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a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A008288.
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%I #17 May 25 2021 08:05:39

%S 1,6,35,196,1093,6090,33991,190152,1066313,5993422,33759851,190538380,

%T 1077316493,6101144722,34603634063,196524445840,1117492252561,

%U 6361505951382,36251199646387,206773994830164,1180452564195797,6744529721551450,38563791929450071,220652949570236760

%N a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A008288.

%H G. C. Greubel, <a href="/A026934/b026934.txt">Table of n, a(n) for n = 1..500</a>

%t A008288[n_, k_]:= Binomial[n, k]*Hypergeometric2F1[-k, k-n, -n, -1];

%t A026934[n_]:= Sum[A008288[n, k]*A008288[n, k+1], {k, 0, n-1}];

%t Table[A026934[n], {n, 1, 40}] (* _G. C. Greubel_, May 25 2021 *)

%o (Sage)

%o @CachedFunction

%o def A008288(n,k): return sum(binomial(n-j, j)*binomial(n-2*j, k-j) for j in (0..k))

%o def A026934(n): return sum(A008288(n, k)*A008288(n, k+1) for k in (0..n-1))

%o [A026934(n) for n in (1..40)] # _G. C. Greubel_, May 25 2021

%Y Cf. A008288.

%K nonn

%O 1,2

%A _Clark Kimberling_

%E More terms from _Sean A. Irvine_, Oct 17 2019