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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":"24c61d8e7dd7d6c9bf6cbbff0e6ce3576a1810b1ce3b7982ee60ca32bf0a4d8f","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-8a65e165240c40e0","role":"sequence","name":"Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","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: 0, 4;\n  2: 0, 4,   9;\n  3: 0, 4,  24,   16;\n  4: 0, 4,  45,   80,    25;\n  5: 0, 4,  72,  240,   200,    36;\n  6: 0, 4, 105,  560,   875,   420,    49;\n  7: 0, 4, 144, 1120,  2800,  2520,   784,    64;\n  8: 0, 4, 189, 2016,  7350, 10584,  6174,  1344,   81;\n  9: 0, 4, 240, 3360, 16800, 35280, 32928, 13440, 2160, 100;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^2+4*n-(n^4+4*n^3+4*n^2)*x)*P(n,x) = -(2*n^2-2-(n^4+2*n^3-13*n^2-10*n-4)*x-(3*n^4+12*n^3+11*n^2+8*n+2)*x^2)*P(n-1,x) - ((4*n+2)*x-(2*n^3+8*n^2+10*n+4)*x^2+(2*n^3+8*n^2+6*n+2)*x^3)*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[(-(2*n^2 - 2 - (n^4 + 2*n^3 - 13*n^2 - 10*n - 4)*x - (3*n^4 + 12*n^3 + 11*n^2 + 8*n + 2)*x^2)*P[n - 1, x] - ((4*n + 2)*x - (2*n^3 + 8*n^2 + 10*n + 4)*x^2 + (2*n^3 + 8*n^2 + 6*n + 2)*x^3)*D[P[n - 1, x], x])/(-2*n^2 - 4*n + (n^4 + 4*n^3 + 4*n^2)*x)]];\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] = (-(2*n^2 - 2 - (n^4 + 2*n^3 - 13*n^2 - 10*n - 4)*x - (3*n^4 + 12*n^3 + 11*n^2 + 8*n + 2)*x^2)*v[i - 1] - ((4*n + 2)*x - (2*n^3 + 8*n^2 + 10*n + 4)*x^2 + (2*n^3 + 8*n^2 + 6*n + 2)*x^3)*deriv(v[i - 1], x))/(-2*n^2 - 4*n + (n^4 + 4*n^3 + 4*n^2)*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-8a65e165240c40e0","steps":["\u03b8+1"],"edges":["e-25592cbd0b4a7710"],"text":"A281260 \u2192 \u03b8+1"},"headline":"A281260 \u2192 \u03b8+1","chain":null}