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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":"dc924e2280bcd3ff87d910ca04bbbf330dab53caeec8b373c40746daeffdd8ec","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-37ec1285a176250d","role":"sequence","name":"veronese_section_2_0 of Riordan array (1-x,x/(1-x)).","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, 4;\n  1: 0, 4;\n  2: 0, 4,   9;\n  3: 0, 4,  27;\n  4: 0, 4,  54,   16;\n  5: 0, 4,  90,   80;\n  6: 0, 4, 135,  240,   25;\n  7: 0, 4, 189,  560,  175;\n  8: 0, 4, 252, 1120,  700,  36;\n  9: 0, 4, 324, 2016, 2100, 324;\n","recurrence_sentence":"The row polynomials satisfy (400*n-400-(58*n^3+152*n^2-62*n-148)*x+(n^5+9*n^4+27*n^3+23*n^2-24*n-36)*x^2)*P(n,x) = (400*n-1200-(58*n^3+36*n^2-1614*n-880)*x+(n^5+7*n^4-83*n^3-611*n^2-1122*n-592)*x^2+(n^5+15*n^4+87*n^3+241*n^2+312*n+144)*x^3)*P(n-1,x) + (800*x-(116*n^2+652*n+1632)*x^2+(2*n^4+28*n^3+262*n^2+988*n+1120)*x^3-(2*n^4+28*n^3+146*n^2+336*n+288)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4*x, P(1,x) = 4*x.","mathematica":"P[0, x_] := 4*x;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((400*n - 1200 - (58*n^3 + 36*n^2 - 1614*n - 880)*x + (n^5 + 7*n^4 - 83*n^3 - 611*n^2 - 1122*n - 592)*x^2 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^3)*P[n - 1, x] + (800*x - (116*n^2 + 652*n + 1632)*x^2 + (2*n^4 + 28*n^3 + 262*n^2 + 988*n + 1120)*x^3 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^4)*D[P[n - 1, x], x])/(400*n - 400 - (58*n^3 + 152*n^2 - 62*n - 148)*x + (n^5 + 9*n^4 + 27*n^3 + 23*n^2 - 24*n - 36)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4*x; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((400*n - 1200 - (58*n^3 + 36*n^2 - 1614*n - 880)*x + (n^5 + 7*n^4 - 83*n^3 - 611*n^2 - 1122*n - 592)*x^2 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^3)*v[i - 1] + (800*x - (116*n^2 + 652*n + 1632)*x^2 + (2*n^4 + 28*n^3 + 262*n^2 + 988*n + 1120)*x^3 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^4)*deriv(v[i - 1], x))/(400*n - 400 - (58*n^3 + 152*n^2 - 62*n - 148)*x + (n^5 + 9*n^4 + 27*n^3 + 23*n^2 - 24*n - 36)*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-7717563017cb9e5f","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-7643b829dc52d64b","e-1903ff43b3046e06"],"text":"A159854 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A159854 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}