A360708
Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.
Original entry on oeis.org
1, 0, 1, 1, 2, 5, 14, 42, 136, 479, 1825, 7433, 32053, 145608, 695081, 3479117, 18209842, 99373513, 563920590, 3320674902, 20255823092, 127799984935, 832807892861, 5597481205009, 38753768384761, 276057156622776, 2021100095469577, 15193591060371577
Offset: 0
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Join[{1},Table[Sum[Binomial[n-k-1,k-1] * k^(n-2*k), {k,0,n/2}], {n,1,40}]] (* Vaclav Kotesovec, Feb 20 2023 *)
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x^2/(1-k*x))^k))
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a(n) = if(n==0, 1, sum(k=1, n\2, k^(n-2*k)*binomial(n-k-1, k-1)));
A360815
Expansion of Sum_{k>=0} x^(3*k) / (1 - k*x)^(k+1).
Original entry on oeis.org
1, 0, 0, 1, 2, 3, 5, 11, 30, 88, 260, 771, 2343, 7474, 25380, 91650, 347988, 1371873, 5570173, 23233703, 99676434, 440931977, 2014619700, 9506385864, 46246356169, 231348803925, 1187212953132, 6239006165820, 33546182775824, 184497923546700
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/(1-k*x)^(k+1)))
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a(n) = sum(k=0, n\3, k^(n-3*k)*binomial(n-2*k, k));
A360819
Expansion of Sum_{k>=0} ( (k*x)^3 / (1 - k*x) )^k.
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 65, 257, 769, 21732, 182268, 1075171, 22120299, 292415838, 2784944366, 52394511682, 914813711338, 12411977351379, 240868108545883, 5024364548461861, 88977315031536205, 1888119425325238979, 44744897995532996819, 971263427084750362992
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, ((k*x)^3/(1-k*x))^k))
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a(n) = sum(k=0, n\3, k^n*binomial(n-2*k-1, n-3*k));
Showing 1-3 of 3 results.