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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":"09bcc17d87f36a7d2058bd26a802cb3b32ae56c62a21fda37d7aae1cf26e47b9","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-20a6b63b1c7531be","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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:         9,          4;\n  2:        81,         72,         12;\n  3:       729,        972,        324,         32;\n  4:      6561,      11664,       5832,       1152,        80;\n  5:     59049,     131220,      87480,      25920,      3600,       192;\n  6:    531441,    1417176,    1180980,     466560,     97200,     10368,      448;\n  7:   4782969,   14880348,   14880348,    7348320,   2041200,    326592,    28224,    1024;\n  8:  43046721,  153055008,  178564176,  105815808,  36741600,   7838208,  1016064,   73728,   2304;\n  9: 387420489, 1549681956, 2066242608, 1428513408, 595213920, 158723712, 27433728, 2985984, 186624, 5120;\n","recurrence_sentence":"The row polynomials satisfy (9+2*n*x)*P(n,x) = (81+(18*n+36)*x+(4*n+4)*x^2)*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[((81 + (18*n + 36)*x + (4*n + 4)*x^2)*P[n - 1, x])/(9 + 2*n*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] = ((81 + (18*n + 36)*x + (4*n + 4)*x^2)*v[i - 1])/(9 + 2*n*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-366b2525683554ec","steps":["basis A013609","\u03b8+1"],"edges":["e-0da8e8e5bf86fd8d","e-5886240293799ea6"],"text":"A038279 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038279 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}