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a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A008288.
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%I #16 May 25 2021 08:05:33

%S 1,10,75,500,3149,19214,115031,680424,3992921,23305234,135514019,

%T 785892316,4549048229,26295995926,151857925039,876366840784,

%U 5055045581745,29148894792730,168045778127355,968679251764676,5583525654107645,32183666525389086,185514611981021959

%N a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A008288.

%H G. C. Greubel, <a href="/A026935/b026935.txt">Table of n, a(n) for n = 2..500</a>

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

%t A026935[n_]:= Sum[A008288[n, k]*A008288[n, k+2], {k, 0, n-2}];

%t Table[A026935[n], {n, 2, 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 A026935(n): return sum(A008288(n, k)*A008288(n, k+2) for k in (0..n-2))

%o [A026935(n) for n in (2..40)] # _G. C. Greubel_, May 25 2021

%Y Cf. A008288.

%K nonn

%O 2,2

%A _Clark Kimberling_

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