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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":"2c4e4ca848029a69a27bb9290f981c36899fdf45105bfe433a6c650eca696a8b","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-0486a9621eaf19a3","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.","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:  2,   6;\n  2:  3,  16,   15;\n  3:  4,  30,   54,   28;\n  4:  5,  48,  126,  128,   45;\n  5:  6,  70,  240,  360,  250,    66;\n  6:  7,  96,  405,  800,  825,   432,   91;\n  7:  8, 126,  630, 1540, 2100,  1638,  686,  120;\n  8:  9, 160,  924, 2688, 4550,  4704, 2940, 1024,  153;\n  9: 10, 198, 1296, 4368, 8820, 11340, 9408, 4896, 1458, 190;\n","recurrence_sentence":"The row polynomials satisfy (n+(n^2+3*n-2)*x+(2*n^2-n)*x^2)*P(n,x) = (n+1+(n^2+6*n+3)*x+(3*n^2+8*n+3)*x^2+(2*n^2+3*n+1)*x^3)*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[((n + 1 + (n^2 + 6*n + 3)*x + (3*n^2 + 8*n + 3)*x^2 + (2*n^2 + 3*n + 1)*x^3)*P[n - 1, x])/(n + (n^2 + 3*n - 2)*x + (2*n^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] = ((n + 1 + (n^2 + 6*n + 3)*x + (3*n^2 + 8*n + 3)*x^2 + (2*n^2 + 3*n + 1)*x^3)*v[i - 1])/(n + (n^2 + 3*n - 2)*x + (2*n^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-c8c19eba9d68881f","steps":["\u03b8+1+n","\u03b8+1"],"edges":["e-dd616c2789f09167","e-1cbaa3463f93c56a"],"text":"A007318 \u2192 \u03b8+1+n \u2192 \u03b8+1"},"headline":"A007318 \u2192 \u03b8+1+n \u2192 \u03b8+1","chain":null}