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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":"7b159a87c30473287f4a0d0322ad3557cf588f379fade70bd9d8da6092ee1214","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-0515531ee2138839","role":"sequence","name":"reversal of Riordan array (1, 3+3x).","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:     3;\n  2:     9,      6;\n  3:    27,     36;\n  4:    81,    162,     27;\n  5:   243,    648,    243;\n  6:   729,   2430,   1458,   108;\n  7:  2187,   8748,   7290,  1296;\n  8:  6561,  30618,  32805,  9720,  405;\n  9: 19683, 104976, 137781, 58320, 6075;\n","recurrence_sentence":"The row polynomials satisfy (3*n+n^2*x)*P(n,x) = (9*n+(3*n^2+12*n-12)*x+(2*n^2+2*n-4)*x^2)*P(n-1,x) - (9*x+(3*n+18)*x^2+(4*n+8)*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[((9*n + (3*n^2 + 12*n - 12)*x + (2*n^2 + 2*n - 4)*x^2)*P[n - 1, x] - (9*x + (3*n + 18)*x^2 + (4*n + 8)*x^3)*D[P[n - 1, x], x])/(3*n + 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] = ((9*n + (3*n^2 + 12*n - 12)*x + (2*n^2 + 2*n - 4)*x^2)*v[i - 1] - (9*x + (3*n + 18)*x^2 + (4*n + 8)*x^3)*deriv(v[i - 1], x))/(3*n + 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-016d7724d51f6416","steps":["reverse","\u03b8+1"],"edges":["e-450f652a77090025","e-fa69f282656e011e"],"text":"A099093 \u2192 reverse \u2192 \u03b8+1"},"headline":"A099093 \u2192 reverse \u2192 \u03b8+1","chain":null}