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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":"14c0c37e98c5dea4984743d5d90abdd8792d2ac5e06d92554861f9b087660924","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c1d69b4cd7e1b71e","role":"sequence","name":"basis_map_A013609 of Pascal-(1,3,1) array.","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,      4;\n  2:     7,     28,      12;\n  3:    20,    120,     144,      32;\n  4:    61,    488,     936,     544,      80;\n  5:   182,   1820,    4848,    4832,    1760,     192;\n  6:   547,   6564,   22356,   31712,   19920,    5184,     448;\n  7:  1640,  22960,   95328,  176320,  160640,   71424,   14336,   1024;\n  8:  4921,  78736,  385680,  880832, 1059680,  687360,  232960,  37888,  2304;\n  9: 14762, 265716, 1500192, 4081728, 6125760, 5288832, 2612736, 709632, 96768, 5120;\n","recurrence_sentence":"The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(8*n-8+(12*n^2+21*n-39)*x+(4*n^3+29*n^2-n-62)*x^2+(7*n^3+10*n^2-29*n-36)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (4+(8*n+28)*x+(4*n^2+44*n+73)*x^2+(16*n^2+82*n+85)*x^3+(17*n^2+55*n+40)*x^4+(2*n^2+6*n+4)*x^5)*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 - 8 + (12*n^2 + 21*n - 39)*x + (4*n^3 + 29*n^2 - n - 62)*x^2 + (7*n^3 + 10*n^2 - 29*n - 36)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 73)*x^2 + (16*n^2 + 82*n + 85)*x^3 + (17*n^2 + 55*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];\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 - 8 + (12*n^2 + 21*n - 39)*x + (4*n^3 + 29*n^2 - n - 62)*x^2 + (7*n^3 + 10*n^2 - 29*n - 36)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 73)*x^2 + (16*n^2 + 82*n + 85)*x^3 + (17*n^2 + 55*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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-d8f9ad9ce0e039d0","steps":["basis A013609","\u03b8+1"],"edges":["e-471a582f60d8a81d","e-e2a66a5b4913db40"],"text":"A081578 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A081578 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}