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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":"b0093b80494678f22dd4bf035ee9f87874d0853ad1f57d461ad764cabec271c6","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-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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,        2;\n  2:       6,       14,         6;\n  3:      24,       92,        87,        24;\n  4:     120,      652,       987,       584,       120;\n  5:     720,     5112,     10812,     10084,      4370,       720;\n  6:    5040,    44424,    121692,    157084,    106220,     36504,      5040;\n  7:   40320,   425952,   1439988,   2392464,   2224245,   1184280,    338436,     40320;\n  8:  362880,  4478688,  18071316,  36892048,  43942845,  32011272,  14099204,   3457152,   362880;\n  9: 3628800, 51318720, 241175448, 585776272, 854463990, 794695446, 477110396, 179591904, 38632464, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(2*n^3-2*n^2)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*P[n - 1, 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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(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-7fd16a32c438d522","steps":["\u03b8+1"],"edges":["e-9cf16697d865df2a"],"text":"A059364 \u2192 \u03b8+1"},"headline":"A059364 \u2192 \u03b8+1","chain":null}