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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":"fe75293eda9fa90cb2076aabf0c53e38af7db3518d90a4c416f9bc84b41578b1","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-6202e291a2627bce","role":"sequence","name":"Triangle, row sums = number of edges in n-dimensional hypercubes.","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: 1,  6;\n  2: 1, 12,  15;\n  3: 1, 18,  45,   28;\n  4: 1, 24,  90,  112,   45;\n  5: 1, 30, 150,  280,  225,   66;\n  6: 1, 36, 225,  560,  675,  396,   91;\n  7: 1, 42, 315,  980, 1575, 1386,  637,  120;\n  8: 1, 48, 420, 1568, 3150, 3696, 2548,  960,  153;\n  9: 1, 54, 540, 2352, 5670, 8316, 7644, 4320, 1377, 190;\n","recurrence_sentence":"The row polynomials satisfy (1+(5*n-3)*x+(2*n^2-n)*x^2)*P(n,x) = (1+(5*n+3)*x+(2*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[((1 + (5*n + 3)*x + (2*n^2 + 8*n + 3)*x^2 + (2*n^2 + 3*n + 1)*x^3)*P[n - 1, x])/(1 + (5*n - 3)*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] = ((1 + (5*n + 3)*x + (2*n^2 + 8*n + 3)*x^2 + (2*n^2 + 3*n + 1)*x^3)*v[i - 1])/(1 + (5*n - 3)*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":["2\u03b8+1","\u03b8+1"],"edges":["e-c4ea45619bc7f60a","e-3ee1b8794cb4ceaf"],"text":"A007318 \u2192 2\u03b8+1 \u2192 \u03b8+1"},"headline":"A007318 \u2192 2\u03b8+1 \u2192 \u03b8+1","chain":null}