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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":"7b40380f92c17cbdb6d739d4526d21f567a9fae102fee0bd89a322ed123891b5","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-a4a705640891e75c","role":"sequence","name":"veronese_section_2_0 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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,   4;\n  1: 1,  16;\n  2: 1,  26,   12;\n  3: 1,  50,   84;\n  4: 1,  66,  198,    32;\n  5: 1, 102,  576,   320;\n  6: 1, 124,  963,   992,    80;\n  7: 1, 172, 2043,  3872,  1040;\n  8: 1, 200, 2955,  7880,  4000,  192;\n  9: 1, 260, 5295, 21344, 19600, 3072;\n","recurrence_sentence":"The row polynomials satisfy 32*P(n,x) = -(706-2280*x+2208*x^2-256*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (888-768*x+384*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (168+192*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 1 + 4*x, P(1,x) = 1 + 16*x, P(2,x) = 1 + 26*x + 12*x^2, P(3,x) = 1 + 50*x + 84*x^2, P(4,x) = 1 + 66*x + 198*x^2 + 32*x^3, P(5,x) = 1 + 102*x + 576*x^2 + 320*x^3.","mathematica":"P[0, x_] := 1 + 4*x;\nP[1, x_] := 1 + 16*x;\nP[2, x_] := 1 + 26*x + 12*x^2;\nP[3, x_] := 1 + 50*x + 84*x^2;\nP[4, x_] := 1 + 66*x + 198*x^2 + 32*x^3;\nP[5, x_] := 1 + 102*x + 576*x^2 + 320*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(706 - 2280*x + 2208*x^2 - 256*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (888 - 768*x + 384*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (168 + 192*x)*P[n - 2, x] + (81 - 144*x)*D[P[n - 2, x], x])/(32)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x; v[2] = 1 + 16*x; v[3] = 1 + 26*x + 12*x^2; v[4] = 1 + 50*x + 84*x^2; v[5] = 1 + 66*x + 198*x^2 + 32*x^3; v[6] = 1 + 102*x + 576*x^2 + 320*x^3; for(n = 6, N, my(i = n + 1); v[i] = (-(706 - 2280*x + 2208*x^2 - 256*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (888 - 768*x + 384*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (168 + 192*x)*v[i - 2] + (81 - 144*x)*deriv(v[i - 2], x))/(32)); 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-5a70a6e97131d437","steps":["Veronese 2","\u03b8+1"],"edges":["e-4995ac39f8921064","e-4d33fa67f4848e90"],"text":"A064861 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A064861 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}