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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":"1227c91f51b7993d8b62908592beaf40c8da01f2756eed53af6f3596cedaf24b","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-4172472abe953cea","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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:          5,          24,           9;\n  3:         45,         236,         135,          16;\n  4:        585,        3248,        2286,         448,         25;\n  5:       9945,       57556,       46170,       11680,       1125,        36;\n  6:     208845,     1248456,     1099071,      327360,      41875,      2376,       49;\n  7:    5221125,    32046780,    30285801,    10137904,    1558375,    119700,     4459,     64;\n  8:  151412625,   950241120,   950393484,   347840640,   61033350,   5715360,   292236,   7680,    81;\n  9: 4996616625, 31963607460, 33501027492, 13168329536, 2557601550, 276494904, 17423028, 635136, 12393, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-56*n+49+(16*n-28)*x-9*x^2)*P(n-2,x) + ((16*n-28)*x^2+4*x^3)*d/dx P(n-2,x) + (8*n-10+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 56*n + 49 + (16*n - 28)*x - 9*x^2)*P[n - 2, x] + ((16*n - 28)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (8*n - 10 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 56*n + 49 + (16*n - 28)*x - 9*x^2)*v[i - 2] + ((16*n - 28)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (8*n - 10 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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-4172472abe953cea","steps":["(\u03b8+1)\u00b2"],"edges":["e-949401e45c61be64"],"text":"A290319 \u2192 (\u03b8+1)\u00b2"},"headline":"A290319 \u2192 (\u03b8+1)\u00b2","chain":null}