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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":"24202623740da6628a708d9b457459df55840a54f08a8307f7ac4a8fd171abda","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-cd0e6a1564dd27fb","role":"sequence","name":"veronese_section_2_1 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= 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:     0;\n  1:     2;\n  2:     2;\n  3:     4,      32;\n  4:    12,     192;\n  5:    48,    1120,     288;\n  6:   240,    7200,    4320;\n  7:  1440,   51968,   50400,    2048;\n  8: 10080,  420224,  564480,   57344;\n  9: 80640, 3779968, 6465312, 1118208, 12800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(16*n^2-128*n+256)*x-144*x^2)*P(n-4,x) - ((16*n^2-128*n+256)*x^2-64*x^3)*d/dx P(n-4,x) + (4*n^3-42*n^2+148*n-175-(32*n-112)*x)*P(n-3,x) + (32*n-112)*x^2*d/dx P(n-3,x) - (6*n^2-36*n+55-16*x)*P(n-2,x) - 16*x^2*d/dx P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 2, P(3,x) = 4 + 32*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 2;\nP[3, x_] := 4 + 32*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (16*n^2 - 128*n + 256)*x - 144*x^2)*P[n - 4, x] - ((16*n^2 - 128*n + 256)*x^2 - 64*x^3)*D[P[n - 4, x], x] + (4*n^3 - 42*n^2 + 148*n - 175 - (32*n - 112)*x)*P[n - 3, x] + (32*n - 112)*x^2*D[P[n - 3, x], x] - (6*n^2 - 36*n + 55 - 16*x)*P[n - 2, x] - 16*x^2*D[P[n - 2, x], x] + (4*n - 10)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 2; v[4] = 4 + 32*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (16*n^2 - 128*n + 256)*x - 144*x^2)*v[i - 4] - ((16*n^2 - 128*n + 256)*x^2 - 64*x^3)*deriv(v[i - 4], x) + (4*n^3 - 42*n^2 + 148*n - 175 - (32*n - 112)*x)*v[i - 3] + (32*n - 112)*x^2*deriv(v[i - 3], x) - (6*n^2 - 36*n + 55 - 16*x)*v[i - 2] - 16*x^2*deriv(v[i - 2], x) + (4*n - 10)*v[i - 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-60c7e36cbe887b7e","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-28dcf565f3f6cf28","e-486d395f9e7e2a8d"],"text":"A137320 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A137320 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}