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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":"6eca9a7fa06dbfe1b8bde14063467f100ce9cf871bd78ddc139e5954d79e2fb4","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-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:       2,         4;\n  2:       6,        28,        18;\n  3:      24,       184,       261,         96;\n  4:     120,      1304,      2961,       2336,        600;\n  5:     720,     10224,     32436,      40336,      21850,       4320;\n  6:    5040,     88848,    365076,     628336,     531100,     219024,      35280;\n  7:   40320,    851904,   4319964,    9569856,   11121225,    7105680,    2369052,     322560;\n  8:  362880,   8957376,  54213948,  147568192,  219714225,  192067632,   98694428,   27657216,   3265920;\n  9: 3628800, 102637440, 723526344, 2343105088, 4272319950, 4768172676, 3339772772, 1436735232, 347692176, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((4*n^3-4*n^2)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(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) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((4*n^3 - 4*n^2)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((4*n^3 - 4*n^2)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 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-7fd16a32c438d522","steps":["(\u03b8+1)\u00b2"],"edges":["e-e38d1a5e5ec84172"],"text":"A059364 \u2192 (\u03b8+1)\u00b2"},"headline":"A059364 \u2192 (\u03b8+1)\u00b2","chain":null}