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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":"1d42fe67a1229c9c16dee84f98f5f2671601c4ba69a6cc05ed6c5f9f988bb3ed","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-5aab339818c7e39d","role":"sequence","name":"basis_map_A013610 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      4,       6;\n  2:     15,      48,       27;\n  3:     56,     276,      324,      108;\n  4:    209,    1392,     2511,     1728,      405;\n  5:    780,    6546,    15984,    16848,     8100,     1458;\n  6:   2911,   29472,    90882,   129600,    95175,    34992,     5103;\n  7:  10864,  128808,   480168,   865080,   858600,   481140,   142884,   17496;\n  8:  40545,  550944,  2409426,  5251392,  6583275,  4980528,  2250423,  559872,   59049;\n  9: 151316, 2318238, 11636784, 29778192, 45167220, 43140762, 26290656, 9937728, 2125764, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(16*n+8*n*x-6*n^3*x^2)*P(n,x) = -(32*n+32+(12*n^2+64*n+52)*x-(12*n^3+12*n^2-24*n-24)*x^2-(9*n^3+18*n^2+9*n)*x^3)*P(n-1,x) - (16+48*x-(12*n^2+12*n-44)*x^2-(24*n^2+24*n-12)*x^3-(9*n^2+9*n)*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[(-(32*n + 32 + (12*n^2 + 64*n + 52)*x - (12*n^3 + 12*n^2 - 24*n - 24)*x^2 - (9*n^3 + 18*n^2 + 9*n)*x^3)*P[n - 1, x] - (16 + 48*x - (12*n^2 + 12*n - 44)*x^2 - (24*n^2 + 24*n - 12)*x^3 - (9*n^2 + 9*n)*x^4)*D[P[n - 1, x], x])/(-16*n - 8*n*x + 6*n^3*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] = (-(32*n + 32 + (12*n^2 + 64*n + 52)*x - (12*n^3 + 12*n^2 - 24*n - 24)*x^2 - (9*n^3 + 18*n^2 + 9*n)*x^3)*v[i - 1] - (16 + 48*x - (12*n^2 + 12*n - 44)*x^2 - (24*n^2 + 24*n - 12)*x^3 - (9*n^2 + 9*n)*x^4)*deriv(v[i - 1], x))/(-16*n - 8*n*x + 6*n^3*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-f287bed89e07609f","steps":["basis A013610","\u03b8+1"],"edges":["e-82aefe014dac12a4","e-ffdd6598ddb81c5f"],"text":"A125662 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A125662 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}