A360727
Expansion of Sum_{k>=0} (k * x * (1 + x^2))^k.
Original entry on oeis.org
1, 1, 4, 28, 264, 3206, 47684, 839249, 17058688, 393216567, 10134918592, 288815780665, 9016571143680, 306027510946208, 11219450971161024, 441846991480590475, 18602901833071633792, 833832341625621777368, 39642569136740054367808
Offset: 0
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nmax = 20; CoefficientList[1 + Series[Sum[(k*x*(1 + x^2))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+x^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^(n-2*k)*binomial(n-2*k, k));
A360731
Expansion of Sum_{k>=0} (k * x * (1 + k*x^3))^k.
Original entry on oeis.org
1, 1, 4, 27, 257, 3141, 46899, 827639, 16855357, 389100834, 10040378183, 286386193685, 8947506702834, 303875954083536, 11146559606379269, 439178938765108083, 18497974976610341624, 829420114454360154295, 39445018962975879216867
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x^3))^k))
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a(n) = sum(k=0, n\4, (n-3*k)^(n-2*k)*binomial(n-3*k, k));
A360748
Expansion of Sum_{k>=0} (x * (1 + k*x^2))^k.
Original entry on oeis.org
1, 1, 1, 2, 5, 10, 21, 53, 133, 327, 861, 2361, 6469, 18168, 52757, 155221, 463077, 1412656, 4379917, 13747504, 43834213, 141866555, 464650309, 1541008295, 5176660997, 17586913779, 60400627453, 209746820056, 735953607173, 2607716976945, 9330605338485
Offset: 0
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Join[{1},Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^k, {k,0,n/3}], {n,1,30}]] (* Vaclav Kotesovec, Feb 20 2023 *)
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k*x^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^k*binomial(n-2*k, k));
A360018
Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^2))^k.
Original entry on oeis.org
1, 1, 4, 28, 288, 3854, 63104, 1220729, 27248128, 689446671, 19501121536, 609753349945, 20883798220800, 777529328875208, 31266494467227648, 1350520199148276667, 62360172065142341632, 3065369553470816704832, 159818389764050045894656
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+(k*x)^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^n*binomial(n-2*k, k));
Showing 1-4 of 4 results.
