A355463
Expansion of Sum_{k>=0} (x/(1 - k^k * x))^k.
Original entry on oeis.org
1, 1, 2, 10, 131, 5656, 869097, 490286392, 1264458639313, 12443651667592768, 681538604797281047489, 153070077563816488157872384, 205935348854901274982393017521537, 1352544986573612111579941739713633174912
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(k*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^k*x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k^(k*(n-k))*binomial(n-1, k-1)));
A193198
G.f.: A(x) = Sum_{n>=0} x^n/(1 - 3^n*x)^n.
Original entry on oeis.org
1, 1, 4, 28, 352, 7696, 296704, 19845568, 2325071872, 472050401536, 167325747134464, 102717666720160768, 109887628080679616512, 203517277347030338768896, 656102983404750860283019264, 3660938644168893995628877692928
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 352*x^4 + 7696*x^5 +...
where:
A(x) = 1 + x/(1-3*x) + x^2/(1-9*x)^2 + x^3/(1-27*x)^3 + x^4/(1-81*x)^4 +...
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{a(n)=local(A=1);A=1+sum(m=1,n,x^m/(1-3^m*x +x*O(x^n))^m);polcoeff(A,n)}
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{a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*3^(k*(n-k))))}
A355440
Expansion of e.g.f. Sum_{k>=0} exp(4^k * x) * x^k/k!.
Original entry on oeis.org
1, 2, 10, 98, 2050, 84482, 7221250, 1218502658, 421846581250, 288641130823682, 403002184457781250, 1112950376623239069698, 6251793960501383945781250, 69503063309910921346390425602, 1568447691296998939150390025781250
Offset: 0
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nn = 6; B[n_] := n! 4^Binomial[n, 2]; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
Table[B[n], {n, 0, nn}] CoefficientList[Series[e[x]^2, {x, 0, nn}], x]
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(4^k*x)*x^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-4^k*x)^(k+1)))
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a(n) = sum(k=0, n, 4^(k*(n-k))*binomial(n, k));
Showing 1-3 of 3 results.
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