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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":"d23bb107cd7330a8c0c242cb8585f75027ed7abd93c255f83265f3fd13eb6bee","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-3a1bf08f9f4a4163","role":"sequence","name":"basis_map_A013610 of Pascal-(1,2,1) array read by antidiagonals.","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:    2,      6;\n  2:    6,     36,      27;\n  3:   16,    144,     270,     108;\n  4:   44,    528,    1566,    1512,      405;\n  5:  120,   1800,    7344,   11664,     7290,     1458;\n  6:  328,   5904,   30672,   68256,    70470,    32076,     5103;\n  7:  896,  18816,  118800,  341280,   498960,   373248,   132678,   17496;\n  8: 2448,  58752,  436752, 1539648,  2930580,  3114288,  1806462,  524880,   59049;\n  9: 6688, 180576, 1544832, 6459264, 15192360, 21117672, 17391024, 8188128, 2007666, 196830;\n","recurrence_sentence":"The row polynomials satisfy (36*n*x+(54*n^2+108*n)*x^2+(27*n^3+81*n^2+54*n)*x^3)*P(n,x) = -(48*n-48+(108*n^2+180*n-360)*x+(54*n^3+378*n^2-72*n-900)*x^2+(135*n^3+162*n^2-729*n-864)*x^3-(162*n^2+486*n+324)*x^4)*P(n-1,x) + (16+(48*n+168)*x+(36*n^2+396*n+660)*x^2+(216*n^2+1116*n+1170)*x^3+(351*n^2+1161*n+864)*x^4+(81*n^2+243*n+162)*x^5)*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[(-(48*n - 48 + (108*n^2 + 180*n - 360)*x + (54*n^3 + 378*n^2 - 72*n - 900)*x^2 + (135*n^3 + 162*n^2 - 729*n - 864)*x^3 - (162*n^2 + 486*n + 324)*x^4)*P[n - 1, x] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 660)*x^2 + (216*n^2 + 1116*n + 1170)*x^3 + (351*n^2 + 1161*n + 864)*x^4 + (81*n^2 + 243*n + 162)*x^5)*D[P[n - 1, x], x])/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*n)*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] = (-(48*n - 48 + (108*n^2 + 180*n - 360)*x + (54*n^3 + 378*n^2 - 72*n - 900)*x^2 + (135*n^3 + 162*n^2 - 729*n - 864)*x^3 - (162*n^2 + 486*n + 324)*x^4)*v[i - 1] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 660)*x^2 + (216*n^2 + 1116*n + 1170)*x^3 + (351*n^2 + 1161*n + 864)*x^4 + (81*n^2 + 243*n + 162)*x^5)*deriv(v[i - 1], x))/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*n)*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-f25cd955a876cde3","steps":["basis A013610","\u03b8+1"],"edges":["e-ce7b55dc0173f12e","e-818a5113544b27fc"],"text":"A081577 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A081577 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}