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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"3df0684102db6bed1bb18651900e01e1b04426c9a07b3b86ef9c77d521fc7b86","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fb40e29cda26e7b9","role":"sequence","name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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:       5,       20,        9;\n  3:      17,       92,       81,       16;\n  4:      74,      480,      585,      224,       25;\n  5:     394,     2896,     4320,     2320,      500,      36;\n  6:    2484,    20048,    34461,    22960,     7000,     972,     49;\n  7:   18108,   157328,   300951,   233744,    89250,   17640,   1715,    64;\n  8:  149904,  1383328,  2879784,  2513280,  1128225,  281232,  39102,  2816,   81;\n  9: 1389456, 13487904, 30093660, 28815680, 14642250, 4319028, 761460, 78720, 4374, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-12*n^3+54*n^2-108*n+81+(4*n^3-36*n^2+108*n-108)*x+(6*n^2-36*n+54)*x^2+(4*n-12)*x^3+x^4)*P(n-4,x) + (4*n^3-30*n^2+76*n-65+(12*n^2-60*n+76)*x+(12*n-30)*x^2+4*x^3)*P(n-3,x) - (6*n^2-24*n+25+(12*n-24)*x+6*x^2)*P(n-2,x) + (4*n-6+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 5 + 20*x + 9*x^2, P(3,x) = 17 + 92*x + 81*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 5 + 20*x + 9*x^2;\nP[3, x_] := 17 + 92*x + 81*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 + (4*n^3 - 36*n^2 + 108*n - 108)*x + (6*n^2 - 36*n + 54)*x^2 + (4*n - 12)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 30*n^2 + 76*n - 65 + (12*n^2 - 60*n + 76)*x + (12*n - 30)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 24*n + 25 + (12*n - 24)*x + 6*x^2)*P[n - 2, x] + (4*n - 6 + 4*x)*P[n - 1, x]];\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; v[3] = 5 + 20*x + 9*x^2; v[4] = 17 + 92*x + 81*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 + (4*n^3 - 36*n^2 + 108*n - 108)*x + (6*n^2 - 36*n + 54)*x^2 + (4*n - 12)*x^3 + x^4)*v[i - 4] + (4*n^3 - 30*n^2 + 76*n - 65 + (12*n^2 - 60*n + 76)*x + (12*n - 30)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 24*n + 25 + (12*n - 24)*x + 6*x^2)*v[i - 2] + (4*n - 6 + 4*x)*v[i - 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-efe23a76953cf69d","steps":["D+1","(\u03b8+1)\u00b2"],"edges":["e-d9ba82797b30d77a","e-e0af12d42b626f71"],"text":"A130534 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A130534 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}