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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":"760e6c74c6cbf67a8d1f0b583b8c86a756c374e2e2662c6fc8d79b5c599989e3","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-e8e64448d4c6498a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 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: 0, 4;\n  2: 0, 2,  9;\n  3: 0, 0, 12, 16;\n  4: 0, 0,  3, 40,  25;\n  5: 0, 0,  0, 24, 100,  36;\n  6: 0, 0,  0,  4, 105, 210,  49;\n  7: 0, 0,  0,  0,  40, 336, 392,   64;\n  8: 0, 0,  0,  0,   5, 216, 882,  672,   81;\n  9: 0, 0,  0,  0,   0,  60, 840, 2016, 1080, 100;\n","recurrence_sentence":"The row polynomials satisfy -(n^3+3*n^2+2*n-(4*n^3+2*n^2)*x)*P(n,x) = (n^3+2*n^2-n-2-(7*n^3+16*n^2+5*n-4)*x+(12*n^3+18*n^2-4*n-2)*x^2)*P(n-1,x) - ((2*n^2+6*n+4)*x-(10*n^2+22*n+8)*x^2+(8*n^2+16*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[((n^3 + 2*n^2 - n - 2 - (7*n^3 + 16*n^2 + 5*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*P[n - 1, x] - ((2*n^2 + 6*n + 4)*x - (10*n^2 + 22*n + 8)*x^2 + (8*n^2 + 16*n + 4)*x^3)*D[P[n - 1, x], x])/(-n^3 - 3*n^2 - 2*n + (4*n^3 + 2*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] = ((n^3 + 2*n^2 - n - 2 - (7*n^3 + 16*n^2 + 5*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*v[i - 1] - ((2*n^2 + 6*n + 4)*x - (10*n^2 + 22*n + 8)*x^2 + (8*n^2 + 16*n + 4)*x^3)*deriv(v[i - 1], x))/(-n^3 - 3*n^2 - 2*n + (4*n^3 + 2*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-e8e64448d4c6498a","steps":["\u03b8+1"],"edges":["e-6f5e35c392b70db4"],"text":"A119900 \u2192 \u03b8+1"},"headline":"A119900 \u2192 \u03b8+1","chain":null}