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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":"dd9d8faca2a1b542f5d441385ebe6f3d74e9334e5136852dfc00c83bd91c9d0e","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-7bf0ce499426260c","role":"sequence","name":"derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 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:   0;\n  1:   2;\n  2:   8,    12;\n  3:  20,    80,     36;\n  4:  40,   300,    360,     80;\n  5:  70,   840,   1890,   1120,    150;\n  6: 112,  1960,   7056,   7840,   2800,    252;\n  7: 168,  4032,  21168,  37632,  25200,   6048,    392;\n  8: 240,  7560,  54432, 141120, 151200,  68040,  11760,   576;\n  9: 330, 13200, 124740, 443520, 693000, 498960, 161700, 21120, 810;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+6*n-8+(n^4+3*n^3-5*n^2-3*n+4)*x)*P(n,x) = (2*n^2+12*n+16+(n^4+6*n^3+19*n^2+10*n-16)*x+(3*n^4+8*n^3+n^2-6*n+4)*x^2)*P(n-1,x) + ((4*n+6)*x+(2*n^3+6*n^2-8)*x^2-(2*n^3+6*n^2+4*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 12*n + 16 + (n^4 + 6*n^3 + 19*n^2 + 10*n - 16)*x + (3*n^4 + 8*n^3 + n^2 - 6*n + 4)*x^2)*P[n - 1, x] + ((4*n + 6)*x + (2*n^3 + 6*n^2 - 8)*x^2 - (2*n^3 + 6*n^2 + 4*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 6*n - 8 + (n^4 + 3*n^3 - 5*n^2 - 3*n + 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 12*n + 16 + (n^4 + 6*n^3 + 19*n^2 + 10*n - 16)*x + (3*n^4 + 8*n^3 + n^2 - 6*n + 4)*x^2)*v[i - 1] + ((4*n + 6)*x + (2*n^3 + 6*n^2 - 8)*x^2 - (2*n^3 + 6*n^2 + 4*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 6*n - 8 + (n^4 + 3*n^3 - 5*n^2 - 3*n + 4)*x)); 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-0e6d847331141ca6","steps":["D","\u03b8+1"],"edges":["e-e77b91d6fbb51ebc","e-c1b7418dbd3deaa6"],"text":"A145596 \u2192 D \u2192 \u03b8+1"},"headline":"A145596 \u2192 D \u2192 \u03b8+1","chain":null}