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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-55f11606ed9ae2bb"]},{"id":194512,"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-55f11606ed9ae2bb"]},{"id":194513,"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 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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"ceada521a35371616d279a9a62ae7b305bc99f1ecdb04b99076714c9ef224c86","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-19327cf22807de6e","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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":[],"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:             529,              1012,               363;\n  1:           12167,             34914,             25047,              5324;\n  2:          279841,           1070696,           1152162,            489808,             73205;\n  3:         6436343,          30782510,          44166210,          28163960,           8418575,            966306;\n  4:       148035889,         849597276,        1523734245,        1295542160,         580881675,         133350228,          12400927;\n  5:      3404825447,       22797526906,       49064242689,       52145571940,       31173983225,       10734693354,        1996549247,         155897368;\n  6:     78310985281,      599249278672,     1504636775796,     1918957047392,     1434003228350,      658394525712,      183682530724,       28685115712,       1929229929;\n  7:   1801152661463,    15505575085638,    44494258941396,    66204018135024,    59367733653690,    34071916705596,    12674094619956,     2968909476192,     399350595303,     23579476910;\n  8:  41426511213649,   396253585521860,  1279209944565135,  2175274881579360,  2275763123391450,  1567308168457416,   728760440647470,   227616393174720,   45925318459845,   5423279689300,   285311670611;\n  9: 952809757913927, 10025215713703058, 35960012886108795, 68793068129947260, 82252581459719550, 66088161103287708, 36875278296761982, 14396736868301040, 3873035190113595, 686044880696450, 72183852664583, 3423740047332;\n","recurrence_sentence":"The row polynomials satisfy (23+(11*n+22)*x)*P(n,x) = (529+(253*n+1012)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1012*x + 363*x^2.","mathematica":"P[0, x_] := 529 + 1012*x + 363*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((529 + (253*n + 1012)*x + (121*n + 363)*x^2)*P[n - 1, x])/(23 + (11*n + 22)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 529 + 1012*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((529 + (253*n + 1012)*x + (121*n + 363)*x^2)*v[i - 1])/(23 + (11*n + 22)*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-344e4bfa9933088a","steps":["basis A007318","\u03b8+1"],"edges":["e-347b6efdbfa5233f","e-55f11606ed9ae2bb"],"text":"A038337 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038337 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}