{"id":"n-f1b46de67f8e4966","revision":0,"first_index":1,"fingerprint":"f6-668e26fcca6da53a","name":"derivative of Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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":false,"rows":[[],["4"],["40","32"],["316","672","192"],["2320","9920","6912","1024"],["16564","127680","163200","56320","5120"],["116920","1536992","3214080","1940480","399360","24576"],["821356","17842272","57433152","54010880","18887680","2580480","114688"],["5758240","202835840","968550912","1332516864","704471040","160333824","15597568","524288"],["40333924","2277260160","15745278720","30483425280","22865418240","7710474240","1231749120","89653248","2359296"],["282416200","25372533152","249842741760","663159900160","678321745920","317936812032","74166435840","8768716800","495452160","10485760"],["1977149596","281357194272","3899876325312","13932532725760","18917199656960","11840238305280","3782864633856","645951651840","58781859840","2652897280","46137344"],["13840755760","3110746333760","60186288045312","285517462191104","504758255738880","410488992595968","172523249074176","39901403283456","5199274967040","375327293440","13841203200","201326592"],["96887416084","34328935717440","921458798682240","5745825317872640","13037027192949760","13506042427637760","7263836020080640","2185226824974336","382328038490112","39246962360320","2301976641536","70665633792","872415232"],["678218289880","378393392220512","14027855594538240","114085127965885440","328580752027207680","427240876072378368","288207114618142720","109689046394470400","24744314213498880","3386855108116480","280879536537600","13648936304640","354200584192","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","72","1180","20176","368884","7232408","151690508","3390588576","80445952228","2018357239784","53363478537340","1482088882808688","43117790920871892","1310635900036271416"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-21","-24","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-12","-16"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188285,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-1b710c9b4637df7b"]},{"id":188286,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-1b710c9b4637df7b"]},{"id":188287,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-1b710c9b4637df7b"]},{"id":188288,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-1b710c9b4637df7b"]}],"region_exact":{},"incoming_edges":[{"id":"e-1b710c9b4637df7b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f1b46de67f8e4966","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":"68f9e4570b94751ed2c2af9f9326f785b57291c3565fd8624a56447602a4a18b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-de5746895014fee3","role":"sequence","name":"Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        0;\n   2:        4;\n   3:       40,         32;\n   4:      316,        672,         192;\n   5:     2320,       9920,        6912,        1024;\n   6:    16564,     127680,      163200,       56320,        5120;\n   7:   116920,    1536992,     3214080,     1940480,      399360,      24576;\n   8:   821356,   17842272,    57433152,    54010880,    18887680,    2580480,     114688;\n   9:  5758240,  202835840,   968550912,  1332516864,   704471040,  160333824,   15597568,   524288;\n  10: 40333924, 2277260160, 15745278720, 30483425280, 22865418240, 7710474240, 1231749120, 89653248, 2359296;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(21+24*x+16*x^2)*P(n-2,x) - (12*x+16*x^2)*d/dx P(n-2,x) + (10+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[-(21 + 24*x + 16*x^2)*P[n - 2, x] - (12*x + 16*x^2)*D[P[n - 2, x], x] + (10 + 8*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = -(21 + 24*x + 16*x^2)*v[i - 2] - (12*x + 16*x^2)*deriv(v[i - 2], x) + (10 + 8*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-de5746895014fee3","steps":["D"],"edges":["e-1b710c9b4637df7b"],"text":"A225467 \u2192 D"},"headline":"A225467 \u2192 D","chain":null}