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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":"48c6bea00126440b5d79e53652d0a8c818ad89525d009d842427966cbb3bbad8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba8de4e02acfb6ca","role":"sequence","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  12;\n   3: 1,  40,    45;\n   4: 1,  84,   315,    112;\n   5: 1, 144,  1134,   1344,     225;\n   6: 1, 220,  2970,   7392,    4125,     396;\n   7: 1, 312,  6435,  27456,   32175,   10296,     637;\n   8: 1, 420, 12285,  80080,  160875,  108108,   22295,    960;\n   9: 1, 544, 21420, 198016,  607750,  700128,  303212,  43520,  1377;\n  10: 1, 684, 34884, 434112, 1889550, 3325608, 2469012, 744192, 78489, 1900;\n","recurrence_sentence":"The row polynomials satisfy (18*n-27-(44*n^3-156*n^2+175*n-60)*x+(8*n^5-36*n^4+62*n^3-51*n^2+20*n-3)*x^2)*P(n,x) = (18*n-27-(44*n^3-156*n^2-41*n+264)*x+(8*n^5-36*n^4-166*n^3+473*n^2-137*n-177)*x^2+(24*n^5-44*n^4-22*n^3+63*n^2-10*n-12)*x^3)*P(n-1,x) + (36*x-(88*n^2-48*n-18)*x^2+(16*n^4+60*n^2-40*n-48)*x^3-(16*n^4-28*n^2+8*n+6)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n - 27 - (44*n^3 - 156*n^2 - 41*n + 264)*x + (8*n^5 - 36*n^4 - 166*n^3 + 473*n^2 - 137*n - 177)*x^2 + (24*n^5 - 44*n^4 - 22*n^3 + 63*n^2 - 10*n - 12)*x^3)*P[n - 1, x] + (36*x - (88*n^2 - 48*n - 18)*x^2 + (16*n^4 + 60*n^2 - 40*n - 48)*x^3 - (16*n^4 - 28*n^2 + 8*n + 6)*x^4)*D[P[n - 1, x], x])/(18*n - 27 - (44*n^3 - 156*n^2 + 175*n - 60)*x + (8*n^5 - 36*n^4 + 62*n^3 - 51*n^2 + 20*n - 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((18*n - 27 - (44*n^3 - 156*n^2 - 41*n + 264)*x + (8*n^5 - 36*n^4 - 166*n^3 + 473*n^2 - 137*n - 177)*x^2 + (24*n^5 - 44*n^4 - 22*n^3 + 63*n^2 - 10*n - 12)*x^3)*v[i - 1] + (36*x - (88*n^2 - 48*n - 18)*x^2 + (16*n^4 + 60*n^2 - 40*n - 48)*x^3 - (16*n^4 - 28*n^2 + 8*n + 6)*x^4)*deriv(v[i - 1], x))/(18*n - 27 - (44*n^3 - 156*n^2 + 175*n - 60)*x + (8*n^5 - 36*n^4 + 62*n^3 - 51*n^2 + 20*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["(\u03b8+1)\u00b2"],"edges":["e-14374e20b2874ad7"],"text":"A091042 \u2192 (\u03b8+1)\u00b2"},"headline":"A091042 \u2192 (\u03b8+1)\u00b2","chain":null}