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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":"028ffb5cc4820d3fd1acbc76032753921390e6c3910046b8d7fc3658af507448","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-4d60c3dd8627c01d","role":"sequence","name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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: 1,  12,   18;\n  3: 1,  24,   99,     96;\n  4: 1,  40,  315,    800,     600;\n  5: 1,  60,  765,   3600,    6850,    4320;\n  6: 1,  84, 1575,  11760,   40600,   63504,    35280;\n  7: 1, 112, 2898,  31360,  169225,  472752,   640332,   322560;\n  8: 1, 144, 4914,  72576,  561225, 2422224,  5788076,  7013376,  3265920;\n  9: 1, 180, 7830, 151200, 1581825, 9695700, 35460320, 75052800, 83152656, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n+(4*n^2-4*n)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((4*n^2-4*n)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (2*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[(-(n + (4*n^2 - 4*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((4*n^2 - 4*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (2*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] = (-(n + (4*n^2 - 4*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((4*n^2 - 4*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (2*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-4d60c3dd8627c01d","steps":["(\u03b8+1)\u00b2"],"edges":["e-85e78a9430ebb6ba"],"text":"A094638 \u2192 (\u03b8+1)\u00b2"},"headline":"A094638 \u2192 (\u03b8+1)\u00b2","chain":null}