A360787
Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).
Original entry on oeis.org
1, 1, 1, 3, 13, 40, 177, 965, 4733, 28103, 184065, 1191888, 8713549, 67005689, 528870257, 4526024267, 40051790333, 368513578472, 3583302492545, 35868588067501, 373781214260749, 4052932682659599, 45218033687522481, 523234757502985824, 6245693941097387773
Offset: 0
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Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^(2*k), {k,0,n/2}], {n,1,30}]] (* Vaclav Kotesovec, Feb 21 2023 *)
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^2)^(k+1)))
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a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n-k, k));
A360835
Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^3)^(k+1).
Original entry on oeis.org
1, 1, 4, 27, 258, 3221, 49572, 905466, 19122502, 458161191, 12275530636, 363646493044, 11801356347294, 416365459777150, 15867258718677348, 649548679156603983, 28426564854590132236, 1324406974148881529057, 65448443631801436742052
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-(k*x)^3)^(k+1)))
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a(n) = sum(k=0, n\3, (n-3*k)^n*binomial(n-2*k, k));
A360795
Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).
Original entry on oeis.org
1, 3, 4, 17, 6, 211, 8, 1929, 7300, 22601, 12, 1724809, 14, 6703047, 223678576, 738787345, 18, 65630598229, 20, 2119646503661, 24448573943662, 3423809253371, 24, 21453113652593665, 12016296386718776, 4240253019018225, 8255251542208471048, 67251293544533119589, 30
Offset: 1
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a[n_] := DivisorSum[n, #^(n-#) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)
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my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)^(k+1)))
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a(n) = sumdiv(n, d, d^(n-d)*binomial(d+n/d-1, d));
A360813
Expansion of Sum_{k>=0} ( x / (1 - (k * x)^3) )^k.
Original entry on oeis.org
1, 1, 1, 1, 2, 17, 82, 258, 818, 5671, 43363, 240520, 1183168, 8547054, 77831681, 596258173, 4031934111, 33313129161, 338733239446, 3187239159511, 27197807726066, 260179611473044, 2918973182685904, 31820249821418229, 324099587971865989
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-(k*x)^3))^k))
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a(n) = sum(k=0, n\3, (n-3*k)^(3*k)*binomial(n-2*k-1, k));
Showing 1-4 of 4 results.