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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":"23c5fc9f9dd5e571e9623dd971e4a1cadaf814045ad6560e11da76c31ca26fd7","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-efce1b6c07b70e1a","role":"sequence","name":"Triangle of coefficients in expansion of (1+6x)^n.","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: 1,  24;\n  2: 1,  48,   324;\n  3: 1,  72,   972,   3456;\n  4: 1,  96,  1944,  13824,   32400;\n  5: 1, 120,  3240,  34560,  162000,   279936;\n  6: 1, 144,  4860,  69120,  486000,  1679616,   2286144;\n  7: 1, 168,  6804, 120960, 1134000,  5878656,  16003008,  17915904;\n  8: 1, 192,  9072, 193536, 2268000, 15676416,  64012032, 143327232,  136048896;\n  9: 1, 216, 11664, 290304, 4082400, 35271936, 192036096, 644972544, 1224440064, 1007769600;\n","recurrence_sentence":"The row polynomials satisfy (1+(18*n-6)*x+36*n^2*x^2)*P(n,x) = (1+(18*n+18)*x+(36*n^2+180*n+108)*x^2+(216*n^2+432*n+216)*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[((1 + (18*n + 18)*x + (36*n^2 + 180*n + 108)*x^2 + (216*n^2 + 432*n + 216)*x^3)*P[n - 1, x])/(1 + (18*n - 6)*x + 36*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] = ((1 + (18*n + 18)*x + (36*n^2 + 180*n + 108)*x^2 + (216*n^2 + 432*n + 216)*x^3)*v[i - 1])/(1 + (18*n - 6)*x + 36*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-efce1b6c07b70e1a","steps":["(\u03b8+1)\u00b2"],"edges":["e-f16bde71e90b2f84"],"text":"A013613 \u2192 (\u03b8+1)\u00b2"},"headline":"A013613 \u2192 (\u03b8+1)\u00b2","chain":null}