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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":"24824316070b987b3882185368285d3cc71e9bd4df81f26a656dfcb8c42d675f","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-3d0ce3ba50a1aaca","role":"sequence","name":"basis_map_A013609 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,       8;\n  2:     16,      64,       36;\n  3:     64,     384,      480,      128;\n  4:    256,    2048,     4032,     2560,       400;\n  5:   1024,   10240,    27648,    28672,     11200,     1152;\n  6:   4096,   49152,   168960,   245760,    161280,    43008,     3136;\n  7:  16384,  229376,   958464,  1802240,   1689600,   774144,   150528,     8192;\n  8:  65536, 1048576,  5160960, 11927552,  14643200,  9732096,  3311616,   491520,   20736;\n  9: 262144, 4718592, 26738688, 73400320, 111820800, 98402304, 49201152, 12976128, 1520640, 51200;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+n+(2*n^3+n^2)*x)*P(n,x) = (8*n^2+4*n+(8*n^3+28*n^2+4*n-4)*x+(12*n^3+18*n^2-4*n-2)*x^2)*P(n-1,x) - ((4*n+2)*x+(4*n^2+16*n+6)*x^2+(8*n^2+16*n+4)*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[((8*n^2 + 4*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*P[n - 1, x] - ((4*n + 2)*x + (4*n^2 + 16*n + 6)*x^2 + (8*n^2 + 16*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^2 + n + (2*n^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] = ((8*n^2 + 4*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*v[i - 1] - ((4*n + 2)*x + (4*n^2 + 16*n + 6)*x^2 + (8*n^2 + 16*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^2 + n + (2*n^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 A013609","\u03b8+1"],"edges":["e-11b86d48c22b376e","e-4436b7c7803c413e"],"text":"A091044 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A091044 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}