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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":"eb8bd64fd0e6e5612707bcc5a736bccd57329f7b1964a7019d293ab16d8b7811","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-eac80c2bf8e566f5","role":"sequence","name":"Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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,  12;\n  2: 1,  24,   27;\n  3: 1,  36,  135,    48;\n  4: 1,  48,  324,   384,    75;\n  5: 1,  60,  594,  1440,   825,    108;\n  6: 1,  72,  945,  3648,  4275,   1512,    147;\n  7: 1,  84, 1377,  7440, 14475,  10044,   2499,   192;\n  8: 1,  96, 1890, 13248, 37500,  43200,  20286,  3840,  243;\n  9: 1, 108, 2484, 21504, 81450, 138888, 106428, 36864, 5589, 300;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+1+(9*n^3+23*n^2+18*n+5)*x+(9*n^4+42*n^3+78*n^2+75*n+23)*x^2+(3*n^3+14*n^2+13*n-5)*x^3+n^2*x^4)*P(n,x) = ((2*n^2+8*n+8)*x+(18*n^3+100*n^2+182*n+110)*x^2+(18*n^4+156*n^3+516*n^2+786*n+454)*x^3+(6*n^3+46*n^2+100*n+50)*x^4+(2*n^2+4*n+2)*x^5)*P(n-2,x) + (n^2+2*n+1+(9*n^3+33*n^2+34*n+9)*x+(9*n^4+69*n^3+161*n^2+160*n+78)*x^2+(9*n^4+63*n^3+197*n^2+328*n+222)*x^3+(3*n^3+21*n^2+50*n+25)*x^4+(n^2+2*n+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 + 8*n + 8)*x + (18*n^3 + 100*n^2 + 182*n + 110)*x^2 + (18*n^4 + 156*n^3 + 516*n^2 + 786*n + 454)*x^3 + (6*n^3 + 46*n^2 + 100*n + 50)*x^4 + (2*n^2 + 4*n + 2)*x^5)*P[n - 2, x] + (n^2 + 2*n + 1 + (9*n^3 + 33*n^2 + 34*n + 9)*x + (9*n^4 + 69*n^3 + 161*n^2 + 160*n + 78)*x^2 + (9*n^4 + 63*n^3 + 197*n^2 + 328*n + 222)*x^3 + (3*n^3 + 21*n^2 + 50*n + 25)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 1, x])/(n^2 + 2*n + 1 + (9*n^3 + 23*n^2 + 18*n + 5)*x + (9*n^4 + 42*n^3 + 78*n^2 + 75*n + 23)*x^2 + (3*n^3 + 14*n^2 + 13*n - 5)*x^3 + n^2*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (((2*n^2 + 8*n + 8)*x + (18*n^3 + 100*n^2 + 182*n + 110)*x^2 + (18*n^4 + 156*n^3 + 516*n^2 + 786*n + 454)*x^3 + (6*n^3 + 46*n^2 + 100*n + 50)*x^4 + (2*n^2 + 4*n + 2)*x^5)*v[i - 2] + (n^2 + 2*n + 1 + (9*n^3 + 33*n^2 + 34*n + 9)*x + (9*n^4 + 69*n^3 + 161*n^2 + 160*n + 78)*x^2 + (9*n^4 + 63*n^3 + 197*n^2 + 328*n + 222)*x^3 + (3*n^3 + 21*n^2 + 50*n + 25)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 1])/(n^2 + 2*n + 1 + (9*n^3 + 23*n^2 + 18*n + 5)*x + (9*n^4 + 42*n^3 + 78*n^2 + 75*n + 23)*x^2 + (3*n^3 + 14*n^2 + 13*n - 5)*x^3 + n^2*x^4)); 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-eac80c2bf8e566f5","steps":["(\u03b8+1)\u00b2"],"edges":["e-5c8788bb6f0033f7"],"text":"A249250 \u2192 (\u03b8+1)\u00b2"},"headline":"A249250 \u2192 (\u03b8+1)\u00b2","chain":null}