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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":"af560ac3ad903afc8a24a9ae23fadfce66c6e403bf95471979b16bd89af2b848","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-e3a9cbbad5cb7ab4","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.","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:      4;\n  2:     15,      2;\n  3:     56,     16;\n  4:    209,     92,     3;\n  5:    780,    464,    36;\n  6:   2911,   2182,   279,    4;\n  7:  10864,   9824,  1776,   64;\n  8:  40545,  42936, 10098,  624,   5;\n  9: 151316, 183648, 53352, 4800, 100;\n","recurrence_sentence":"The row polynomials satisfy -(48*n-(24*n^2+20*n)*x-(2*n^3+2*n^2)*x^2)*P(n,x) = -(168*n+24-(78*n^2+52*n+46)*x-(7*n^3+32*n^2-3*n-20)*x^2-(n^3+2*n^2-n-2)*x^3)*P(n-1,x) - (72-(72*n+108)*x-(6*n^2-30*n-4)*x^2+(4*n^2+36*n+28)*x^3+(2*n^2+6*n+4)*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[(-(168*n + 24 - (78*n^2 + 52*n + 46)*x - (7*n^3 + 32*n^2 - 3*n - 20)*x^2 - (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] - (72 - (72*n + 108)*x - (6*n^2 - 30*n - 4)*x^2 + (4*n^2 + 36*n + 28)*x^3 + (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(-48*n + (24*n^2 + 20*n)*x + (2*n^3 + 2*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] = (-(168*n + 24 - (78*n^2 + 52*n + 46)*x - (7*n^3 + 32*n^2 - 3*n - 20)*x^2 - (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] - (72 - (72*n + 108)*x - (6*n^2 - 30*n - 4)*x^2 + (4*n^2 + 36*n + 28)*x^3 + (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(-48*n + (24*n^2 + 20*n)*x + (2*n^3 + 2*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-e3a9cbbad5cb7ab4","steps":["\u03b8+1"],"edges":["e-7fb0594f63763599"],"text":"A261711 \u2192 \u03b8+1"},"headline":"A261711 \u2192 \u03b8+1","chain":null}