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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":"15d646c9053419f2188a5bab850504ea622f041a3c1b5d91e3dba06f17f589ca","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-9813839be9bee901","role":"sequence","name":"basis_map_A013610 of Pascal-(1,3,1) array.","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:     7,     42,      27;\n  3:    20,    180,     324,      108;\n  4:    61,    732,    2106,     1836,      405;\n  5:   182,   2730,   10908,    16308,     8910,     1458;\n  6:   547,   9846,   50301,   107028,   100845,    39366,     5103;\n  7:  1640,  34440,  214488,   595080,   813240,   542376,   163296,    17496;\n  8:  4921, 118104,  867780,  2972808,  5364630,  5219640,  2653560,   647352,   59049;\n  9: 14762, 398574, 3375432, 13775832, 31011660, 40162068, 29760696, 12124728, 2480058, 196830;\n","recurrence_sentence":"The row polynomials satisfy (108*n*x+(162*n^2+324*n)*x^2+(81*n^3+243*n^2+162*n)*x^3)*P(n,x) = -(192*n-192+(432*n^2+756*n-1404)*x+(216*n^3+1566*n^2-54*n-3348)*x^2+(567*n^3+810*n^2-2349*n-2916)*x^3-(486*n^2+1458*n+972)*x^4)*P(n-1,x) + (64+(192*n+672)*x+(144*n^2+1584*n+2628)*x^2+(864*n^2+4428*n+4590)*x^3+(1377*n^2+4455*n+3240)*x^4+(243*n^2+729*n+486)*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[(-(192*n - 192 + (432*n^2 + 756*n - 1404)*x + (216*n^3 + 1566*n^2 - 54*n - 3348)*x^2 + (567*n^3 + 810*n^2 - 2349*n - 2916)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*P[n - 1, x] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2628)*x^2 + (864*n^2 + 4428*n + 4590)*x^3 + (1377*n^2 + 4455*n + 3240)*x^4 + (243*n^2 + 729*n + 486)*x^5)*D[P[n - 1, x], x])/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*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] = (-(192*n - 192 + (432*n^2 + 756*n - 1404)*x + (216*n^3 + 1566*n^2 - 54*n - 3348)*x^2 + (567*n^3 + 810*n^2 - 2349*n - 2916)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*v[i - 1] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2628)*x^2 + (864*n^2 + 4428*n + 4590)*x^3 + (1377*n^2 + 4455*n + 3240)*x^4 + (243*n^2 + 729*n + 486)*x^5)*deriv(v[i - 1], x))/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*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-d8f9ad9ce0e039d0","steps":["basis A013610","\u03b8+1"],"edges":["e-a15ab36c8834192d","e-49b6985980c4bac9"],"text":"A081578 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A081578 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}