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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":"6c3d8d4bb5da2d2e20461f7a49a0b3afa5f25df9e00f8bcbc4de1ef5d2fc99e6","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-37ca9f2607503703","role":"sequence","name":"basis_map_A013609 of Pascal-(1,2,1) array read by antidiagonals.","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:    6,     24,     12;\n  3:   16,     96,    120,      32;\n  4:   44,    352,    696,     448,      80;\n  5:  120,   1200,   3264,    3456,    1440,     192;\n  6:  328,   3936,  13632,   20224,   13920,    4224,     448;\n  7:  896,  12544,  52800,  101120,   98560,   49152,   11648,   1024;\n  8: 2448,  39168, 194112,  456192,  578880,  410112,  158592,  30720,  2304;\n  9: 6688, 120384, 686592, 1913856, 3000960, 2780928, 1526784, 479232, 78336, 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) = -(6*n-6+(9*n^2+15*n-30)*x+(3*n^3+21*n^2-4*n-50)*x^2+(5*n^3+6*n^2-27*n-32)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (3+(6*n+21)*x+(3*n^2+33*n+55)*x^2+(12*n^2+62*n+65)*x^3+(13*n^2+43*n+32)*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[(-(6*n - 6 + (9*n^2 + 15*n - 30)*x + (3*n^3 + 21*n^2 - 4*n - 50)*x^2 + (5*n^3 + 6*n^2 - 27*n - 32)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 55)*x^2 + (12*n^2 + 62*n + 65)*x^3 + (13*n^2 + 43*n + 32)*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] = (-(6*n - 6 + (9*n^2 + 15*n - 30)*x + (3*n^3 + 21*n^2 - 4*n - 50)*x^2 + (5*n^3 + 6*n^2 - 27*n - 32)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 55)*x^2 + (12*n^2 + 62*n + 65)*x^3 + (13*n^2 + 43*n + 32)*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-f25cd955a876cde3","steps":["basis A013609","\u03b8+1"],"edges":["e-144ed6eafdc58374","e-3cf2dab5ace4e977"],"text":"A081577 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A081577 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}