{"id":"n-2b363192a9cb3eca","revision":0,"first_index":0,"fingerprint":"f6-6b726760bd05165a","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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"],["15","12"],["225","360","144"],["3375","8100","6480","1728"],["50625","162000","194400","103680","20736"],["759375","3037500","4860000","3888000","1555200","248832"],["11390625","54675000","109350000","116640000","69984000","22394880","2985984"],["170859375","956812500","2296350000","3061800000","2449440000","1175731200","313528320","35831808"],["2562890625","16402500000","45927000000","73483200000","73483200000","47029248000","18811699200","4299816960","429981696"],["38443359375","276792187500","885735000000","1653372000000","1984046400000","1587237120000","846526464000","290237644800","58047528960","5159780352"],["576650390625","4613203125000","16607531250000","35429400000000","49601160000000","47617113600000","31744742400000","14511882240000","4353564672000","773967052800","61917364224"],["8649755859375","76117851562500","304471406250000","730731375000000","1169170200000000","1309470624000000","1047576499200000","598615142400000","239446056960000","63852281856000","10216365096960","743008370688"],["129746337890625","1245564843750000","5480485312500000","14614627500000000","26306329500000000","33672101760000000","31427294976000000","21550145126400000","10775072563200000","3831136911360000","919472858726400","133741506723840","8916100448256"],["1946195068359375","20240428710937500","97154057812500000","284985236250000000","569970472500000000","820757480400000000","875474645760000000","700379716608000000","420227829964800000","186767924428800000","59765735817216000","13039796905574400","1738639587409920","106993205379072"],["29192926025390625","326960771484375000","1700196011718750000","5440627237500000000","11969379922500000000","19151007876000000000","22981209451200000000","21011391498240000000","14707974048768000000","7844252826009600000","3137701130403840000","912785783390208000","182557156678041600","22468573129605120","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","27","729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987","205891132094649","5559060566555523","150094635296999121","4052555153018976267","109418989131512359209"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["15","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192106,"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-c497246ce9a2c8d7","e-f0fcc23e3861ffce"]},{"id":192107,"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-c497246ce9a2c8d7","e-f0fcc23e3861ffce"]},{"id":192108,"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-c497246ce9a2c8d7","e-f0fcc23e3861ffce"]}],"region_exact":{},"incoming_edges":[{"id":"e-f0fcc23e3861ffce","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-2b363192a9cb3eca","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":"1e26cb603c3e033920383053d37c4b66ac859a024434833e936620bfd4c30ecb","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-e03395d80bc5749d","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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]"}]},{"id":"e-c497246ce9a2c8d7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-2b363192a9cb3eca","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"07f2dbc01cc55f812c78860696131b34a84f0dec67a2a6aeba4a8250fd91b939","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-d7df91eaf8843b88","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-0f8c8146c57fbc93","kind":"fingerprint_related","node_a_id":"n-a041f87d41cb6b7b","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-a041f87d41cb6b7b","other":{"id":"n-a041f87d41cb6b7b","name":"shift_x_plus_one 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":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["25","40","16"]],"card_rows":[["1"],["5","4"],["25","40","16"],["125","300","240","64"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-f93ad378f9476e41","e-74c6877a5ab7eefe"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-20caf49c7fde39bb","kind":"fingerprint_related","node_a_id":"n-d7df91eaf8843b88","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-d7df91eaf8843b88","other":{"id":"n-d7df91eaf8843b88","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","15"],["144","360","225"]],"card_rows":[["1"],["12","15"],["144","360","225"],["1728","6480","8100","3375"]],"row_sums_preview":["1","27","729","19683","531441","14348907","387420489","10460353203"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610"],"edges":["e-cc1debec4a411a9c"],"text":"A038271 \u2192 basis A013610"},"headline":"A038271 \u2192 basis A013610","chain":null}},{"id":"l-24abb43506d140ad","kind":"fingerprint_related","node_a_id":"n-506262231820dae3","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-506262231820dae3","other":{"id":"n-506262231820dae3","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","5"],["16","40","25"]],"card_rows":[["1"],["4","5"],["16","40","25"],["64","240","300","125"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["x \u21a6 x/(x+1)"],"edges":["e-7e32de33a5202f3a"],"text":"A038231 \u2192 x \u21a6 x/(x+1)"},"headline":"A038231 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-25a4fe4e583cb717","kind":"fingerprint_related","node_a_id":"n-10a5ec77b5e8ca54","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-10a5ec77b5e8ca54","other":{"id":"n-10a5ec77b5e8ca54","name":"reversal