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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":"eaf9faddccb8d4bbad9a0f56e3f3ac57ed76a081d5bb81c9b4631a5e4517e643","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-50226352e9cd0ce4","role":"sequence","name":"Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.","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,   16,     18;\n  4: 1,   44,    162,      96;\n  5: 1,  104,    882,    1536,      600;\n  6: 1,  228,   3816,   13984,    15000,     4320;\n  7: 1,  480,  14526,   96640,   206100,   155520,    35280;\n  8: 1,  988,  51138,  567328,  2087500,  3010608,  1728720,   322560;\n  9: 1, 2008, 171198, 2996608, 17529100, 42111648, 44883216, 20643840, 3265920;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -((4*n^2-12*n+8)*x-(9*n^3-45*n^2+72*n-36)*x^2)*P(n-2,x) - ((2*n-2)*x-(4*n^2-12*n+8)*x^2-(4*n^3-20*n^2+32*n-16)*x^3)*d/dx P(n-2,x) - ((n-1)*x^2-(4*n^2-12*n+8)*x^3)*d^2/dx^2 P(n-2,x) + (n-2+(4*n^2-12*n+8)*x)*P(n-1,x) + ((2*n-3)*x-(4*n^2-12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 - 12*n + 8)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*P[n - 2, x] - ((2*n - 2)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*D[P[n - 2, x], x] - ((n - 1)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], {x, 2}] + (n - 2 + (4*n^2 - 12*n + 8)*x)*P[n - 1, x] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n - 2)]];\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; for(n = 3, N, my(i = n + 1); v[i] = (-((4*n^2 - 12*n + 8)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*v[i - 2] - ((2*n - 2)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*deriv(v[i - 2], x) - ((n - 1)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(deriv(v[i - 2], x), x) + (n - 2 + (4*n^2 - 12*n + 8)*x)*v[i - 1] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n - 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-50226352e9cd0ce4","steps":["(\u03b8+1)\u00b2"],"edges":["e-00470390dfbbae49"],"text":"A055356 \u2192 (\u03b8+1)\u00b2"},"headline":"A055356 \u2192 (\u03b8+1)\u00b2","chain":null}