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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":"6f21384d09d2725364035119e08876025fa997c11ad02d3a988fe8fe784212b4","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-a8ddb226edd55d5f","role":"sequence","name":"basis_map_A013611 of Pascal-(1,2,1) array read by antidiagonals.","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:    2,      8;\n  2:    6,     48,      48;\n  3:   16,    192,     480,      256;\n  4:   44,    704,    2784,     3584,     1280;\n  5:  120,   2400,   13056,    27648,    23040,     6144;\n  6:  328,   7872,   54528,   161792,   222720,   135168,    28672;\n  7:  896,  25088,  211200,   808960,  1576960,  1572864,   745472,   131072;\n  8: 2448,  78336,  776448,  3649536,  9262080, 13123584, 10149888,  3932160,   589824;\n  9: 6688, 240768, 2746368, 15310848, 48015360, 88989696, 97714176, 61341696, 20054016, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (12*n*x+(24*n^2+48*n)*x^2+(16*n^3+48*n^2+32*n)*x^3)*P(n,x) = -(12*n-12+(36*n^2+60*n-120)*x+(24*n^3+168*n^2-32*n-400)*x^2+(80*n^3+96*n^2-432*n-512)*x^3-(128*n^2+384*n+256)*x^4)*P(n-1,x) + (3+(12*n+42)*x+(12*n^2+132*n+220)*x^2+(96*n^2+496*n+520)*x^3+(208*n^2+688*n+512)*x^4+(64*n^2+192*n+128)*x^5)*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[(-(12*n - 12 + (36*n^2 + 60*n - 120)*x + (24*n^3 + 168*n^2 - 32*n - 400)*x^2 + (80*n^3 + 96*n^2 - 432*n - 512)*x^3 - (128*n^2 + 384*n + 256)*x^4)*P[n - 1, x] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 220)*x^2 + (96*n^2 + 496*n + 520)*x^3 + (208*n^2 + 688*n + 512)*x^4 + (64*n^2 + 192*n + 128)*x^5)*D[P[n - 1, x], x])/(12*n*x + (24*n^2 + 48*n)*x^2 + (16*n^3 + 48*n^2 + 32*n)*x^3)]];\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] = (-(12*n - 12 + (36*n^2 + 60*n - 120)*x + (24*n^3 + 168*n^2 - 32*n - 400)*x^2 + (80*n^3 + 96*n^2 - 432*n - 512)*x^3 - (128*n^2 + 384*n + 256)*x^4)*v[i - 1] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 220)*x^2 + (96*n^2 + 496*n + 520)*x^3 + (208*n^2 + 688*n + 512)*x^4 + (64*n^2 + 192*n + 128)*x^5)*deriv(v[i - 1], x))/(12*n*x + (24*n^2 + 48*n)*x^2 + (16*n^3 + 48*n^2 + 32*n)*x^3)); 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-f25cd955a876cde3","steps":["basis A013611","\u03b8+1"],"edges":["e-84c91f2034e460dd","e-f1d4866057272c99"],"text":"A081577 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A081577 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}