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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":"7f30ee012712ba1dbee7b830c3b00595fecc906b244aa9a202e696d0218f30b7","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-08fe62d9c34c34e9","role":"sequence","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:      4,      12;\n  2:     16,      96,       81;\n  3:     64,     576,     1080,       432;\n  4:    256,    3072,     9072,      8640,      2025;\n  5:   1024,   15360,    62208,     96768,     56700,      8748;\n  6:   4096,   73728,   380160,    829440,    816480,    326592,     35721;\n  7:  16384,  344064,  2156544,   6082560,   8553600,   5878656,   1714608,    139968;\n  8:  65536, 1572864, 11612160,  40255488,  74131200,  73903104,  37721376,   8398080,   531441;\n  9: 262144, 7077888, 60162048, 247726080, 566092800, 747242496, 560431872, 221709312, 38972340, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(6*n^3+3*n^2)*x)*P(n,x) = (16*n^2+8*n+(24*n^3+84*n^2+12*n-12)*x+(54*n^3+81*n^2-18*n-9)*x^2)*P(n-1,x) - ((8*n+4)*x+(12*n^2+48*n+18)*x^2+(36*n^2+72*n+18)*x^3)*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[((16*n^2 + 8*n + (24*n^3 + 84*n^2 + 12*n - 12)*x + (54*n^3 + 81*n^2 - 18*n - 9)*x^2)*P[n - 1, x] - ((8*n + 4)*x + (12*n^2 + 48*n + 18)*x^2 + (36*n^2 + 72*n + 18)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (6*n^3 + 3*n^2)*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] = ((16*n^2 + 8*n + (24*n^3 + 84*n^2 + 12*n - 12)*x + (54*n^3 + 81*n^2 - 18*n - 9)*x^2)*v[i - 1] - ((8*n + 4)*x + (12*n^2 + 48*n + 18)*x^2 + (36*n^2 + 72*n + 18)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (6*n^3 + 3*n^2)*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-2245b35ec7cd4857","steps":["basis A013610","\u03b8+1"],"edges":["e-0cc36dd83eba3acf","e-8994821a56af0495"],"text":"A091044 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A091044 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}