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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":"a24bc2ad880666446ba46b875b0d19abbd571716d1d544259ef0a712ea78b297","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-10f70e3ea8b205cf","role":"sequence","name":"basis_map_A013609 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]"}]}],"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:          1;\n  1:         11,          4;\n  2:        121,         88,         12;\n  3:       1331,       1452,        396,         32;\n  4:      14641,      21296,       8712,       1408,         80;\n  5:     161051,     292820,     159720,      38720,       4400,       192;\n  6:    1771561,    3865224,    2635380,     851840,     145200,     12672,      448;\n  7:   19487171,   49603708,   40584852,   16397920,    3726800,    487872,    34496,    1024;\n  8:  214358881,  623589472,  595244496,  288603392,   81989600,  14310912,  1517824,   90112,   2304;\n  9: 2357947691, 7716919716, 8418457872, 4761955968, 1623394080, 354195072, 50088192, 4460544, 228096, 5120;\n","recurrence_sentence":"The row polynomials satisfy (11+2*n*x)*P(n,x) = (121+(22*n+44)*x+(4*n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((121 + (22*n + 44)*x + (4*n + 4)*x^2)*P[n - 1, x])/(11 + 2*n*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] = ((121 + (22*n + 44)*x + (4*n + 4)*x^2)*v[i - 1])/(11 + 2*n*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-a920b56c9b3ef4d1","steps":["basis A013609","\u03b8+1"],"edges":["e-71aa91d8d70f875b","e-a99bb9226b8ec246"],"text":"A038303 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038303 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}