A355718 -id:A355718 - cp's OEIS frontend

A355719 Expansion of e.g.f. exp( x/(1 - log(1+x)) ).

1, 1, 3, 10, 45, 231, 1405, 9472, 72177, 596845, 5442631, 53052726, 561826309, 6286949787, 75704999721, 954108249676, 12862823623393, 179921659771257, 2683989118991467, 41178997678745506, 673670267643931581, 11223738258484213519, 200027545794685345749

Offset: 0

Seiichi Manyama, Jul 15 2022

sign

a(43) is negative. - Vaclav Kotesovec, Jul 15 2022

Cf. A355718, A355720.

Cf. A006252, A108125.

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

a006252(n) = sum(k=0, n, k!*stirling(n, k, 1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*a006252(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A108125(k) * binomial(n-1,k-1) * a(n-k).

A355720 Expansion of e.g.f. exp( x/(2 - exp(x)) ).

Original entry on oeis.org

1, 1, 3, 16, 113, 986, 10237, 123096, 1680737, 25668766, 433329461, 8009178596, 160802065393, 3483842906610, 80992799730221, 2010720004254856, 53081510001375041, 1484613248976841958, 43846812123456425221, 1363477059263944382604

Offset: 0

Seiichi Manyama, Jul 15 2022

nonn

Cf. A000248, A355718, A355719.

Cf. A000670, A052882, A075729.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(2-exp(x)))))

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*a000670(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A052882(k) * binomial(n-1,k-1) * a(n-k).

a(n) ~ n^(n - 1/4) * exp(sqrt(2*n) - 1/4 - n) / (sqrt(2) * log(2)^n). - Vaclav Kotesovec, Jul 15 2022

cp's OEIS Frontend

A355719 Expansion of e.g.f. exp( x/(1 - log(1+x)) ).

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A355720 Expansion of e.g.f. exp( x/(2 - exp(x)) ).

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