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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":"99223a8cb533346cb8bd1b4ea07b6c10ea4a9f91622c3b8536b7fd4aacdab325","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-3b748f1f6d80b05e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:      3,       12;\n  2:     11,       84,       108;\n  3:     45,      504,      1296,       864;\n  4:    197,     2904,     11232,     15552,       6480;\n  5:    903,    16500,     85320,    181440,     162000,      46656;\n  6:   4279,    93276,    604260,   1740960,    2430000,    1539648,     326592;\n  7:  20793,   526512,   4100544,  14938560,   28576800,   28740096,   13716864,    2239488;\n  8: 103049,  2972208,  27046656, 119266560,  290848320,  404974080,  310915584,  116453376,  15116544;\n  9: 518859, 16791516, 174847248, 905603328, 2689485120, 4808740608, 5157540864, 3144241152, 952342272, 100776960;\n","recurrence_sentence":"The row polynomials satisfy (3*n+6+(18*n^2+54*n+36)*x+(27*n^3+81*n^2+54*n)*x^2)*P(n,x) = (21*n+6+(144*n^2+252*n+36)*x+(189*n^3+891*n^2+702*n)*x^2+(486*n^3+1134*n^2+324*n)*x^3)*P(n-1,x) - (1+(12*n+21)*x+(18*n^2+153*n+144)*x^2+(162*n^2+621*n+378)*x^3+(324*n^2+810*n+324)*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[((21*n + 6 + (144*n^2 + 252*n + 36)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (486*n^3 + 1134*n^2 + 324*n)*x^3)*P[n - 1, x] - (1 + (12*n + 21)*x + (18*n^2 + 153*n + 144)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (324*n^2 + 810*n + 324)*x^4)*D[P[n - 1, x], x])/(3*n + 6 + (18*n^2 + 54*n + 36)*x + (27*n^3 + 81*n^2 + 54*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] = ((21*n + 6 + (144*n^2 + 252*n + 36)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (486*n^3 + 1134*n^2 + 324*n)*x^3)*v[i - 1] - (1 + (12*n + 21)*x + (18*n^2 + 153*n + 144)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (324*n^2 + 810*n + 324)*x^4)*deriv(v[i - 1], x))/(3*n + 6 + (18*n^2 + 54*n + 36)*x + (27*n^3 + 81*n^2 + 54*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-8ce61d302cec295b","steps":["basis A013610","\u03b8+1"],"edges":["e-72b4d9a78d1bd227","e-b8a214b18584b818"],"text":"A114687 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A114687 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}