A349893
a(n) = Sum_{k=0..n} k^(k*(n-k)).
Original entry on oeis.org
1, 2, 3, 7, 46, 1052, 88603, 27121965, 37004504306, 198705527223758, 5595513387083114571, 686714367475480207331583, 468422339816915120237104999422, 1664212116512828935888786624225704856, 31295654819650678010096952493864470025103251
Offset: 0
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Table[1 + Sum[k^(k*(n - k)), {k, 1, n}], {n, 0, 16}] (* Vaclav Kotesovec, Dec 05 2021 *)
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a(n) = sum(k=0, n, k^(k*(n-k)));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^k*x)))
A234568
Sum_{k=0..n} (n-k)^(2*k).
Original entry on oeis.org
1, 1, 2, 6, 27, 163, 1268, 12344, 145653, 2036149, 33192790, 622384730, 13263528351, 318121600695, 8517247764136, 252725694989612, 8258153081400857, 295515712276222953, 11523986940937975402, 487562536078882116718, 22291094729329088403299, 1097336766599161926448779
Offset: 0
O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 163*x^5 + 1268*x^6 +...
O.g.f.: A(x) = 1 + x/(1-x) + x^2/(1-4*x) + x^3/(1-9*x) + x^4/(1-16*x) +...
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 27*x^4/4! + 163*x^5/5! +...
where the e.g.f. is a series involving iterated integration:
E(x) = 1 + Integral exp(x) dx + Integral^2 exp(4*x) dx^2 + Integral^3 exp(9*x) dx^3 + Integral^4 exp(16*x) dx^4 +...
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Flatten[{1,Table[Sum[(n-k)^(2*k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 23 2014 *)
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a(n)=sum(k=0, n, (n-k)^(2*k))
for(n=0, 20, print1(a(n), ", "))
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/* From o.g.f. Sum_{n>=0} x^n/(1-n^2*x): */
{a(n)=polcoeff(sum(m=0, n, x^m/(1-m^2*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
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/* From e.g.f. involving iterated integration: */
INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G
a(n)=my(A=1+x);A=1+sum(k=1,n,INTEGRATE(k,exp(k^2*x+x*O(x^n))));n!*polcoeff(A,n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Dec 28 2013
A349880
Expansion of Sum_{k>=0} x^k/(1 - k^3 * x).
Original entry on oeis.org
1, 1, 2, 10, 93, 1307, 28002, 842196, 33388393, 1717595949, 111931584098, 8979468552886, 872315432217509, 101425775048588759, 13924209725224120770, 2229705716369149960592, 412760812611799202662609, 87644186710319273062637625, 21180850892383599137766296770
Offset: 0
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a(n, s=0, t=3) = sum(k=0, n, k^(t*(n-k)+s));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^3*x)))
Showing 1-3 of 3 results.