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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":"419bba055ac097b522af72dd0303f4361e7146b5b5f2d03649e1087fcc041484","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-4e0759372c05411f","role":"sequence","name":"Scaled array A078740 ((3,2)-Stirling2).","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: 1,  12,    18;\n  2: 1,  28,   144,    240,     125;\n  3: 1,  48,   459,   1680,    2850,     2268,       686;\n  4: 1,  72,  1062,   6336,   19275,    32760,     31556,     16128,       3402;\n  5: 1, 100,  2070,  17760,   80875,   218988,    371420,    399360,     264870,      99000,      15972;\n  6: 1, 132,  3618,  41600,  259125,   989388,   2472736,   4181184,    4829625,    3762000,    1892682,     555984,      72501;\n  7: 1, 168,  5859,  86240,  695625,  3498768,  11796799,  27807552,   46873890,   56941500,   49529172,   30126096,   12180168,    2942940,     321750;\n  8: 1, 208,  8964, 163520, 1645700, 10435824,  45005716, 138322432,  311826915,  524106000,  660910712,  623278656,  433410978,  215815600,   72844200,   14936064,   1405118;\n  9: 1, 252, 13122, 289536, 3538800, 27433728, 145986288, 560454912, 1603306467, 3489568500, 5849735562, 7592180544, 7617310974, 5855594472, 3386026800, 1425715200, 412721478, 73465704, 6063356;\n","recurrence_sentence":"The row polynomials satisfy -(n^3+3*n^2+2*n-(n^5+4*n^4+5*n^3-6*n^2-12*n+8)*x-(10*n^4+15*n^3-5*n^2+6*n-8)*x^2-(16*n^3+16*n^2-28*n+8)*x^3)*P(n,x) = -(n^3+3*n^2+2*n-(n^5+4*n^4+3*n^3-28*n^2-28*n-24)*x-(n^5+24*n^4+34*n^3+21*n^2-122*n-48)*x^2-(20*n^4+126*n^3+94*n^2-12*n)*x^3-(96*n^3+144*n^2+180*n+48)*x^4-(96*n^2+96*n+24)*x^5)*P(n-1,x) - ((2*n^2+3*n+4)*x-(2*n^4+6*n^3+10*n^2-16*n-8)*x^2-(4*n^4+32*n^3+46*n^2+18*n+8)*x^3-(2*n^4+46*n^3+86*n^2+104*n+32)*x^4-(20*n^3+84*n^2+105*n+28)*x^5-(32*n^2+32*n+8)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 2*n - (n^5 + 4*n^4 + 3*n^3 - 28*n^2 - 28*n - 24)*x - (n^5 + 24*n^4 + 34*n^3 + 21*n^2 - 122*n - 48)*x^2 - (20*n^4 + 126*n^3 + 94*n^2 - 12*n)*x^3 - (96*n^3 + 144*n^2 + 180*n + 48)*x^4 - (96*n^2 + 96*n + 24)*x^5)*P[n - 1, x] - ((2*n^2 + 3*n + 4)*x - (2*n^4 + 6*n^3 + 10*n^2 - 16*n - 8)*x^2 - (4*n^4 + 32*n^3 + 46*n^2 + 18*n + 8)*x^3 - (2*n^4 + 46*n^3 + 86*n^2 + 104*n + 32)*x^4 - (20*n^3 + 84*n^2 + 105*n + 28)*x^5 - (32*n^2 + 32*n + 8)*x^6)*D[P[n - 1, x], x])/(-n^3 - 3*n^2 - 2*n + (n^5 + 4*n^4 + 5*n^3 - 6*n^2 - 12*n + 8)*x + (10*n^4 + 15*n^3 - 5*n^2 + 6*n - 8)*x^2 + (16*n^3 + 16*n^2 - 28*n + 8)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 2*n - (n^5 + 4*n^4 + 3*n^3 - 28*n^2 - 28*n - 24)*x - (n^5 + 24*n^4 + 34*n^3 + 21*n^2 - 122*n - 48)*x^2 - (20*n^4 + 126*n^3 + 94*n^2 - 12*n)*x^3 - (96*n^3 + 144*n^2 + 180*n + 48)*x^4 - (96*n^2 + 96*n + 24)*x^5)*v[i - 1] - ((2*n^2 + 3*n + 4)*x - (2*n^4 + 6*n^3 + 10*n^2 - 16*n - 8)*x^2 - (4*n^4 + 32*n^3 + 46*n^2 + 18*n + 8)*x^3 - (2*n^4 + 46*n^3 + 86*n^2 + 104*n + 32)*x^4 - (20*n^3 + 84*n^2 + 105*n + 28)*x^5 - (32*n^2 + 32*n + 8)*x^6)*deriv(v[i - 1], x))/(-n^3 - 3*n^2 - 2*n + (n^5 + 4*n^4 + 5*n^3 - 6*n^2 - 12*n + 8)*x + (10*n^4 + 15*n^3 - 5*n^2 + 6*n - 8)*x^2 + (16*n^3 + 16*n^2 - 28*n + 8)*x^3)); 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-4e0759372c05411f","steps":["(\u03b8+1)\u00b2"],"edges":["e-31b2ab2fb2ee0af1"],"text":"A090452 \u2192 (\u03b8+1)\u00b2"},"headline":"A090452 \u2192 (\u03b8+1)\u00b2","chain":null}