of basis_map_A007318 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","5"],["16","40","25"]],"card_rows":[["1"],["4","5"],["16","40","25"],["64","240","300","125"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A007318","reverse"],"edges":["e-13c78408b476aecd","e-edec282618df57f0"],"text":"A013611 \u2192 basis A007318 \u2192 reverse"},"headline":"A013611 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-2611676d060f1ed0","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-cfe169c7cfe15942","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-cfe169c7cfe15942","other":{"id":"n-cfe169c7cfe15942","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["640","800"],["15360","38400","24000"],["327680","1228800","1536000","640000"]],"card_rows":[["640","800"],["15360","38400","24000"],["327680","1228800","1536000","640000"],["6553600","32768000","61440000","51200000","16000000"]],"row_sums_preview":["1440","77760","3732480","167961600","7255941120","304749527040","12538266255360","507799783342080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013611","D"],"edges":["e-fd91849598d32af1","e-5998fe0f53363e2d"],"text":"A038319 \u2192 basis A013611 \u2192 D"},"headline":"A038319 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-35ad88f67d40fd39","kind":"fingerprint_related","node_a_id":"n-244a69427003c906","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-244a69427003c906","other":{"id":"n-244a69427003c906","name":"shift_x_plus_one of derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["40","32"]],"card_rows":[["4"],["40","32"],["300","480","192"],["2000","4800","3840","1024"]],"row_sums_preview":["0","4","72","972","11664","131220","1417176","14880348"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D","basis A007318"],"edges":["e-b04b02754d5cebbf","e-d85043437d73496a"],"text":"A013611 \u2192 D \u2192 basis A007318"},"headline":"A013611 \u2192 D \u2192 basis A007318","chain":null}},{"id":"l-3eeea7057c691398","kind":"fingerprint_related","node_a_id":"n-09320ea6fe78bcb6","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-09320ea6fe78bcb6","other":{"id":"n-09320ea6fe78bcb6","name":"basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","10"],["64","160","100"]],"card_rows":[["1"],["8","10"],["64","160","100"],["512","1920","2400","1000"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609"],"edges":["e-c5caeaebd0a50686"],"text":"A013622 \u2192 basis A013609"},"headline":"A013622 \u2192 basis A013609","chain":null}},{"id":"l-45abae4794c3be14","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-5ea11ecf1dd9fff0","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-5ea11ecf1dd9fff0","other":{"id":"n-5ea11ecf1dd9fff0","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","360","225"],["1728","6480","8100","3375"],["20736","103680","194400","162000","50625"]],"card_rows":[["144","360","225"],["1728","6480","8100","3375"],["20736","103680","194400","162000","50625"],["248832","1555200","3888000","4860000","3037500","759375"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013611","reverse"],"edges":["e-a5d7492b5780a0c3","e-5d85ac50fe9eaea9"],"text":"A038329 \u2192 basis A013611 \u2192 reverse"},"headline":"A038329 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-48c626d0dccd3634","kind":"fingerprint_related","node_a_id":"n-7ad950efd156275d","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-7ad950efd156275d","other":{"id":"n-7ad950efd156275d","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["10"],["160","200"]],"card_rows":[["10"],["160","200"],["1920","4800","3000"],["20480","76800","96000","40000"]],"row_sums_preview":["0","10","360","9720","233280","5248800","113374080","2380855680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609","D"],"edges":["e-c5caeaebd0a50686","e-a7ec95bdf6e12bb3"],"text":"A013622 \u2192 basis A013609 \u2192 D"},"headline":"A013622 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-4afc80adb147ad9b","kind":"fingerprint_related","node_a_id":"n-d04994959b21388c","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-d04994959b21388c","other":{"id":"n-d04994959b21388c","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["25","40","16"]],"card_rows":[["1"],["5","4"],["25","40","16"],["125","300","240","64"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A007318"],"edges":["e-13c78408b476aecd"],"text":"A013611 \u2192 basis A007318"},"headline":"A013611 \u2192 basis A007318","chain":null}},{"id":"l-5a080e14f9323e94","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-6d2899d43e7720c2","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-6d2899d43e7720c2","other":{"id":"n-6d2899d43e7720c2","name":"basis_map_A013609 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":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","12"],["75","120","48"]],"card_rows":[["3"],["15","12"],["75","120","48"],["375","900","720","192"]],"row_sums_preview":["3","27","243","2187","19683","177147","1594323","14348907"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","basis A013609"],"edges":["e-618e7f7547136f14","e-22d3037eb57e6ab8"],"text":"A120910 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A120910 