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%I #7 Apr 11 2017 11:46:02
%S 1,-1,13,-32,1666,-13426,515194,-1432000,1447711256,-4097653768,
%T 256749458824,-2204786032640,11533922227138736,-33268276510233104,
%U 577462439822785168,-1674851096410984448,6621155504764033947008,-34711497070334170000000
%N Numerators of exponential expansion of (3/(2*log(1+x)))*(1 - 1/(1+x)^(2/3)).
%C This is one half of the numerator of the z-sequence of the Sheffer triangle S2[3,2] given in A225466. See the W. Lang link of A006232 for a- and z- sequences for Sheffer triangles and for references.
%C The denominators are given in A284863.
%C The nontrivial recurrence for the column m=0 entries A225466(n, 0) = 2^n from the z-sequence z(n) = 2*a(n)/A284863(n) is: T(n,0) = n*Sum_{k=0..n-1} z(k)*A225466(n-1,k), n >= 1, T(0, 0) = 1.
%F a(n) = numerator(r(n)), with the rationals (in lowest terms) r(n) = [x^n/n!] (3/(2*log(1+x)))*(1 - 1/(1+x)^(2/3)).
%e The rationals r(n) begin: 1, -1/3, 13/27, -32/27, 1666/405, -13426/729, 515194/5103, -1432000/2187, 1447711256/295245, -4097653768/98415, 256749458824/649539, ...
%e The z-sequence is {2*r(n)}, n >= 0.
%e The nontrivial recurrence for A225466(4, 0) = 16 from this z-sequence is: 16 = 8*(1*8 + (-1/3)*117 + (13/27)*135 + (-32/27)*27).
%Y Cf. A006232, A225466, A284863.
%K sign,easy
%O 0,3
%A _Wolfdieter Lang_, Apr 09 2017
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