A193198 -id:A193198 - cp's OEIS frontend

A355463 Expansion of Sum_{k>=0} (x/(1 - k^k * x))^k.

1, 1, 2, 10, 131, 5656, 869097, 490286392, 1264458639313, 12443651667592768, 681538604797281047489, 153070077563816488157872384, 205935348854901274982393017521537, 1352544986573612111579941739713633174912

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..52

Cf. A193198, A193199, A349893, A355464.

Cf. A080108, A355471, A355472.

Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(k*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^(k*(n-k))*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k^(k*(n-k)) * binomial(n-1,k-1) for n > 0.

A193199 G.f.: A(x) = Sum_{n>=0} x^n/(1 - 4^n*x)^n.

Original entry on oeis.org

1, 1, 5, 49, 1025, 42241, 3610625, 609251329, 210923290625, 144320565411841, 201501092228890625, 556475188311619534849, 3125896980250691972890625, 34751531654955460673195212801, 784223845648499469575195012890625

Offset: 0

Paul D. Hanna, Jul 17 2011

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 49*x^3 + 1025*x^4 + 42241*x^5 +...
where:
A(x) = 1 + x/(1-4*x) + x^2/(1-16*x)^2 + x^3/(1-64*x)^3 + x^4/(1-256*x)^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A000684, A193198.

{a(n)=local(A=1);A=1+sum(m=1,n,x^m/(1-4^m*x +x*O(x^n))^m);polcoeff(A,n)}

{a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*4^(k*(n-k))))}

a(n) = Sum_{k=0..n-1} binomial(n-1,k)*4^(k*(n-k)) for n>0 with a(0)=1.

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A355463 Expansion of Sum_{k>=0} (x/(1 - k^k * x))^k.

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A193199 G.f.: A(x) = Sum_{n>=0} x^n/(1 - 4^n*x)^n.

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