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-60c4610e151715c4","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-e46ced92cad8408a","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-e46ced92cad8408a","other":{"id":"n-e46ced92cad8408a","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["640","512"],["19200","30720","12288"],["512000","1228800","983040","262144"]],"card_rows":[["640","512"],["19200","30720","12288"],["512000","1228800","983040","262144"],["12800000","40960000","49152000","26214400","5242880"]],"row_sums_preview":["1152","62208","2985984","134369280","5804752896","243799621632","10030613004288","406239826673664"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013609","D"],"edges":["e-4eca5e8e987e069c","e-4e1ddc54bc21ae1f"],"text":"A038334 \u2192 basis A013609 \u2192 D"},"headline":"A038334 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-6e02e4319117cba1","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-17c78c88c7e7c0f0","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-17c78c88c7e7c0f0","other":{"id":"n-17c78c88c7e7c0f0","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","40"],["1024","2560","1600"]],"card_rows":[["1"],["32","40"],["1024","2560","1600"],["32768","122880","153600","64000"]],"row_sums_preview":["1","72","5184","373248","26873856","1934917632","139314069504","10030613004288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","basis A013609"],"edges":["e-62a1b0a86f60f435","e-35a4f20be57215a6"],"text":"A038271 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038271 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-70fb2b20fcc870e7","kind":"fingerprint_related","node_a_id":"n-82073868f14b935d","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-82073868f14b935d","other":{"id":"n-82073868f14b935d","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","20"],["256","640","400"]],"card_rows":[["1"],["16","20"],["256","640","400"],["4096","15360","19200","8000"]],"row_sums_preview":["1","36","1296","46656","1679616","60466176","2176782336","78364164096"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-9c08db68889738c4","e-3b7a60d3a406d50b"],"text":"A027467 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A027467 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-8e24d5f1ec2debab","kind":"fingerprint_related","node_a_id":"n-dbabf32666f7e9da","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-dbabf32666f7e9da","other":{"id":"n-dbabf32666f7e9da","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["50","80","32"]],"card_rows":[["1"],["5","4"],["50","80","32"],["750","1800","1440","384"]],"row_sums_preview":["1","9","162","4374","157464","7085880","382637520","24106163760"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","basis A007318"],"edges":["e-3656191e13afb7f3","e-80ddf6dcdab57466"],"text":"A136572 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A136572 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-9381970f49edda3d","kind":"fingerprint_related","node_a_id":"n-03ac2a4394de29ac","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-03ac2a4394de29ac","other":{"id":"n-03ac2a4394de29ac","name":"shift_x_plus_one of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["25","40","16"],["125","300","240","64"],["625","2000","2400","1280","256"]],"card_rows":[["25","40","16"],["125","300","240","64"],["625","2000","2400","1280","256"],["3125","12500","20000","16000","6400","1024"]],"row_sums_preview":["81","729","6561","59049","531441","4782969","43046721","387420489"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-b91bf4c52fd06745","e-de1bc984e3f3a6a7"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a9d9cc24d1f1cafa","kind":"fingerprint_related","node_a_id":"n-881b87f0bbbf8ba6","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-881b87f0bbbf8ba6","other":{"id":"n-881b87f0bbbf8ba6","name":"shift_x_plus_one 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":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","8"],["100","160","64"]],"card_rows":[["1"],["10","8"],["100","160","64"],["1000","2400","1920","512"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-d7d2422771947afc","e-1d7b3825834f6b24"],"text":"A130123 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A130123 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-b5cd845072feadb3","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-0007c3affb4bff57","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-0007c3affb4bff57","other":{"id":"n-0007c3affb4bff57","name":"basis_map_A013611 of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["10","8"]],"card_rows":[["1"],["10","8"],["75","120","48"],["500","1200","960","256"]],"row_sums_preview":["0","1","18","243","2916","32805","354294","3720087"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","basis A013611"],"edges":["e-62aaaa59f91d42c1","e-f50126f61818ff49"],"text":"A038231 \u2192 D \u2192 basis A013611"},"headline":"A038231 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-c6015c1693f67380","kind":"fingerprint_related","node_a_id":"n-93016690ff7736f4","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-93016690ff7736f4","other":{"id":"n-93016690ff7736f4","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","640","400"],["4096","15360","19200","8000"],["65536","327680","614400","512000","160000"]],"card_rows":[["256","640","400"],["4096","15360","19200","8000"],["65536","327680","614400","512000","160000"],["1048576","6553600","16384000","20480000","12800000","3200000"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013611"],"edges":["e-fd91849598d32af1"],"text":"A038319 \u2192 basis A013611"},"headline":"A038319 \u2192 basis A013611","chain":null}},{"id":"l-d94d190adc17cd66","kind":"fingerprint_related","node_a_id":"n-92221d6c8c131732","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-92221d6c8c131732","other":{"id":"n-92221d6c8c131732","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["15"],["360","450"]],"card_rows":[["15"],["360","450"],["6480","16200","10125"],["103680","388800","486000","202500"]],"row_sums_preview":["0","15","810","32805","1180980","39858075","1291401630","40679151345"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610","D"],"edges":["e-cc1debec4a411a9c","e-70306f529a48621b"],"text":"A038271 \u2192 basis A013610 \u2192 D"},"headline":"A038271 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-db4c774700eec33a","kind":"fingerprint_related","node_a_id":"n-16c35c7477b7e922","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-16c35c7477b7e922","other":{"id":"n-16c35c7477b7e922","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","360","144"],["3375","8100","6480","1728"],["50625","162000","194400","103680","20736"]],"card_rows":[["225","360","144"],["3375","8100","6480","1728"],["50625","162000","194400","103680","20736"],["759375","3037500","4860000","3888000","1555200","248832"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013611"],"edges":["e-a5d7492b5780a0c3"],"text":"A038329 \u2192 basis A013611"},"headline":"A038329 \u2192 basis A013611","chain":null}},{"id":"l-ec4e12e54c460c2e","kind":"fingerprint_related","node_a_id":"n-7e007679d5feba96","node_b_id":"n-2b363192a9cb3eca","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-7e007679d5feba96","other":{"id":"n-7e007679d5feba96","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","640","256"],["8000","19200","15360","4096"],["160000","512000","614400","327680","65536"]],"card_rows":[["400","640","256"],["8000","19200","15360","4096"],["160000","512000","614400","327680","65536"],["3200000","12800000","20480000","16384000","6553600","1048576"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013609"],"edges":["e-4eca5e8e987e069c"],"text":"A038334 \u2192 basis A013609"},"headline":"A038334 \u2192 basis A013609","chain":null}},{"id":"l-f5361e426e067a02","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-e6b61b230f1a291b","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-e6b61b230f1a291b","other":{"id":"n-e6b61b230f1a291b","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["360","288"],["8100","12960","5184"],["162000","388800","311040","82944"]],"card_rows":[["360","288"],["8100","12960","5184"],["162000","388800","311040","82944"],["3037500","9720000","11664000","6220800","1244160"]],"row_sums_preview":["648","26244","944784","31886460","1033121304","32543321076","1004193907488","30502389939948"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013611","D"],"edges":["e-a5d7492b5780a0c3","e-45ed6ed04dc131e0"],"text":"A038329 \u2192 basis A013611 \u2192 D"},"headline":"A038329 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-fd90a398675b8c92","kind":"fingerprint_related","node_a_id":"n-2b363192a9cb3eca","node_b_id":"n-b190b660c2f1fbab","fingerprint":"f6-6b726760bd05165a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b726760bd05165a"},"other_id":"n-b190b660c2f1fbab","other":{"id":"n-b190b660c2f1fbab","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6b726760bd05165a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","360","225"],["1728","6480","8100","3375"],["20736","103680","194400","162000","50625"]],"card_rows":[["144","360","225"],["1728","6480","8100","3375"],["20736","103680","194400","162000","50625"],["248832","1555200","3888000","4860000","3037500","759375"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x/(x+1)"],"edges":["e-78a63a57aa10b4bc"],"text":"A038329 \u2192 x \u21a6 x/(x+1)"},"headline":"A038329 \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:          15,           12;\n  2:         225,          360,          144;\n  3:        3375,         8100,         6480,          1728;\n  4:       50625,       162000,       194400,        103680,         20736;\n  5:      759375,      3037500,      4860000,       3888000,       1555200,        248832;\n  6:    11390625,     54675000,    109350000,     116640000,      69984000,      22394880,      2985984;\n  7:   170859375,    956812500,   2296350000,    3061800000,    2449440000,    1175731200,    313528320,     35831808;\n  8:  2562890625,  16402500000,  45927000000,   73483200000,   73483200000,   47029248000,  18811699200,   4299816960,   429981696;\n  9: 38443359375, 276792187500, 885735000000, 1653372000000, 1984046400000, 1587237120000, 846526464000, 290237644800, 58047528960, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (15+12*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[(15 + 12*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] = (15 + 12*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-a6a633b989da1879","steps":["basis A013610","reverse"],"edges":["e-cc1debec4a411a9c","e-c497246ce9a2c8d7"],"text":"A038271 \u2192 basis A013610 \u2192 reverse"},"headline":"A038271 \u2192 basis A013610 \u2192 reverse","chain":null}