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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":"25be1d1f44b8717d999fee0ceb7f395dab72800516e42a95388e7a4b04e1f0a6","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-7d7dad3bd149bf81","role":"sequence","name":"basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      4,       8;\n  2:     15,      64,       36;\n  3:     56,     368,      432,      128;\n  4:    209,    1856,     3348,     2048,      400;\n  5:    780,    8728,    21312,    19968,     8000,     1152;\n  6:   2911,   39296,   121176,   153600,    94000,    27648,     3136;\n  7:  10864,  171744,   640224,  1025280,   848000,   380160,    87808,    8192;\n  8:  40545,  734592,  3212568,  6223872,  6502000,  3935232,  1382976,  262144,  20736;\n  9: 151316, 3090984, 15515712, 35292672, 44609600, 34086528, 16156672, 4653056, 746496, 51200;\n","recurrence_sentence":"The row polynomials satisfy (36*n+6*n*x-(30*n^3-4*n)*x^2-10*n^3*x^3+n^5*x^4)*P(n,x) = (72*n+72+(63*n^2+156*n+93)*x-(60*n^3+66*n^2-32*n-38)*x^2-(6*n^4+71*n^3+116*n^2+53*n+2)*x^3+(2*n^5+2*n^4-18*n^3-36*n^2-22*n-4)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (27+54*x-(63*n^2+63*n-27)*x^2-(114*n^2+114*n+8)*x^3+(3*n^4+6*n^3-58*n^2-61*n-10)*x^4+(4*n^4+8*n^3-6*n^2-10*n-2)*x^5+(n^4+2*n^3+n^2)*x^6)*d/dx 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[((72*n + 72 + (63*n^2 + 156*n + 93)*x - (60*n^3 + 66*n^2 - 32*n - 38)*x^2 - (6*n^4 + 71*n^3 + 116*n^2 + 53*n + 2)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (27 + 54*x - (63*n^2 + 63*n - 27)*x^2 - (114*n^2 + 114*n + 8)*x^3 + (3*n^4 + 6*n^3 - 58*n^2 - 61*n - 10)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(36*n + 6*n*x - (30*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*x^4)]];\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] = ((72*n + 72 + (63*n^2 + 156*n + 93)*x - (60*n^3 + 66*n^2 - 32*n - 38)*x^2 - (6*n^4 + 71*n^3 + 116*n^2 + 53*n + 2)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (27 + 54*x - (63*n^2 + 63*n - 27)*x^2 - (114*n^2 + 114*n + 8)*x^3 + (3*n^4 + 6*n^3 - 58*n^2 - 61*n - 10)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(36*n + 6*n*x - (30*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*x^4)); 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-f287bed89e07609f","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-7797a63e71923f20","e-22c2c549b1bf13ab"],"text":"A125662 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A125662 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}