A360788
Expansion of Sum_{k>=0} x^k / (1 - (k*x)^3)^(k+1).
Original entry on oeis.org
1, 1, 1, 1, 3, 25, 109, 324, 1135, 8803, 64189, 337854, 1707319, 13421410, 121248893, 894378619, 6082868725, 53046554917, 543432115477, 4989423130739, 42565774604131, 421544374075072, 4781440892689533, 51342685464272591, 522295380717090265
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^3)^(k+1)))
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a(n) = sum(k=0, n\3, (n-3*k)^(3*k)*binomial(n-2*k, k));
A360834
Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).
Original entry on oeis.org
1, 1, 4, 29, 304, 4100, 67520, 1314167, 29520128, 751658635, 21393444864, 673046604600, 23192501108736, 868730852002205, 35145114836811776, 1527192185786650417, 70941146068492943360, 3508043437942077557884, 183989995827118805352448
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)^2)^(k+1)))
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a(n) = sum(k=0, n\2, (n-2*k)^n*binomial(n-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));
A360812
Expansion of Sum_{k>=0} ( x / (1 - (k * x)^2) )^k.
Original entry on oeis.org
1, 1, 1, 2, 9, 29, 113, 613, 3033, 17010, 110929, 713249, 5061097, 38762873, 302389553, 2544613578, 22404995001, 203762678941, 1960880744337, 19509713674397, 201306862742217, 2166901479447194, 24018963506471921, 275731857268608673, 3271769647891351705
Offset: 0
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my(N=30, 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\2, (n-2*k)^(2*k)*binomial(n-k-1, k));
Showing 1-4 of 4 results.