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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":"e8a70d0026258d9b0ba2e31386741c8453e5885be1a7a12d84f66443c6d6dc82","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-6a6677e3386580cd","role":"sequence","name":"basis_map_A013609 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","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:    1;\n  2:    2,     4;\n  3:    4,    12;\n  4:   10,    48,     36;\n  5:   26,   160,    180;\n  6:   76,   600,   1080,    480;\n  7:  232,  2184,   5040,   3360;\n  8:  764,  8512,  25200,  26880,  8400;\n  9: 2620, 33408, 117936, 161280, 75600;\n","recurrence_sentence":"The row polynomials satisfy (2+(2*n+4)*x+(n^2+n)*x^2)*P(n,x) = (2*n+(3*n^2+9*n-6)*x+(n^3+9*n^2+8*n-16)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (1+(2*n+8)*x+(n^2+11*n+22)*x^2+(4*n^2+20*n+24)*x^3+(4*n^2+12*n+8)*x^4)*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[((2*n + (3*n^2 + 9*n - 6)*x + (n^3 + 9*n^2 + 8*n - 16)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (1 + (2*n + 8)*x + (n^2 + 11*n + 22)*x^2 + (4*n^2 + 20*n + 24)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 4)*x + (n^2 + n)*x^2)]];\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] = ((2*n + (3*n^2 + 9*n - 6)*x + (n^3 + 9*n^2 + 8*n - 16)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (1 + (2*n + 8)*x + (n^2 + 11*n + 22)*x^2 + (4*n^2 + 20*n + 24)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 4)*x + (n^2 + n)*x^2)); 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-97b51734d39ede6c","steps":["basis A013609","\u03b8+1"],"edges":["e-4139d19ff9e22325","e-7bb9eee7e94d069a"],"text":"A100861 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A100861 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}