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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":"3a8d831f0cb1e840acd2e47c3a516262483e5c384fcbf7422c62baaa49858a82","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fac4d4a000d744e8","role":"sequence","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:    2,     2;\n  2:    5,    10,      3;\n  3:   13,    40,     27,      4;\n  4:   34,   142,    153,     56,      5;\n  5:   89,   470,    699,    420,    100,     6;\n  6:  233,  1488,   2826,   2376,    950,   162,     7;\n  7:  610,  4570,  10566,  11440,   6475,  1890,   245,    8;\n  8: 1597, 13730,  37419,  49608,  36425, 15204,  3430,  352,   9;\n  9: 4181, 40568, 127341, 199852, 180610, 98442, 32046, 5808, 486, 10;\n","recurrence_sentence":"The row polynomials satisfy (20*n-10+(20*n^2-6*n-2)*x+(8*n^3-8*n^2+2*n)*x^2)*P(n,x) = (80*n-60+(100*n^2+26*n-82)*x+(32*n^3+80*n^2-64*n-20)*x^2+(24*n^3-26*n+6)*x^3)*P(n-1,x) - (25+(50*n+55)*x+(20*n^2+110*n+33)*x^2+(36*n^2+76*n-1)*x^3+(16*n^2+16*n-4)*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[((80*n - 60 + (100*n^2 + 26*n - 82)*x + (32*n^3 + 80*n^2 - 64*n - 20)*x^2 + (24*n^3 - 26*n + 6)*x^3)*P[n - 1, x] - (25 + (50*n + 55)*x + (20*n^2 + 110*n + 33)*x^2 + (36*n^2 + 76*n - 1)*x^3 + (16*n^2 + 16*n - 4)*x^4)*D[P[n - 1, x], x])/(20*n - 10 + (20*n^2 - 6*n - 2)*x + (8*n^3 - 8*n^2 + 2*n)*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] = ((80*n - 60 + (100*n^2 + 26*n - 82)*x + (32*n^3 + 80*n^2 - 64*n - 20)*x^2 + (24*n^3 - 26*n + 6)*x^3)*v[i - 1] - (25 + (50*n + 55)*x + (20*n^2 + 110*n + 33)*x^2 + (36*n^2 + 76*n - 1)*x^3 + (16*n^2 + 16*n - 4)*x^4)*deriv(v[i - 1], x))/(20*n - 10 + (20*n^2 - 6*n - 2)*x + (8*n^3 - 8*n^2 + 2*n)*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-fac4d4a000d744e8","steps":["\u03b8+1"],"edges":["e-2b404cb8e7cd55aa"],"text":"A060920 \u2192 \u03b8+1"},"headline":"A060920 \u2192 \u03b8+1","chain":null}