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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":"c0f74d8bec30aaa23bdad472bf8b578de1fbd67ea32c06e12f4bdfd4af3e8163","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-125f84bee6f03b65","role":"sequence","name":"one_plus_derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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,       9;\n  3:     2,      32,      54,      16;\n  4:     6,     112,     261,     160,      25;\n  5:    24,     496,    1395,    1200,     375,      36;\n  6:   120,    2672,    8541,    9040,    4000,     756,     49;\n  7:   720,   16992,   59724,   73024,   40250,   10836,   1372,    64;\n  8:  5040,  124704,  472176,  643328,  414225,  140112,  25382,  2304,   81;\n  9: 40320, 1037952, 4175604, 6196160, 4488225, 1787940, 409542, 53376, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256+(4*n^3-48*n^2+192*n-256)*x+(6*n^2-48*n+96)*x^2+(4*n-16)*x^3+x^4)*P(n-4,x) + (4*n^3-42*n^2+148*n-175+(12*n^2-84*n+148)*x+(12*n-42)*x^2+4*x^3)*P(n-3,x) - (6*n^2-36*n+55+(12*n-36)*x+6*x^2)*P(n-2,x) + (4*n-10+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 12*x + 9*x^2, P(3,x) = 2 + 32*x + 54*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 12*x + 9*x^2;\nP[3, x_] := 2 + 32*x + 54*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*P[n - 2, x] + (4*n - 10 + 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] = 1 + 4*x; v[3] = 1 + 12*x + 9*x^2; v[4] = 2 + 32*x + 54*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*v[i - 2] + (4*n - 10 + 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-b9fbfb57d9269e2f","steps":["D+1","(\u03b8+1)\u00b2"],"edges":["e-649ce40fa3feb4c2","e-c6cc9a9367a08ef6"],"text":"A132393 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A132393 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}