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a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A008288.
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%I #16 May 25 2021 08:04:46

%S 1,14,131,1028,7333,49362,319943,2020552,12526153,76603094,463676491,

%T 2784550604,16619355501,98706690714,583943891087,3443573355152,

%U 20254044891793,118871039217822,696401860980499,4073713439171732,23799679604123189,138895001298962786,809854583626750039

%N a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A008288.

%H G. C. Greubel, <a href="/A026936/b026936.txt">Table of n, a(n) for n = 3..1000</a>

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

%t A026936[n_]:= Sum[A008288[n, k]*A008288[n, k+3], {k, 0, n-3}];

%t Table[A026936[n], {n, 3, 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 A026936(n): return sum(A008288(n, k)*A008288(n, k+3) for k in (0..n-3))

%o [A026936(n) for n in (3..40)] # _G. C. Greubel_, May 25 2021

%Y Cf. A008288.

%K nonn

%O 3,2

%A _Clark Kimberling_

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