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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":"9e1154d337d022723830218c4c932b6d4add633b3f3b2751e333a81c087497f6","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-5b4c394836f6a454","role":"sequence","name":"theta_plus_one of Riordan array (1, 3+3x).","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: 0, 24;\n  2: 0, 24, 243;\n  3: 0,  0, 486, 1728;\n  4: 0,  0, 243, 5184, 10125;\n  5: 0,  0,   0, 5184, 40500,  52488;\n  6: 0,  0,   0, 1728, 60750, 262440,  250047;\n  7: 0,  0,   0,    0, 40500, 524880, 1500282,  1119744;\n  8: 0,  0,   0,    0, 10125, 524880, 3750705,  7838208,  4782969;\n  9: 0,  0,   0,    0,     0, 262440, 5000940, 23514624, 38263752, 19683000;\n","recurrence_sentence":"The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(9*n^4+36*n^3+27*n^2-6*n)*x+9*n^3*x^2)*P(n,x) = -(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8-(30*n^4+162*n^3+300*n^2+264*n+108)*x+(99*n^3+459*n^2+702*n+324)*x^2-(54*n^2+108*n+54)*x^3)*P(n-1,x) + ((4*n^4+24*n^3+52*n^2+48*n+16)*x+(3*n^4-18*n^3-141*n^2-228*n-108)*x^2-(27*n^3+99*n^2+126*n+54)*x^3+(27*n^2+54*n+27)*x^4)*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[(-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (30*n^4 + 162*n^3 + 300*n^2 + 264*n + 108)*x + (99*n^3 + 459*n^2 + 702*n + 324)*x^2 - (54*n^2 + 108*n + 54)*x^3)*P[n - 1, x] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (3*n^4 - 18*n^3 - 141*n^2 - 228*n - 108)*x^2 - (27*n^3 + 99*n^2 + 126*n + 54)*x^3 + (27*n^2 + 54*n + 27)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (9*n^4 + 36*n^3 + 27*n^2 - 6*n)*x + 9*n^3*x^2)]];\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] = (-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (30*n^4 + 162*n^3 + 300*n^2 + 264*n + 108)*x + (99*n^3 + 459*n^2 + 702*n + 324)*x^2 - (54*n^2 + 108*n + 54)*x^3)*v[i - 1] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (3*n^4 - 18*n^3 - 141*n^2 - 228*n - 108)*x^2 - (27*n^3 + 99*n^2 + 126*n + 54)*x^3 + (27*n^2 + 54*n + 27)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (9*n^4 + 36*n^3 + 27*n^2 - 6*n)*x + 9*n^3*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-016d7724d51f6416","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-3c1a49e974ad23c3","e-53cb7f765f08e122"],"text":"A099093 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A099093 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}