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A264409 a(n) = Sum_{k=0..n} binomial(n, k) * binomial((n-k)*k, k).

%I A264409 #10 Jan 10 2019 11:18:28
%S A264409 1,1,3,10,53,376,3187,31312,348833,4318804,58583231,862021084,
%T A264409 13650998473,231123405124,4160680867085,79272259679386,
%U A264409 1592221255517713,33599025754872240,742661269363444447,17149370461633306924,412742027009797487561,10331628852664232678356,268469799828424474556585,7229580560131818394109850,201438863633591604857727001
%N A264409 a(n) = Sum_{k=0..n} binomial(n, k) * binomial((n-k)*k, k).
%H A264409 Seiichi Manyama, <a href="/A264409/b264409.txt">Table of n, a(n) for n = 0..468</a>
%e A264409 a(0) = 1;
%e A264409 a(1) = 1*C(0,0) + 1*C(0,1) = 1;
%e A264409 a(2) = 1*C(0,0) + 2*C(1,1) + 1*C(0,2) = 1 + 2*1 + 0 = 3;
%e A264409 a(3) = 1*C(0,0) + 3*C(2,1) + 3*C(2,2) + 1*C(0,3) = 1 + 3*2 + 3*1 + 0 = 10;
%e A264409 a(4) = 1*C(0,0) + 4*C(3,1) + 6*C(4,2) + 4*C(3,3) + 1*C(0,4) = 1 + 4*3 + 6*6 + 4 + 0 = 53; ...
%t A264409 Table[Sum[Binomial[n, k] * Binomial[(n-k)*k, k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 22 2017 *)
%o A264409 (PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial((n-k)*k, k))}
%o A264409 for(n=0,30,print1(a(n),", "))
%K A264409 nonn
%O A264409 0,3
%A A264409 _Paul D. Hanna_, Nov 18 2015