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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":"10a3891060f0b659a5ebc8fbd3e136a7c03b7fd9cadf13b9a8ec534be9087169","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9d2e609ab19d361f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (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: 1,  2;\n  2: 1,  8,   3;\n  3: 1, 12,  18,    4;\n  4: 1, 16,  48,   32,    5;\n  5: 1, 20,  90,  120,   50,    6;\n  6: 1, 24, 144,  304,  240,   72,    7;\n  7: 1, 28, 210,  616,  770,  420,   98,    8;\n  8: 1, 32, 288, 1088, 1920, 1632,  672,  128,   9;\n  9: 1, 36, 378, 1752, 4050, 4860, 3066, 1008, 162, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n^2-n+(2*n^3-3*n^2+n)*x-(2*n^3+n^2+n-4)*x^2-(n^2-n)*x^3)*P(n,x) = -(n^2-n+(2*n^3+5*n^2-7*n)*x+(6*n^3+5*n^2-3*n-8)*x^2+(n^2-n)*x^3)*P(n-1,x) + (n*x+(2*n^2+10*n+3)*x^2+(12*n^2+26*n+19)*x^3+(2*n^2+10*n+9)*x^4+(n+1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^3 + 5*n^2 - 7*n)*x + (6*n^3 + 5*n^2 - 3*n - 8)*x^2 + (n^2 - n)*x^3)*P[n - 1, x] + (n*x + (2*n^2 + 10*n + 3)*x^2 + (12*n^2 + 26*n + 19)*x^3 + (2*n^2 + 10*n + 9)*x^4 + (n + 1)*x^5)*D[P[n - 1, x], x])/(-n^2 + n - (2*n^3 - 3*n^2 + n)*x + (2*n^3 + n^2 + n - 4)*x^2 + (n^2 - n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - n + (2*n^3 + 5*n^2 - 7*n)*x + (6*n^3 + 5*n^2 - 3*n - 8)*x^2 + (n^2 - n)*x^3)*v[i - 1] + (n*x + (2*n^2 + 10*n + 3)*x^2 + (12*n^2 + 26*n + 19)*x^3 + (2*n^2 + 10*n + 9)*x^4 + (n + 1)*x^5)*deriv(v[i - 1], x))/(-n^2 + n - (2*n^3 - 3*n^2 + n)*x + (2*n^3 + n^2 + n - 4)*x^2 + (n^2 - n)*x^3)); 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-9d2e609ab19d361f","steps":["\u03b8+1"],"edges":["e-79f1cfe47acb6394"],"text":"A102413 \u2192 \u03b8+1"},"headline":"A102413 \u2192 \u03b8+1","chain":null}