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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":"8928d19e0c6a1919172309209d74ec0e393b969fbbf8b0715eb742d8bb596288","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-1fe6eede339bc6ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= 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:    8,     2;\n  3:   21,    12;\n  4:   55,    50,     3;\n  5:  144,   180,    27;\n  6:  377,   600,   153,    4;\n  7:  987,  1908,   702,   48;\n  8: 2584,  5878,  2853,  344,  5;\n  9: 6765, 17700, 10719, 1920, 75;\n","recurrence_sentence":"The row polynomials satisfy -(30*n-(15*n^2+9*n)*x-(3*n^3+3*n^2)*x^2)*P(n,x) = -(70*n+20-(30*n^2+6*n+36)*x-(7*n^3+29*n^2-6*n-12)*x^2-(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (25-(25*n+20)*x-(5*n^2+10*n+27)*x^2+(n^2+23*n+14)*x^3+(4*n^2+12*n+8)*x^4)*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[(-(70*n + 20 - (30*n^2 + 6*n + 36)*x - (7*n^3 + 29*n^2 - 6*n - 12)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (25 - (25*n + 20)*x - (5*n^2 + 10*n + 27)*x^2 + (n^2 + 23*n + 14)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(-30*n + (15*n^2 + 9*n)*x + (3*n^3 + 3*n^2)*x^2)]];\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] = (-(70*n + 20 - (30*n^2 + 6*n + 36)*x - (7*n^3 + 29*n^2 - 6*n - 12)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (25 - (25*n + 20)*x - (5*n^2 + 10*n + 27)*x^2 + (n^2 + 23*n + 14)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(-30*n + (15*n^2 + 9*n)*x + (3*n^3 + 3*n^2)*x^2)); 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-1fe6eede339bc6ea","steps":["\u03b8+1"],"edges":["e-93708f9994ff9930"],"text":"A181371 \u2192 \u03b8+1"},"headline":"A181371 \u2192 \u03b8+1","chain":null}