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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":"6ced3e55a26cd577d75ca2dd2f0b5b4322ef328eae25f651e3589dbe671072d3","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-e63fe40b997d2bc1","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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,        16;\n  2:       6,       112,         288;\n  3:      24,       736,        4176,         6144;\n  4:     120,      5216,       47376,       149504,        153600;\n  5:     720,     40896,      518976,      2581504,       5593600,       4423680;\n  6:    5040,    355392,     5841216,     40213504,     135961600,     224280576,      144506880;\n  7:   40320,   3407616,    69119424,    612470784,    2847033600,    7276216320,     9703636992,     5284823040;\n  8:  362880,  35829504,   867423168,   9444364288,   56246841600,  196677255168,   404252377088,   453135826944,   214035333120;\n  9: 3628800, 410549760, 11576421504, 149958725632, 1093713907200, 4882608820224, 13679709274112, 23539470041088, 22786354446336, 9512681472000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(16*n^3-16*n^2)*x-(144*n^3-288*n^2+144*n)*x^2)*P(n-2,x) + ((16*n^3-16*n^2)*x^2+(64*n^3-128*n^2+64*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(16*n^2-16*n)*x)*P(n-1,x) - (16*n^2-16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (16*n^3 - 16*n^2)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*P[n - 2, x] + ((16*n^3 - 16*n^2)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 1 + (16*n^2 - 16*n)*x)*P[n - 1, x] - (16*n^2 - 16*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (16*n^3 - 16*n^2)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*v[i - 2] + ((16*n^3 - 16*n^2)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 1 + (16*n^2 - 16*n)*x)*v[i - 1] - (16*n^2 - 16*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-4d60c3dd8627c01d","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-60ae4909e1eb1035","e-a8eaa204e7680b8b"],"text":"A094638 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A094638 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}