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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":"737dfbc67b5b1842b6a8ae994066e237153b8343e77b4130a498cddd52496b67","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-d8f9ad9ce0e039d0","role":"sequence","name":"Pascal-(1,3,1) 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: 1,  2;\n  2: 1, 10,    3;\n  3: 1, 18,   27,    4;\n  4: 1, 26,   99,   52,     5;\n  5: 1, 34,  219,  292,    85,     6;\n  6: 1, 42,  387,  980,   645,   126,     7;\n  7: 1, 50,  603, 2372,  2965,  1206,   175,    8;\n  8: 1, 58,  867, 4724,  9605,  7086,  2023,  232,   9;\n  9: 1, 66, 1179, 8292, 24405, 29286, 14511, 3144, 297, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n+(4*n^2-2*n)*x-(4*n^2+7*n)*x^2)*P(n,x) = -(n+(4*n^2+14*n-14)*x+(28*n^2+17*n-52)*x^2-(8*n+14)*x^3)*P(n-1,x) + (x+(4*n+21)*x^2+(56*n+99)*x^3+(4*n+7)*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[(-(n + (4*n^2 + 14*n - 14)*x + (28*n^2 + 17*n - 52)*x^2 - (8*n + 14)*x^3)*P[n - 1, x] + (x + (4*n + 21)*x^2 + (56*n + 99)*x^3 + (4*n + 7)*x^4)*D[P[n - 1, x], x])/(-n - (4*n^2 - 2*n)*x + (4*n^2 + 7*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] = (-(n + (4*n^2 + 14*n - 14)*x + (28*n^2 + 17*n - 52)*x^2 - (8*n + 14)*x^3)*v[i - 1] + (x + (4*n + 21)*x^2 + (56*n + 99)*x^3 + (4*n + 7)*x^4)*deriv(v[i - 1], x))/(-n - (4*n^2 - 2*n)*x + (4*n^2 + 7*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-d8f9ad9ce0e039d0","steps":["\u03b8+1"],"edges":["e-1706bc42be491225"],"text":"A081578 \u2192 \u03b8+1"},"headline":"A081578 \u2192 \u03b8+1","chain":null}