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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":"144322c4497e88950d503ab2aac78422dd008d2cab9bd33e391664067f4c708d","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-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:      5,       4;\n  2:     22,      40,        9;\n  3:     95,     276,      135,       16;\n  4:    409,    1640,     1269,      320,      25;\n  5:   1760,    9008,     9630,     3808,     625,      36;\n  6:   7573,   47160,    64611,    35200,    9000,    1080,     49;\n  7:  32585,  239068,   400275,   279440,   98125,   18252,   1715,    64;\n  8: 140206, 1184200,  2345445,  2003200,  900250,  229320,  33271,  2560,   81;\n  9: 603275, 5764620, 13190850, 13347360, 7336875, 2388204, 473340, 56064, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy (1352*n+130*n*x-(260*n^3-50*n)*x^2-50*n^3*x^3+2*n^5*x^4)*P(n,x) = (3380*n+3380+(1183*n^2+3029*n+1846)*x-(650*n^3+715*n^2-385*n-450)*x^2-(26*n^4+346*n^3+541*n^2+199*n-22)*x^3+(5*n^5+5*n^4-45*n^3-90*n^2-55*n-10)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (2197+2535*x-(1183*n^2+1183*n-806)*x^2-(1235*n^2+1235*n+10)*x^3+(13*n^4+26*n^3-328*n^2-341*n-51)*x^4+(10*n^4+20*n^3-15*n^2-25*n-5)*x^5+(n^4+2*n^3+n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3380*n + 3380 + (1183*n^2 + 3029*n + 1846)*x - (650*n^3 + 715*n^2 - 385*n - 450)*x^2 - (26*n^4 + 346*n^3 + 541*n^2 + 199*n - 22)*x^3 + (5*n^5 + 5*n^4 - 45*n^3 - 90*n^2 - 55*n - 10)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (2197 + 2535*x - (1183*n^2 + 1183*n - 806)*x^2 - (1235*n^2 + 1235*n + 10)*x^3 + (13*n^4 + 26*n^3 - 328*n^2 - 341*n - 51)*x^4 + (10*n^4 + 20*n^3 - 15*n^2 - 25*n - 5)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(1352*n + 130*n*x - (260*n^3 - 50*n)*x^2 - 50*n^3*x^3 + 2*n^5*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3380*n + 3380 + (1183*n^2 + 3029*n + 1846)*x - (650*n^3 + 715*n^2 - 385*n - 450)*x^2 - (26*n^4 + 346*n^3 + 541*n^2 + 199*n - 22)*x^3 + (5*n^5 + 5*n^4 - 45*n^3 - 90*n^2 - 55*n - 10)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (2197 + 2535*x - (1183*n^2 + 1183*n - 806)*x^2 - (1235*n^2 + 1235*n + 10)*x^3 + (13*n^4 + 26*n^3 - 328*n^2 - 341*n - 51)*x^4 + (10*n^4 + 20*n^3 - 15*n^2 - 25*n - 5)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(1352*n + 130*n*x - (260*n^3 - 50*n)*x^2 - 50*n^3*x^3 + 2*n^5*x^4)); 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-6466a8550c08a52f","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-235ea1c8ecbb7de6","e-ce61d3d0a8d17f3a"],"text":"A116414 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A116414 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}