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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":"82c6284784002555a9596983a59f8eb2f382db3727845af3bb2e5b8b2d0f9f80","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2a657d152d421a92","role":"sequence","name":"Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  12;\n   3: 1,  32,   108;\n   4: 1,  60,   495,    880;\n   5: 1,  96,  1404,   5824,    6825;\n   6: 1, 140,  3150,  22400,   59500,    51408;\n   7: 1, 192,  6120,  65280,  290700,   558144,    379848;\n   8: 1, 252, 10773, 159600, 1047375,  3318084,   4946403,   2768832;\n   9: 1, 320, 17640, 344960, 3099250, 14281344,  34711600,  42099200,  19980675;\n  10: 1, 396, 27324, 680064, 7969500, 49729680, 174053880, 341003520, 347663745, 143071500;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^3-2*n^2-2*n-(8*n^5-16*n^4+6*n^3+4*n^2-2*n)*x)*P(n,x) = -(4*n^3-2*n^2-2*n-(8*n^5-16*n^4-30*n^3+16*n^2-2*n)*x-(36*n^5-30*n^4+6*n^3)*x^2)*P(n-1,x) - ((8*n^2-n-1)*x-(16*n^4-8*n^3-15*n^2+8*n-1)*x^2-(18*n^4-15*n^3+3*n^2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 2*n^2 - 2*n - (8*n^5 - 16*n^4 - 30*n^3 + 16*n^2 - 2*n)*x - (36*n^5 - 30*n^4 + 6*n^3)*x^2)*P[n - 1, x] - ((8*n^2 - n - 1)*x - (16*n^4 - 8*n^3 - 15*n^2 + 8*n - 1)*x^2 - (18*n^4 - 15*n^3 + 3*n^2)*x^3)*D[P[n - 1, x], x])/(-4*n^3 + 2*n^2 + 2*n + (8*n^5 - 16*n^4 + 6*n^3 + 4*n^2 - 2*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n^3 - 2*n^2 - 2*n - (8*n^5 - 16*n^4 - 30*n^3 + 16*n^2 - 2*n)*x - (36*n^5 - 30*n^4 + 6*n^3)*x^2)*v[i - 1] - ((8*n^2 - n - 1)*x - (16*n^4 - 8*n^3 - 15*n^2 + 8*n - 1)*x^2 - (18*n^4 - 15*n^3 + 3*n^2)*x^3)*deriv(v[i - 1], x))/(-4*n^3 + 2*n^2 + 2*n + (8*n^5 - 16*n^4 + 6*n^3 + 4*n^2 - 2*n)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2a657d152d421a92","steps":["(\u03b8+1)\u00b2"],"edges":["e-4055a12c596ba57a"],"text":"A102537 \u2192 (\u03b8+1)\u00b2"},"headline":"A102537 \u2192 (\u03b8+1)\u00b2","chain":null}