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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":"6dbae4a3bc81acd10348089470d32a00010cdda3b888aea40a0a03c629725707","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-3c983d83f25bade9","role":"sequence","name":"shift_x_plus_one 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:     1;\n  1:     4,      4;\n  2:    14,     28,       9;\n  3:    48,    144,      96,      16;\n  4:   165,    660,     675,     240,      25;\n  5:   572,   2860,    3960,    2200,     500,      36;\n  6:  2002,  12012,   21021,   16016,    5775,     924,     49;\n  7:  7072,  49504,  104832,  101920,   50960,   13104,   1568,    64;\n  8: 25194, 201552,  501228,  594048,  382200,  137592,  26754,  2496,   81;\n  9: 90440, 813960, 2325600, 3255840, 2570400, 1199520, 329280, 50400, 3780, 100;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+12*n^2+16*n+(n^4+6*n^3+8*n^2)*x)*P(n,x) = (8*n^3+36*n^2+52*n+24+(4*n^4+30*n^3+70*n^2+60*n+16)*x+(3*n^4+15*n^3+23*n^2+15*n+4)*x^2)*P(n-1,x) - ((4*n^2+12*n+8)*x+(2*n^3+13*n^2+23*n+12)*x^2+(2*n^3+9*n^2+11*n+4)*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[((8*n^3 + 36*n^2 + 52*n + 24 + (4*n^4 + 30*n^3 + 70*n^2 + 60*n + 16)*x + (3*n^4 + 15*n^3 + 23*n^2 + 15*n + 4)*x^2)*P[n - 1, x] - ((4*n^2 + 12*n + 8)*x + (2*n^3 + 13*n^2 + 23*n + 12)*x^2 + (2*n^3 + 9*n^2 + 11*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 12*n^2 + 16*n + (n^4 + 6*n^3 + 8*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] = ((8*n^3 + 36*n^2 + 52*n + 24 + (4*n^4 + 30*n^3 + 70*n^2 + 60*n + 16)*x + (3*n^4 + 15*n^3 + 23*n^2 + 15*n + 4)*x^2)*v[i - 1] - ((4*n^2 + 12*n + 8)*x + (2*n^3 + 13*n^2 + 23*n + 12)*x^2 + (2*n^3 + 9*n^2 + 11*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 12*n^2 + 16*n + (n^4 + 6*n^3 + 8*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-0e6d847331141ca6","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-1d39f03bcbd1a8b9","e-1f9b231e6abf6872"],"text":"A145596 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A145596 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}