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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":"b7cc15471030f603c2b51b8ee43ee3efb6522e5338f490c444441c414a16a56f","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-97b51734d39ede6c","role":"sequence","name":"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: 1,   4;\n  3: 1,  12;\n  4: 1,  24,   27;\n  5: 1,  40,  135;\n  6: 1,  60,  405,   240;\n  7: 1,  84,  945,  1680;\n  8: 1, 112, 1890,  6720,  2625;\n  9: 1, 144, 3402, 20160, 23625;\n","recurrence_sentence":"The row polynomials satisfy (4+(6*n^2-2*n-4)*x+(n^4+2*n^3+n^2)*x^2)*P(n,x) = (4+(6*n^2+14*n-20)*x+(n^4+12*n^3+5*n^2-18*n+4)*x^2+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^3)*P(n-1,x) - (8*x^2+(12*n^2+20*n)*x^3+(2*n^4+12*n^3+26*n^2+24*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[((4 + (6*n^2 + 14*n - 20)*x + (n^4 + 12*n^3 + 5*n^2 - 18*n + 4)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*P[n - 1, x] - (8*x^2 + (12*n^2 + 20*n)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(4 + (6*n^2 - 2*n - 4)*x + (n^4 + 2*n^3 + n^2)*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] = ((4 + (6*n^2 + 14*n - 20)*x + (n^4 + 12*n^3 + 5*n^2 - 18*n + 4)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*v[i - 1] - (8*x^2 + (12*n^2 + 20*n)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(4 + (6*n^2 - 2*n - 4)*x + (n^4 + 2*n^3 + n^2)*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":["(\u03b8+1)\u00b2"],"edges":["e-b214721870127413"],"text":"A100861 \u2192 (\u03b8+1)\u00b2"},"headline":"A100861 \u2192 (\u03b8+1)\u00b2","chain":null}