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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":"f26016c2d640fb16e680da2f3a1b4f9c3a7ce3739d41a0e4f73c2b8351a0729e","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-e5924eef99dd90a5","role":"sequence","name":"basis_map_A013610 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,     6;\n  3:    4,    18;\n  4:   10,    72,     81;\n  5:   26,   240,    405;\n  6:   76,   900,   2430,   1620;\n  7:  232,  3276,  11340,  11340;\n  8:  764, 12768,  56700,  90720,  42525;\n  9: 2620, 50112, 265356, 544320, 382725;\n","recurrence_sentence":"The row polynomials satisfy (24+(36*n+72)*x+(27*n^2+27*n)*x^2)*P(n,x) = (24*n+(54*n^2+162*n-108)*x+(27*n^3+243*n^2+216*n-432)*x^2+(81*n^3+162*n^2-81*n-162)*x^3)*P(n-1,x) - (8+(24*n+96)*x+(18*n^2+198*n+396)*x^2+(108*n^2+540*n+648)*x^3+(162*n^2+486*n+324)*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[((24*n + (54*n^2 + 162*n - 108)*x + (27*n^3 + 243*n^2 + 216*n - 432)*x^2 + (81*n^3 + 162*n^2 - 81*n - 162)*x^3)*P[n - 1, x] - (8 + (24*n + 96)*x + (18*n^2 + 198*n + 396)*x^2 + (108*n^2 + 540*n + 648)*x^3 + (162*n^2 + 486*n + 324)*x^4)*D[P[n - 1, x], x])/(24 + (36*n + 72)*x + (27*n^2 + 27*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] = ((24*n + (54*n^2 + 162*n - 108)*x + (27*n^3 + 243*n^2 + 216*n - 432)*x^2 + (81*n^3 + 162*n^2 - 81*n - 162)*x^3)*v[i - 1] - (8 + (24*n + 96)*x + (18*n^2 + 198*n + 396)*x^2 + (108*n^2 + 540*n + 648)*x^3 + (162*n^2 + 486*n + 324)*x^4)*deriv(v[i - 1], x))/(24 + (36*n + 72)*x + (27*n^2 + 27*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 A013610","\u03b8+1"],"edges":["e-d718d4a3c4d020a8","e-ba4f66a696735767"],"text":"A100861 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A100861 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}