000000","149094400000000000","23296000000000000","2240000000000000","100000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","26","676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976","141167095653376","3670344486987776","95428956661682176","2481152873203736576","64509974703297150976"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["16","10"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200706,"position":0,"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-1173b7dd4d399d87"]},{"id":200707,"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-1173b7dd4d399d87"]},{"id":200708,"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-1173b7dd4d399d87"]}],"region_exact":{},"incoming_edges":[{"id":"e-1173b7dd4d399d87","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-d742a8e69ac54bc3","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":"8bd86f46805abb6dcb13e4d60e793ba19b4935144d4a1ad0daea2f87211ae474","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0195873a37fb074a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","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-0d84e7a48d5a34ab","kind":"fingerprint_related","node_a_id":"n-09968aa1d1f31352","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-09968aa1d1f31352","other":{"id":"n-09968aa1d1f31352","name":"basis_map_A013609 of derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["40","64"]],"card_rows":[["4"],["40","64"],["300","960","768"],["2000","9600","15360","8192"]],"row_sums_preview":["0","4","104","2028","35152","571220","8911032","135150652"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D","basis A013609"],"edges":["e-b04b02754d5cebbf","e-7074b4308a8a7d9f"],"text":"A013611 \u2192 D \u2192 basis A013609"},"headline":"A013611 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-161068a5054c524f","kind":"fingerprint_related","node_a_id":"n-2b4f883be76475ef","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-2b4f883be76475ef","other":{"id":"n-2b4f883be76475ef","name":"basis_map_A013609 of basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","8"],["25","80","64"]],"card_rows":[["1"],["5","8"],["25","80","64"],["125","600","960","512"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","basis A013609"],"edges":["e-f93ad378f9476e41","e-b53b57adb9e27eea"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-202459cc3b78e10a","kind":"fingerprint_related","node_a_id":"n-210f23d82ddfabd8","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-210f23d82ddfabd8","other":{"id":"n-210f23d82ddfabd8","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","8"],["25","80","64"]],"card_rows":[["1"],["5","8"],["25","80","64"],["125","600","960","512"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013611"],"edges":["e-391376c704e767eb"],"text":"A038220 \u2192 basis A013611"},"headline":"A038220 \u2192 basis A013611","chain":null}},{"id":"l-21f7c94d205d5643","kind":"fingerprint_related","node_a_id":"n-b7c94357354b5f47","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-b7c94357354b5f47","other":{"id":"n-b7c94357354b5f47","name":"basis_map_A013609 of basis_map_A013611 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","16"],["100","320","256"]],"card_rows":[["1"],["10","16"],["100","320","256"],["1000","4800","7680","4096"]],"row_sums_preview":["1","26","676","17576","456976","11881376","308915776","8031810176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","basis A013609"],"edges":["e-d7d2422771947afc","e-a0db54daf440939a"],"text":"A130123 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A130123 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-64ef27b08edfc68c","kind":"fingerprint_related","node_a_id":"n-d742a8e69ac54bc3","node_b_id":"n-6590a4d05a99a1a1","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-6590a4d05a99a1a1","other":{"id":"n-6590a4d05a99a1a1","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","1280","400"],["32768","61440","38400","8000"],["1048576","2621440","2457600","1024000","160000"]],"card_rows":[["1024","1280","400"],["32768","61440","38400","8000"],["1048576","2621440","2457600","1024000","160000"],["33554432","104857600","131072000","81920000","25600000","3200000"]],"row_sums_preview":["2704","140608","7311616","380204032","19770609664","1028071702528","53459728531456","2779905883635712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-0c3feb8686271cde","e-a2060598c6593a76"],"text":"A038336 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038336 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-6778a9cf7f991a62","kind":"fingerprint_related","node_a_id":"n-14b3e47d70263f07","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-14b3e47d70263f07","other":{"id":"n-14b3e47d70263f07","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","1280","1024"],["8000","38400","61440","32768"],["160000","1024000","2457600","2621440","1048576"]],"card_rows":[["400","1280","1024"],["8000","38400","61440","32768"],["160000","1024000","2457600","2621440","1048576"],["3200000","25600000","81920000","131072000","104857600","33554432"]],"row_sums_preview":["2704","140608","7311616","380204032","19770609664","1028071702528","53459728531456","2779905883635712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013611"],"edges":["e-bd5ba1e8beeebdaa"],"text":"A038334 \u2192 basis A013611"},"headline":"A038334 \u2192 basis A013611","chain":null}},{"id":"l-6d2b0f19da6e429d","kind":"fingerprint_related","node_a_id":"n-b8de3d8ed94eb089","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-b8de3d8ed94eb089","other":{"id":"n-b8de3d8ed94eb089","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","320","256"],["1000","4800","7680","4096"],["10000","64000","153600","163840","65536"]],"card_rows":[["100","320","256"],["1000","4800","7680","4096"],["10000","64000","153600","163840","65536"],["100000","800000","2560000","4096000","3276800","1048576"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013609","reverse"],"edges":["e-772e88d8a50e6734","e-a6302483b9b43e3b"],"text":"A038319 \u2192 basis A013609 \u2192 reverse"},"headline":"A038319 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-71a7dcdcbb525240","kind":"fingerprint_related","node_a_id":"n-92ad416bcd953046","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-92ad416bcd953046","other":{"id":"n-92ad416bcd953046","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","20"],["1024","1280","400"]],"card_rows":[["1"],["32","20"],["1024","1280","400"],["32768","61440","38400","8000"]],"row_sums_preview":["1","52","2704","140608","7311616","380204032","19770609664","1028071702528"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","basis A007318"],"edges":["e-62a1b0a86f60f435","e-afd11eeb385c1714"],"text":"A038271 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A038271 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-8035cca07ceecef3","kind":"fingerprint_related","node_a_id":"n-d742a8e69ac54bc3","node_b_id":"n-f78f746dffc131f6","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-f78f746dffc131f6","other":{"id":"n-f78f746dffc131f6","name":"basis_map_A013611 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":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","24"],["75","240","192"]],"card_rows":[["3"],["15","24"],["75","240","192"],["375","1800","2880","1536"]],"row_sums_preview":["3","39","507","6591","85683","1113879","14480427","188245551"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","basis A013611"],"edges":["e-618e7f7547136f14","e-bc747a5b4dad45b7"],"text":"A120910 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A120910 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-8cbba8f016296093","kind":"fingerprint_related","node_a_id":"n-d742a8e69ac54bc3","node_b_id":"n-773d2d719c60037e","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-773d2d719c60037e","other":{"id":"n-773d2d719c60037e","name":"basis_map_A013609 of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","8"],["50","160","128"]],"card_rows":[["1"],["5","8"],["50","160","128"],["750","3600","5760","3072"]],"row_sums_preview":["1","13","338","13182","685464","44555160","3475302480","316252525680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","basis A013609"],"edges":["e-3656191e13afb7f3","e-90d1776260a7942d"],"text":"A136572 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A136572 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-96ddc29ac4db278c","kind":"fingerprint_related","node_a_id":"n-f1ec1de6b4eefdf6","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-f1ec1de6b4eefdf6","other":{"id":"n-f1ec1de6b4eefdf6","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["80","128"]],"card_rows":[["8"],["80","128"],["600","1920","1536"],["4000","19200","30720","16384"]],"row_sums_preview":["0","8","208","4056","70304","1142440","17822064","270301304"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013611","D"],"edges":["e-391376c704e767eb","e-6718e8127697d5a2"],"text":"A038220 \u2192 basis A013611 \u2192 D"},"headline":"A038220 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-afda8501ee16cee5","kind":"fingerprint_related","node_a_id":"n-10059bbf81fe8153","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-10059bbf81fe8153","other":{"id":"n-10059bbf81fe8153","name":"basis_map_A013609 of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["25","80","64"],["125","600","960","512"],["625","4000","9600","10240","4096"]],"card_rows":[["25","80","64"],["125","600","960","512"],["625","4000","9600","10240","4096"],["3125","25000","80000","128000","102400","32768"]],"row_sums_preview":["169","2197","28561","371293","4826809","62748517","815730721","10604499373"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","basis