{"id":"n-8c5c43d91328f213","revision":0,"first_index":0,"fingerprint":"f6-b29ddf49d7bb6a39","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["720","448","64"],["17600","17280","5376","512"],["416000","563200","276480","57344","4096"],["9600000","16640000","11264000","3686400","573440","32768"],["217600000","460800000","399360000","180224000","44236800","5505024","262144"],["4864000000","12185600000","12902400000","7454720000","2523136000","495452160","51380224","2097152"],["107520000000","311296000000","389939200000","275251200000","119275520000","32296140800","5284823040","469762048","16777216"],["2355200000000","7741440000000","11206656000000","9358540800000","4954521600000","1717567488000","387553689600","54358179840","4227858432","134217728"],["51200000000000","188416000000000","309657600000000","298844160000000","187170816000000","79272345600000","22900899840000","4429185024000","543581798400","37580963840","1073741824"],["1105920000000000","4505600000000000","8290304000000000","9083289600000000","6574571520000000","3294206361600000","1162661068800000","287897026560000","48721035264000","5315022028800","330712481792","8589934592"],["23756800000000000","106168320000000000","216268800000000000","265289728000000000","217998950400000000","126231773184000000","52707301785600000","15945066086400000","3454764318720000","519691042816000","51024211476480","2886218022912","68719476736"],["507904000000000000","2470707200000000000","5520752640000000000","7497318400000000000","6897532928000000000","4534378168320000000","2188017401856000000","783079912243200000","207285859123200000","39921721016320000","5404786845286400","482410726686720","25013889531904","549755813888"],["10813440000000000000","56885248000000000000","138359603200000000000","206108098560000000000","209924915200000000000","154504737587200000000","84641725808640000000","35008278429696000000","10963118771404800000","2579557357977600000","447123275382784000","55030556970188800","4502500115742720","215504279044096","4398046511104"],["229376000000000000000","1297612800000000000000","3413114880000000000000","5534384128000000000000","6183242956800000000000","5038197964800000000000","3090094751744000000000","1451001013862400000000","525124176445440000000","146174916952064000000","30954688295731200000","4877708458721280000","550305569701888000","41561539529932800","1847179534663680","35184372088832"],["4849664000000000000000","29360128000000000000000","83047219200000000000000","145626234880000000000000","177100292096000000000000","158291019694080000000000","107481556582400000000000","56504589746176000000000","23216016221798400000000","7468432731668480000000","1871038936986419200000","360200009259417600000","52028890226360320000","5418393301680128000","379991218559385600","15762598695796736","281474976710656"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1232","40768","1317120","41796608","1307987968","40478785536","1241349423104","37780199833600","1142473242968064","34358824936669184","1028395173966512128","30652810991776038912","910295599149712670720","26944749734831495053312","795234235417188989140992"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["720","448","64"]],"recurrence":{"denominator":{"coeffs":[["7","2"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["180","112","16"],["40","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194266,"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-110f8a7ea2cc5a01"]},{"id":194267,"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-110f8a7ea2cc5a01"]},{"id":194268,"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-110f8a7ea2cc5a01"]},{"id":194269,"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-110f8a7ea2cc5a01"]}],"region_exact":{},"incoming_edges":[{"id":"e-110f8a7ea2cc5a01","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-8c5c43d91328f213","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"72e6f620e4b24f872ebf8811c467ec56d0f4ccab2919d6b683008e2b53003347","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-64a6f5553b7b7b7f","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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-462e606f7a1728b2","kind":"fingerprint_related","node_a_id":"n-665db32c3a2797ef","node_b_id":"n-8c5c43d91328f213","fingerprint":"f6-b29ddf49d7bb6a39","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b29ddf49d7bb6a39"},"other_id":"n-665db32c3a2797ef","other":{"id":"n-665db32c3a2797ef","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b29ddf49d7bb6a39","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","4"],["180","112","16"]],"card_rows":[["1"],["14","4"],["180","112","16"],["2200","2160","672","64"]],"row_sums_preview":["1","18","308","5096","82320","1306144","20437312","316240512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013611","D+1"],"edges":["e-8c5f302cdd82df95","e-22a16d6d4d1e905f"],"text":"A038291 \u2192 basis A013611 \u2192 D+1"},"headline":"A038291 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-47bddafbd1b3195b","kind":"fingerprint_related","node_a_id":"n-5cfb4e705168cb18","node_b_id":"n-8c5c43d91328f213","fingerprint":"f6-b29ddf49d7bb6a39","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b29ddf49d7bb6a39"},"other_id":"n-5cfb4e705168cb18","other":{"id":"n-5cfb4e705168cb18","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-b29ddf49d7bb6a39","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","2"],["45","28","4"]],"card_rows":[["1"],["7","2"],["45","28","4"],["275","270","84","8"]],"row_sums_preview":["1","9","77","637","5145","40817","319333","2470629"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A007318","D+1"],"edges":["e-7f6faf86ff30c5bb","e-3aea408ee6d5e382"],"text":"A038220 \u2192 basis A007318 \u2192 D+1"},"headline":"A038220 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-fa1a7918a34c8198","kind":"fingerprint_related","node_a_id":"n-10d2cdd1c05d2862","node_b_id":"n-8c5c43d91328f213","fingerprint":"f6-b29ddf49d7bb6a39","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b29ddf49d7bb6a39"},"other_id":"n-10d2cdd1c05d2862","other":{"id":"n-10d2cdd1c05d2862","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b29ddf49d7bb6a39","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["405","252","36"],["7425","7290","2268","216"],["131625","178200","87480","18144","1296"]],"card_rows":[["405","252","36"],["7425","7290","2268","216"],["131625","178200","87480","18144","1296"],["2278125","3948750","2673000","874800","136080","7776"]],"row_sums_preview":["693","17199","416745","9918531","232793757","5403265623","124275109329","2836714452075"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013609","D+1"],"edges":["e-f3e6c91e6b1bdc2d","e-8377d163e02bd8e0"],"text":"A038329 \u2192 basis A013609 \u2192 D+1"},"headline":"A038329 \u2192 basis A013609 \u2192 D+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:              720,              448,               64;\n  1:            17600,            17280,             5376,              512;\n  2:           416000,           563200,           276480,            57344,             4096;\n  3:          9600000,         16640000,         11264000,          3686400,           573440,            32768;\n  4:        217600000,        460800000,        399360000,        180224000,         44236800,          5505024,           262144;\n  5:       4864000000,      12185600000,      12902400000,       7454720000,       2523136000,        495452160,         51380224,         2097152;\n  6:     107520000000,     311296000000,     389939200000,     275251200000,     119275520000,      32296140800,       5284823040,       469762048,       16777216;\n  7:    2355200000000,    7741440000000,   11206656000000,    9358540800000,    4954521600000,    1717567488000,     387553689600,     54358179840,     4227858432,     134217728;\n  8:   51200000000000,  188416000000000,  309657600000000,  298844160000000,  187170816000000,   79272345600000,   22900899840000,   4429185024000,   543581798400,   37580963840,   1073741824;\n  9: 1105920000000000, 4505600000000000, 8290304000000000, 9083289600000000, 6574571520000000, 3294206361600000, 1162661068800000, 287897026560000, 48721035264000, 5315022028800, 330712481792, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy (2*n+7+2*x)*P(n,x) = (40*n+180+(16*n+112)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 720 + 448*x + 64*x^2.","mathematica":"P[0, x_] := 720 + 448*x + 64*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((40*n + 180 + (16*n + 112)*x + 16*x^2)*P[n - 1, x])/(2*n + 7 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 720 + 448*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((40*n + 180 + (16*n + 112)*x + 16*x^2)*v[i - 1])/(2*n + 7 + 2*x)); 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-6874f381e9c7e0a2","steps":["x \u21a6 x+1","D+1"],"edges":["e-4a3b4bf6a80699a3","e-110f8a7ea2cc5a01"],"text":"A038334 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038334 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}