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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":"6028331fff4b84072b3ec513279bdb7266b544f1e1624676ae62801003c47b78","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-e88a150cc5b6bb0f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks 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,   16;\n  4: 1,   44,    27;\n  5: 1,  104,   225;\n  6: 1,  228,  1170,    240;\n  7: 1,  480,  4914,   3360;\n  8: 1,  988, 18333,  28000,  2625;\n  9: 1, 2008, 63639, 181888, 55125;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (9*n^3-54*n^2+99*n-54)*x^2*P(n-4,x) + (4*n^3-24*n^2+44*n-24)*x^3*d/dx P(n-4,x) - (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) + (4*n^2-12*n+8)*x^3*d^2/dx^2 P(n-3,x) + (4*n^2-12*n+8)*x*P(n-2,x) - ((2*n-2)*x+(4*n^2-12*n+8)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (n-2)*P(n-1,x) + (2*n-3)*x*d/dx 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 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 1 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 54*n^2 + 99*n - 54)*x^2*P[n - 4, x] + (4*n^3 - 24*n^2 + 44*n - 24)*x^3*D[P[n - 4, x], x] - (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] + (4*n^2 - 12*n + 8)*x^3*D[P[n - 3, x], {x, 2}] + (4*n^2 - 12*n + 8)*x*P[n - 2, x] - ((2*n - 2)*x + (4*n^2 - 12*n + 8)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (n - 2)*P[n - 1, x] + (2*n - 3)*x*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; v[4] = 1 + 16*x; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 - 54*n^2 + 99*n - 54)*x^2*v[i - 4] + (4*n^3 - 24*n^2 + 44*n - 24)*x^3*deriv(v[i - 4], x) - (4*n^2 - 12*n + 8)*x*v[i - 3] + (4*n^2 - 12*n + 8)*x^2*deriv(v[i - 3], x) + (4*n^2 - 12*n + 8)*x^3*deriv(deriv(v[i - 3], x), x) + (4*n^2 - 12*n + 8)*x*v[i - 2] - ((2*n - 2)*x + (4*n^2 - 12*n + 8)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (n - 2)*v[i - 1] + (2*n - 3)*x*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-e88a150cc5b6bb0f","steps":["(\u03b8+1)\u00b2"],"edges":["e-93d4d6b093b51129"],"text":"A124324 \u2192 (\u03b8+1)\u00b2"},"headline":"A124324 \u2192 (\u03b8+1)\u00b2","chain":null}