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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":"c9023a84c46e9648540ebf9d09a32bd8ade542689d7bc92543ada1968d47b4e3","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-8a9c0145328414df","role":"sequence","name":"basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 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:       2,       12;\n  2:       6,       60,       108;\n  3:      24,      312,       972,        864;\n  4:     120,     1848,      7668,      12096,       6480;\n  5:     720,    12528,     62640,     133920,     129600,      46656;\n  6:    5040,    96336,    551232,    1438560,    1911600,    1259712,     326592;\n  7:   40320,   831168,   5276880,   15918336,   26082000,   23841216,   11430720,    2239488;\n  8:  362880,  7964352,  54972432,  185480064,  354125520,  402361344,  269764992,   98537472,  15116544;\n  9: 3628800, 83998080, 621403488, 2294580096, 4932355680, 6573317184, 5514179328, 2835191808, 816293376, 100776960;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+12*n*x+36*x^2)*P(n-2,x) + (2*n+1+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 12*n*x + 36*x^2)*P[n - 2, x] + (2*n + 1 + 12*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] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 12*n*x + 36*x^2)*v[i - 2] + (2*n + 1 + 12*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-60c7e36cbe887b7e","steps":["basis A013610","\u03b8+1"],"edges":["e-881cf76aa24e5971","e-cc69fa0eb27f035d"],"text":"A137320 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A137320 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}