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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":"781d15663ce9e16004c0d581d4b47117839588941edb77856e74cda0cf024a69","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-5b112963159320a9","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 peaks (1 <= k <= n).","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:      3,        6;\n   3:     11,       48,       27;\n   4:     45,      306,      405,       108;\n   5:    197,     1824,     3942,      2592,       405;\n   6:    903,    10590,    32130,     35640,     14175,      1458;\n   7:   4279,    60768,   238545,    380160,    261225,     69984,      5103;\n   8:  20793,   346794,  1674351,   3511620,   3529575,   1663578,    321489,    17496;\n   9: 103049,  1974144, 11327148,  29598912,  39695670,  27760320,   9573228,  1399680,   59049;\n  10: 518859, 11226870, 74683620, 234284400, 395737650, 373478364, 193322052, 51018336, 5845851, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(6*n+6-(18*n^2+18*n)*x-(27*n^3-27*n)*x^2)*P(n,x) = -(30*n-6-(72*n^2-72*n+72)*x-(135*n^3+162*n^2-297*n)*x^2-(243*n^3-162*n^2-243*n+162)*x^3)*P(n-1,x) - (2-12*n*x-(18*n^2+9*n+9)*x^2+108*n*x^3+(162*n^2+81*n-81)*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[(-(30*n - 6 - (72*n^2 - 72*n + 72)*x - (135*n^3 + 162*n^2 - 297*n)*x^2 - (243*n^3 - 162*n^2 - 243*n + 162)*x^3)*P[n - 1, x] - (2 - 12*n*x - (18*n^2 + 9*n + 9)*x^2 + 108*n*x^3 + (162*n^2 + 81*n - 81)*x^4)*D[P[n - 1, x], x])/(-6*n - 6 + (18*n^2 + 18*n)*x + (27*n^3 - 27*n)*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] = (-(30*n - 6 - (72*n^2 - 72*n + 72)*x - (135*n^3 + 162*n^2 - 297*n)*x^2 - (243*n^3 - 162*n^2 - 243*n + 162)*x^3)*v[i - 1] - (2 - 12*n*x - (18*n^2 + 9*n + 9)*x^2 + 108*n*x^3 + (162*n^2 + 81*n - 81)*x^4)*deriv(v[i - 1], x))/(-6*n - 6 + (18*n^2 + 18*n)*x + (27*n^3 - 27*n)*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-0d75f6272d2a2a84","steps":["basis A013610","\u03b8+1"],"edges":["e-edd649892838aaa7","e-a0b4112cadd1d71d"],"text":"A114656 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A114656 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}