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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":"999a46454d29cb681849977a7282fccf336a4f5cd3f10eeeac78310c1cbf2a06","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-84fc1826f57fd756","role":"sequence","name":"A triangular decomposition of the double factorial numbers A001147.","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: 0,    4;\n  2: 0,    8,      9;\n  3: 0,   16,     90,       16;\n  4: 0,   32,    540,      576,        25;\n  5: 0,   64,   2664,     8256,      2900,        36;\n  6: 0,  128,  11952,    82688,     87700,     12888,        49;\n  7: 0,  256,  50976,   687360,   1606000,    760320,     53214,       64;\n  8: 0,  512, 211392,  5121024,  22514000,  23791680,   5812968,   209408,     81;\n  9: 0, 1024, 862848, 35611648, 268129600, 529698240, 296437456, 40893952, 796392, 100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = ((64*n-96)*x+(144*n^4-320*n^3+152*n^2+16*n-15)*x^2)*P(n-2,x) + (16*x+(64*n^2+16*n-104)*x^2+(64*n^4-352*n^3+224*n^2+8*n-20)*x^3)*d/dx P(n-2,x) - ((16*n^2-20)*x^2-(64*n^3-32*n^2-16)*x^3+(64*n^3-48*n^2+4)*x^4)*d^2/dx^2 P(n-2,x) - (8*n^2-2-(32*n^3-48*n^2+24*n-4)*x)*P(n-1,x) + ((16*n^2-8*n-8)*x-(32*n^3-16*n^2-8*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((64*n - 96)*x + (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*P[n - 2, x] + (16*x + (64*n^2 + 16*n - 104)*x^2 + (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 20)*x^2 - (64*n^3 - 32*n^2 - 16)*x^3 + (64*n^3 - 48*n^2 + 4)*x^4)*D[P[n - 2, x], {x, 2}] - (8*n^2 - 2 - (32*n^3 - 48*n^2 + 24*n - 4)*x)*P[n - 1, x] + ((16*n^2 - 8*n - 8)*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((64*n - 96)*x + (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*v[i - 2] + (16*x + (64*n^2 + 16*n - 104)*x^2 + (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 20)*x^2 - (64*n^3 - 32*n^2 - 16)*x^3 + (64*n^3 - 48*n^2 + 4)*x^4)*deriv(deriv(v[i - 2], x), x) - (8*n^2 - 2 - (32*n^3 - 48*n^2 + 24*n - 4)*x)*v[i - 1] + ((16*n^2 - 8*n - 8)*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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-54b58a0ddb0e34fe","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-5c085bd25a6bcbdf","e-c5bd42f89e0a97a9"],"text":"A185410 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A185410 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}