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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":"ec7545c9bf09b5bb28b064e59e8b5337cb35eee84ca2cee179d07f8e687d3c7f","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-3951e6b9a140d595","role":"sequence","name":"Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).","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:        3;\n  2:       12,         6;\n  3:       60,        90;\n  4:      360,      1080,       135;\n  5:     2520,     12600,      4725;\n  6:    20160,    151200,    113400,      6300;\n  7:   181440,   1905120,   2381400,    396900;\n  8:  1814400,  25401600,  47628000,  15876000,   496125;\n  9: 19958400, 359251200, 943034400, 523908000, 49116375;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n^2+2*n+(n^2+n-2)*x)*P(n-1,x) + ((2*n+4)*x-(2*n+4)*x^2)*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[((n^2 + 2*n + (n^2 + n - 2)*x)*P[n - 1, x] + ((2*n + 4)*x - (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n)]];\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] = ((n^2 + 2*n + (n^2 + n - 2)*x)*v[i - 1] + ((2*n + 4)*x - (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n)); 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-3951e6b9a140d595","steps":["\u03b8+1"],"edges":["e-571de044d461c027"],"text":"A306364 \u2192 \u03b8+1"},"headline":"A306364 \u2192 \u03b8+1","chain":null}