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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":"747fef001c649a906221a7bf977962a9ef4a61eaab3be4ff58726e92f3afcbce","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c1716beb5dbd6d37","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,       2;\n   3:      8,      16,       3;\n   4:     32,      96,      54,       4;\n   5:    128,     512,     480,     128,       5;\n   6:    512,    2560,    3360,    1600,     250,       6;\n   7:   2048,   12288,   20736,   14336,    4200,     432,      7;\n   8:   8192,   57344,  118272,  107520,   47040,    9408,    686,     8;\n   9:  32768,  262144,  638976,  720896,  422400,  129024,  18816,  1024,    9;\n  10: 131072, 1179648, 3317760, 4472832, 3294720, 1368576, 310464, 34560, 1458, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12+(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = (16*n^2-56*n+48+(8*n^3-12*n^2-32*n+48)*x+(6*n^3-19*n^2+12*n+4)*x^2)*P(n-1,x) - ((8*n-12)*x+(4*n^2+4*n-14)*x^2+(4*n^2-4*n-2)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 56*n + 48 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (6*n^3 - 19*n^2 + 12*n + 4)*x^2)*P[n - 1, x] - ((8*n - 12)*x + (4*n^2 + 4*n - 14)*x^2 + (4*n^2 - 4*n - 2)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((16*n^2 - 56*n + 48 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (6*n^3 - 19*n^2 + 12*n + 4)*x^2)*v[i - 1] - ((8*n - 12)*x + (4*n^2 + 4*n - 14)*x^2 + (4*n^2 - 4*n - 2)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A007318","\u03b8+1"],"edges":["e-219ad3693217b36c","e-dc622ee0bf7ccaad"],"text":"A086645 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A086645 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}