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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":"104d194217248e8e41950e131c93459c15f2c793199a3200bd42e8c37d58386a","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-497753d6192915db","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).","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;\n  2: 1,      4;\n  3: 1,     20;\n  4: 1,     80,      27;\n  5: 1,    336,     315;\n  6: 1,   1636,    2655,    240;\n  7: 1,   9460,   21231,   5040;\n  8: 1,  64256,  177030,  71680,  2625;\n  9: 1, 502656, 1593702, 906752, 86625;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+3-(4*n^2-16*n+12)*x-(9*n^2-36*n+27)*x^2)*P(n-4,x) - ((4*n^2-16*n+12)*x^2-(4*n^2-16*n+12)*x^3)*d/dx P(n-4,x) + (2*n^2-6*n+4-(4*n^2-12*n+8)*x)*P(n-3,x) + (4*n^2-12*n+8)*x^2*d/dx P(n-3,x) - (n^2-1-(4*n-4)*x)*P(n-2,x) - (4*n-4)*x^2*d/dx P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 20*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 1 + 20*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (4*n^2 - 16*n + 12)*x - (9*n^2 - 36*n + 27)*x^2)*P[n - 4, x] - ((4*n^2 - 16*n + 12)*x^2 - (4*n^2 - 16*n + 12)*x^3)*D[P[n - 4, x], x] + (2*n^2 - 6*n + 4 - (4*n^2 - 12*n + 8)*x)*P[n - 3, x] + (4*n^2 - 12*n + 8)*x^2*D[P[n - 3, x], x] - (n^2 - 1 - (4*n - 4)*x)*P[n - 2, x] - (4*n - 4)*x^2*D[P[n - 2, x], x] + (2*n - 1)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 4*x; v[4] = 1 + 20*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 3 - (4*n^2 - 16*n + 12)*x - (9*n^2 - 36*n + 27)*x^2)*v[i - 4] - ((4*n^2 - 16*n + 12)*x^2 - (4*n^2 - 16*n + 12)*x^3)*deriv(v[i - 4], x) + (2*n^2 - 6*n + 4 - (4*n^2 - 12*n + 8)*x)*v[i - 3] + (4*n^2 - 12*n + 8)*x^2*deriv(v[i - 3], x) - (n^2 - 1 - (4*n - 4)*x)*v[i - 2] - (4*n - 4)*x^2*deriv(v[i - 2], x) + (2*n - 1)*v[i - 1]); 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-497753d6192915db","steps":["(\u03b8+1)\u00b2"],"edges":["e-8c5840686417f4d7"],"text":"A136394 \u2192 (\u03b8+1)\u00b2"},"headline":"A136394 \u2192 (\u03b8+1)\u00b2","chain":null}