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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":"a610709be0fbdc07e570aa5223db0036bbae00c994d2bccc8172a6d83e8a0d1d","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-1f922d5ae3a4fc22","role":"sequence","name":"veronese_section_2_1 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;\n  1: 0,  4;\n  2: 0,  8;\n  3: 0, 12,   9;\n  4: 0, 16,  36;\n  5: 0, 20,  90,   16;\n  6: 0, 24, 180,   96;\n  7: 0, 28, 315,  336,  25;\n  8: 0, 32, 504,  896, 200;\n  9: 0, 36, 756, 2016, 900, 36;\n","recurrence_sentence":"The row polynomials satisfy (144*n-144-(38*n^3+28*n^2-62*n-4)*x+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^2)*P(n,x) = (144*n-288-(38*n^3-10*n^2-604*n+144)*x+(n^5+4*n^4-61*n^3-228*n^2-148*n)*x^2+(n^5+10*n^4+37*n^3+60*n^2+36*n)*x^3)*P(n-1,x) + (288*x-(76*n^2+284*n+504)*x^2+(2*n^4+20*n^3+150*n^2+404*n+288)*x^3-(2*n^4+20*n^3+74*n^2+120*n+72)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((144*n - 288 - (38*n^3 - 10*n^2 - 604*n + 144)*x + (n^5 + 4*n^4 - 61*n^3 - 228*n^2 - 148*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*P[n - 1, x] + (288*x - (76*n^2 + 284*n + 504)*x^2 + (2*n^4 + 20*n^3 + 150*n^2 + 404*n + 288)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*D[P[n - 1, x], x])/(144*n - 144 - (38*n^3 + 28*n^2 - 62*n - 4)*x + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((144*n - 288 - (38*n^3 - 10*n^2 - 604*n + 144)*x + (n^5 + 4*n^4 - 61*n^3 - 228*n^2 - 148*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*v[i - 1] + (288*x - (76*n^2 + 284*n + 504)*x^2 + (2*n^4 + 20*n^3 + 150*n^2 + 404*n + 288)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*deriv(v[i - 1], x))/(144*n - 144 - (38*n^3 + 28*n^2 - 62*n - 4)*x + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*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 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-6fdf1329e5e61ed8","e-92478556d56a104a"],"text":"A159854 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A159854 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}