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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":"35ea622e77d27b33b957455bec3babe4a1f93207ece3b41c3f2b3bec44b2fd78","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-a0e88202514b92bb","role":"sequence","name":"reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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,  4;\n  2: 1, 10,  12;\n  3: 1, 16,  48,   32;\n  4: 1, 22, 111,  176,    80;\n  5: 1, 28, 201,  536,   560,   192;\n  6: 1, 34, 318, 1220,  2120,  1632,   448;\n  7: 1, 40, 462, 2336,  5840,  7392,  4480,  1024;\n  8: 1, 46, 633, 3992, 13205, 23952, 23632, 11776,  2304;\n  9: 1, 52, 831, 6296, 26105, 62652, 88144, 70912, 29952, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(n+(3*n^2-4*n)*x-(6*n^2+8*n)*x^2)*P(n,x) = -(n+(3*n^2+8*n-8)*x+(12*n^2-2*n-28)*x^2-(24*n+32)*x^3)*P(n-1,x) + (x+(3*n+12)*x^2+(24*n+36)*x^3+(12*n+16)*x^4)*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[(-(n + (3*n^2 + 8*n - 8)*x + (12*n^2 - 2*n - 28)*x^2 - (24*n + 32)*x^3)*P[n - 1, x] + (x + (3*n + 12)*x^2 + (24*n + 36)*x^3 + (12*n + 16)*x^4)*D[P[n - 1, x], x])/(-n - (3*n^2 - 4*n)*x + (6*n^2 + 8*n)*x^2)]];\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] = (-(n + (3*n^2 + 8*n - 8)*x + (12*n^2 - 2*n - 28)*x^2 - (24*n + 32)*x^3)*v[i - 1] + (x + (3*n + 12)*x^2 + (24*n + 36)*x^3 + (12*n + 16)*x^4)*deriv(v[i - 1], x))/(-n - (3*n^2 - 4*n)*x + (6*n^2 + 8*n)*x^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-d5c14b9fec875ecc","steps":["reverse","\u03b8+1"],"edges":["e-635d5bcbc56b7dbf","e-94cf3ec7410388f9"],"text":"A121574 \u2192 reverse \u2192 \u03b8+1"},"headline":"A121574 \u2192 reverse \u2192 \u03b8+1","chain":null}