A013609"],"edges":["e-b91bf4c52fd06745","e-089867ae4dfa5bca"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-b9f7dbbe5170d621","kind":"fingerprint_related","node_a_id":"n-729eaa8d4ea8435a","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-729eaa8d4ea8435a","other":{"id":"n-729eaa8d4ea8435a","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["320","200"],["7680","9600","3000"],["163840","307200","192000","40000"]],"card_rows":[["320","200"],["7680","9600","3000"],["163840","307200","192000","40000"],["3276800","8192000","7680000","3200000","500000"]],"row_sums_preview":["520","20280","703040","22848800","712882560","21624104320","642544814080","18794435811840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013609","D"],"edges":["e-772e88d8a50e6734","e-eee25ed65b73174b"],"text":"A038319 \u2192 basis A013609 \u2192 D"},"headline":"A038319 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-bbb22c8a4c1ad9af","kind":"fingerprint_related","node_a_id":"n-cf1de4f27da354f5","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-cf1de4f27da354f5","other":{"id":"n-cf1de4f27da354f5","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","320","100"],["4096","7680","4800","1000"],["65536","163840","153600","64000","10000"]],"card_rows":[["256","320","100"],["4096","7680","4800","1000"],["65536","163840","153600","64000","10000"],["1048576","3276800","4096000","2560000","800000","100000"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013609"],"edges":["e-772e88d8a50e6734"],"text":"A038319 \u2192 basis A013609"},"headline":"A038319 \u2192 basis A013609","chain":null}},{"id":"l-d12a7a14257670b0","kind":"fingerprint_related","node_a_id":"n-dffda7b8609ca0a7","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-dffda7b8609ca0a7","other":{"id":"n-dffda7b8609ca0a7","name":"derivative of basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["80","50"]],"card_rows":[["5"],["80","50"],["960","1200","375"],["10240","19200","12000","2500"]],"row_sums_preview":["0","5","130","2535","43940","714025","11138790","168938315"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A007318","D"],"edges":["e-31008ec5d02c6da5","e-09c75155f5df7285"],"text":"A013622 \u2192 basis A007318 \u2192 D"},"headline":"A013622 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-e455a45d339ea232","kind":"fingerprint_related","node_a_id":"n-93086473ff31e9a0","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-93086473ff31e9a0","other":{"id":"n-93086473ff31e9a0","name":"basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","5"],["64","80","25"]],"card_rows":[["1"],["8","5"],["64","80","25"],["512","960","600","125"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A007318"],"edges":["e-31008ec5d02c6da5"],"text":"A013622 \u2192 basis A007318"},"headline":"A013622 \u2192 basis A007318","chain":null}},{"id":"l-e645e43d861231c3","kind":"fingerprint_related","node_a_id":"n-bb08c7dea0976170","node_b_id":"n-d742a8e69ac54bc3","fingerprint":"f6-4b28493176e02e0b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b28493176e02e0b"},"other_id":"n-bb08c7dea0976170","other":{"id":"n-bb08c7dea0976170","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-4b28493176e02e0b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","720","576"],["3375","16200","25920","13824"],["50625","324000","777600","829440","331776"]],"card_rows":[["225","720","576"],["3375","16200","25920","13824"],["50625","324000","777600","829440","331776"],["759375","6075000","19440000","31104000","24883200","7962624"]],"row_sums_preview":["1521","59319","2313441","90224199","3518743761","137231006679","5352009260481","208728361158759"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-a617f0e181e2ec6f","e-f5c60de33938dfd4"],"text":"A038329 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A038329 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","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:          16,           10;\n  2:         256,          320,          100;\n  3:        4096,         7680,         4800,          1000;\n  4:       65536,       163840,       153600,         64000,         10000;\n  5:     1048576,      3276800,      4096000,       2560000,        800000,       100000;\n  6:    16777216,     62914560,     98304000,      81920000,      38400000,      9600000,      1000000;\n  7:   268435456,   1174405120,   2202009600,    2293760000,    1433600000,    537600000,    112000000,    10000000;\n  8:  4294967296,  21474836480,  46976204800,   58720256000,   45875200000,  22937600000,   7168000000,  1280000000,   100000000;\n  9: 68719476736, 386547056640, 966367641600, 1409286144000, 1321205760000, 825753600000, 344064000000, 92160000000, 14400000000, 1000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+10*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[(16 + 10*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] = (16 + 10*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-e6ebe74518d42cbe","steps":["basis A007318","basis A013609"],"edges":["e-1194f7dc6f45a71b","e-1173b7dd4d399d87"],"text":"A038259 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A038259 \u2192 basis A007318 \u2192 basis A013609","chain":null}