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Partial sums of the squares of the numbers in sequence A080253.
5

%I #11 Nov 27 2017 11:47:34

%S 1,10,299,21908,2901717,602319006,180257075455,73470070612264,

%T 39124516839956393,26373727254869321522,21951183825927218885331,

%U 22108623093930072226980540,26501124576166085360405809789,37282620382364481062065321327558

%N Partial sums of the squares of the numbers in sequence A080253.

%F a(n) = sum(c(k)^2,k=0..n), where c(n) = A080253(n).

%F a(n) ~ (n!)^2 * 2^(2*n-1) / (log(2))^(2*n + 2). - _Vaclav Kotesovec_, Nov 27 2017

%t t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[c[k]^2, {k,0,n}], {n,0,100}]

%o (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);

%o c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);

%o makelist(sum(c(k)^2,k,0,n),n,0,40);

%Y Cf. A080253, A000670, A217483, A217484, A217485, A217486, A217488.

%K nonn

%O 0,2

%A _Emanuele Munarini_, Oct 04 2012