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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":"d567453e29a9ba7d1ad07ed7fa4bb794ac408c3971e474a1f7d008d7d6ee646c","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-bf2f08bd0370af8d","role":"sequence","name":"basis_map_A013609 of Pascal-(1,4,1) 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:     8,     32,      12;\n  3:    24,    144,     168,      32;\n  4:    80,    640,    1200,     640,       80;\n  5:   256,   2560,    6720,    6400,     2080,     192;\n  6:   832,   9984,   33600,   46080,    26880,    6144,     448;\n  7:  2688,  37632,  154752,  279040,   241280,   97536,   17024,   1024;\n  8:  8704, 139264,  676608, 1513472,  1748480, 1056768,  320768,  45056,   2304;\n  9: 28160, 506880, 2841600, 7598080, 11036160, 9016320, 4085760, 983040, 115200, 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) = -(10*n-10+(15*n^2+27*n-48)*x+(5*n^3+37*n^2+2*n-74)*x^2+(9*n^3+14*n^2-31*n-40)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (5+(10*n+35)*x+(5*n^2+55*n+91)*x^2+(20*n^2+102*n+105)*x^3+(21*n^2+67*n+48)*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[(-(10*n - 10 + (15*n^2 + 27*n - 48)*x + (5*n^3 + 37*n^2 + 2*n - 74)*x^2 + (9*n^3 + 14*n^2 - 31*n - 40)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (5 + (10*n + 35)*x + (5*n^2 + 55*n + 91)*x^2 + (20*n^2 + 102*n + 105)*x^3 + (21*n^2 + 67*n + 48)*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] = (-(10*n - 10 + (15*n^2 + 27*n - 48)*x + (5*n^3 + 37*n^2 + 2*n - 74)*x^2 + (9*n^3 + 14*n^2 - 31*n - 40)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (5 + (10*n + 35)*x + (5*n^2 + 55*n + 91)*x^2 + (20*n^2 + 102*n + 105)*x^3 + (21*n^2 + 67*n + 48)*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-ed9434bfd889c409","steps":["basis A013609","\u03b8+1"],"edges":["e-175e767cc5e2f6ad","e-f12f24771b8ba4a2"],"text":"A081579 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A081579 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}