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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":"24ed1c1b38983a2d8614e068967e943896995b992ba19ce28c8cc16191125153","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-67ad51294793de36","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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,         4;\n  2:        3,        20,         18;\n  3:       15,       136,        225,          96;\n  4:      105,      1192,       2799,        2272,         600;\n  5:      945,     12828,      38601,       45328,       23150,        4320;\n  6:    10395,    163788,     597789,      910352,      679600,      247536,      35280;\n  7:   135135,   2420304,   10350918,    19273728,    18791775,    10068336,    2817108,     322560;\n  8:  2027025,  40628880,  198829242,   436318976,   522798225,   367506288,  151889612,   34274304,   3265920;\n  9: 34459425, 763663860, 4202831934, 10598690464, 15023305425, 13023071892, 7084075432, 2368141056, 445926384, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(8*n^3-20*n^2+12*n)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((8*n^3-20*n^2+12*n)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+1+(4*n^2-4*n)*x)*P(n-1,x) - (4*n^2-4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((8*n^3 - 20*n^2 + 12*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x] - (4*n^2 - 4*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((8*n^3 - 20*n^2 + 12*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1] - (4*n^2 - 4*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-849de7332df28657","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-34da37650456524d","e-ca724ce5b2e9cc37"],"text":"A088996 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A088996 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}