A354314 -id:A354314 - cp's OEIS frontend

A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).

1, 0, 2, 6, 40, 280, 2496, 25424, 297984, 3920256, 57349120, 922611712, 16193375232, 307896882176, 6304666798080, 138318662000640, 3236895083167744, 80483201605795840, 2118875812456366080, 58882581280649117696, 1722441885524719042560

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A052848, A354314.

Cf. A216794, A353998, A354311.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/2*(exp(2*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-2)*binomial(i, j)*v[i-j+1])); v;

a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);

a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-2) * binomial(n,k) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.

A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 9, 48, 315, 2496, 22491, 223728, 2437371, 28931040, 371291283, 5111412120, 75014135235, 1168157451384, 19228202401635, 333378840718944, 6069073767712587, 115683487658404272, 2303091818149762899, 47784447190060311240, 1031179733234906055507

Offset: 0

Seiichi Manyama, May 23 2022

sign

Cf. A052506, A354311.

Cf. A351734, A351737, A354310, A354314.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/3*(exp(3*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;

a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

cp's OEIS Frontend

A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).

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A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula