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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":"9a9f78dd0efa98ed28c6c1449bbfd83f367c72788edb1c615d21ef724c8f5775","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-81e6fb4f9f226d77","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(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:         12,           4;\n  2:        144,          96,           9;\n  3:       1728,        1728,         324,         16;\n  4:      20736,       27648,        7776,        768,        25;\n  5:     248832,      414720,      155520,      23040,      1500,       36;\n  6:    2985984,     5971968,     2799360,     552960,     54000,     2592,      49;\n  7:   35831808,    83607552,    47029248,   11612160,   1512000,   108864,    4116,     64;\n  8:  429981696,  1146617856,   752467968,  222953472,  36288000,  3483648,  197568,   6144,   81;\n  9: 5159780352, 15479341056, 11609505792, 4013162496, 783820800, 94058496, 7112448, 331776, 8748, 100;\n","recurrence_sentence":"The row polynomials satisfy (144+(36*n-12)*x+n^2*x^2)*P(n,x) = (1728+(432*n+432)*x+(12*n^2+60*n+36)*x^2+(n^2+2*n+1)*x^3)*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[((1728 + (432*n + 432)*x + (12*n^2 + 60*n + 36)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(144 + (36*n - 12)*x + n^2*x^2)]];\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] = ((1728 + (432*n + 432)*x + (12*n^2 + 60*n + 36)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(144 + (36*n - 12)*x + n^2*x^2)); 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-81e6fb4f9f226d77","steps":["(\u03b8+1)\u00b2"],"edges":["e-b4afc11aa432f4c7"],"text":"A038327 \u2192 (\u03b8+1)\u00b2"},"headline":"A038327 \u2192 (\u03b8+1)\u00b2","chain":null}