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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":"f6b562b1726b0b29ca122054facac875853914d63fc336152ca0336093eb75e0","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-b6cda967a092feb1","role":"sequence","name":"basis_map_A007318 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,       4;\n  2:     11,      28,       12;\n  3:     45,     168,      144,       32;\n  4:    197,     968,     1248,      576,       80;\n  5:    903,    5500,     9480,     6720,     2000,      192;\n  6:   4279,   31092,    67140,    64480,    30000,     6336,     448;\n  7:  20793,  175504,   455616,   553280,   352800,   118272,   18816,    1024;\n  8: 103049,  990736,  3005184,  4417280,  3590720,  1666560,  426496,   53248,   2304;\n  9: 518859, 5597172, 19427472, 33540864, 33203520, 19789056, 7074816, 1437696, 145152, 5120;\n","recurrence_sentence":"The row polynomials satisfy (n+2+(2*n^2+6*n+4)*x+(n^3+3*n^2+2*n)*x^2)*P(n,x) = (7*n+2+(16*n^2+28*n+4)*x+(7*n^3+33*n^2+26*n)*x^2+(6*n^3+14*n^2+4*n)*x^3)*P(n-1,x) - (1+(4*n+7)*x+(2*n^2+17*n+16)*x^2+(6*n^2+23*n+14)*x^3+(4*n^2+10*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[((7*n + 2 + (16*n^2 + 28*n + 4)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (6*n^3 + 14*n^2 + 4*n)*x^3)*P[n - 1, x] - (1 + (4*n + 7)*x + (2*n^2 + 17*n + 16)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (4*n^2 + 10*n + 4)*x^4)*D[P[n - 1, x], x])/(n + 2 + (2*n^2 + 6*n + 4)*x + (n^3 + 3*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] = ((7*n + 2 + (16*n^2 + 28*n + 4)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (6*n^3 + 14*n^2 + 4*n)*x^3)*v[i - 1] - (1 + (4*n + 7)*x + (2*n^2 + 17*n + 16)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (4*n^2 + 10*n + 4)*x^4)*deriv(v[i - 1], x))/(n + 2 + (2*n^2 + 6*n + 4)*x + (n^3 + 3*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-8ce61d302cec295b","steps":["basis A007318","\u03b8+1"],"edges":["e-6804282af6df94ec","e-70f50e6dcdc8d2df"],"text":"A114687 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A114687 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}