{"id":"n-b0a8157e41423534","revision":0,"first_index":0,"fingerprint":"f6-c96f7338872c62e8","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["3"],["24","18"],["276","324","81"],["4224","6192","2592","324"],["81072","136800","74520","16200","1215"],["1874304","3496032","2255040","654480","87480","4374"],["50727744","102142656","74363184","26036640","4728780","428652","15309"],["1573724160","3368646144","2690537472","1078551936","239500800","29883168","1959552","52488"],["55063376640","123976313856","106637181696","47432915712","12187596960","1878230592","171215856","8503056","177147"],["2145337989120","5041480186368","4610106316800","2228999523840","641002118400","115248111552","13080009600","910141920","35429400","590490"],["92115523261440","224614195762176","216311126144256","112058405268480","35280837187200","7148028311424","952728793632","83071707840","4559763780","142898580","1948617"],["4321789834199040","10884946144628736","10961075683565568","6019931146927104","2043137530368000","455820316199424","68843623597056","7087797281664","490116147840","21772547280","561201696","6377292"],["219975376861900800","570178046236631040","597036041829107712","344840860080617472","124731618319376640","30146310919296000","5037552286550784","590425144987392","48427123217616","2722347856800","99909490824","2155524696","20726199"],["12072964608140083200","32109466757949480960","34805747466836656128","21009550611669774336","8028660614548654080","2076934615591739904","377539911691278336","49154565668921856","4607749518083712","308430528325920","14378860431936","443374849008","8124670008","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","42","681","13332","309807","8371710","258442965","8982855720","347355511515","14796200417490","688568621047425","34763144533040700","1892587483235283975","110534950162569910950"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-36","36","-9"],["48","-24","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","6"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189041,"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-b62ef521b65b51e5"]},{"id":189042,"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-b62ef521b65b51e5"]},{"id":189043,"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-b62ef521b65b51e5"]},{"id":189044,"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-b62ef521b65b51e5"]}],"region_exact":{},"incoming_edges":[{"id":"e-b62ef521b65b51e5","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-b0a8157e41423534","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":"2a472e7b4dbca6170524de6e7e566b4888f2f584ed5f06325efe32223c919775","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-519925749fed9cdb","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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":[],"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:           0;\n  1:           3;\n  2:          24,           18;\n  3:         276,          324,           81;\n  4:        4224,         6192,         2592,         324;\n  5:       81072,       136800,        74520,       16200,        1215;\n  6:     1874304,      3496032,      2255040,      654480,       87480,       4374;\n  7:    50727744,    102142656,     74363184,    26036640,     4728780,     428652,     15309;\n  8:  1573724160,   3368646144,   2690537472,  1078551936,   239500800,   29883168,   1959552,   52488;\n  9: 55063376640, 123976313856, 106637181696, 47432915712, 12187596960, 1878230592, 171215856, 8503056, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-48*n+36+(24*n-36)*x+9*x^2)*P(n-2,x) + (8*n-8+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 48*n + 36 + (24*n - 36)*x + 9*x^2)*P[n - 2, x] + (8*n - 8 + 6*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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 48*n + 36 + (24*n - 36)*x + 9*x^2)*v[i - 2] + (8*n - 8 + 6*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-4172472abe953cea","steps":["basis A013610","D"],"edges":["e-b01e725a7bd6431f","e-b62ef521b65b51e5"],"text":"A290319 \u2192 basis A013610 \u2192 D"},"headline":"A290319 \u2192 basis A013610 \u2192 D","chain":null}