A372320 -id:A372320 - cp's OEIS frontend

A372315 Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).

1, 2, 8, 68, 960, 18832, 471136, 14324480, 512733696, 21119803136, 984029612544, 51169331031040, 2937675286583296, 184560174104465408, 12594824112085327872, 927757127285523243008, 73369903633161123397632, 6200198958236463387836416

Offset: 0

Seiichi Manyama, Apr 27 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A360193, A372316, A372320, A372321.

Cf. A362694.

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

a(n) = sum(k=0, n, (2*k+1)^(k-1)*binomial(n, k));

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 2^(n-1) * n^(n-1) * exp((exp(-1) + 1)/2). - Vaclav Kotesovec, May 04 2024

A372316 Expansion of e.g.f. exp( x - LambertW(-3*x)/3 ).

Original entry on oeis.org

1, 2, 10, 125, 2644, 77597, 2904382, 132169403, 7083715240, 437031850841, 30506442905194, 2377038378159359, 204521399708464252, 19259006462435865413, 1970114326513629358654, 217556451608123850352523, 25794252755430105917806288, 3268152272130255473300883377

Offset: 0

Seiichi Manyama, Apr 27 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A360193, A372315, A372320, A372321.

Cf. A362734.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x)/3)))

a(n) = sum(k=0, n, (3*k+1)^(k-1)*binomial(n, k));

a(n) = Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 3^(n-1) * n^(n-1) * exp((exp(-1) + 1)/3). - Vaclav Kotesovec, May 04 2024

A372321 Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

Original entry on oeis.org

-1, 0, 6, 81, 1620, 45765, 1671678, 74794671, 3958829640, 241898775273, 16756621904970, 1297547591499819, 111065107263415308, 10412999996499836541, 1061234184094567585326, 116812280111404106348415, 13810631408232372091755792, 1745470697932523785587735249

Offset: 0

Seiichi Manyama, Apr 27 2024

sign

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A360193, A372315, A372316, A372320.

my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-3*x)/3)))

a(n) = sum(k=0, n, (3*k-1)^(k-1)*binomial(n, k));

a(n) = Sum_{k=0..n} (3*k-1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k-1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 3^(n-1) * n^(n-1) * exp((exp(-1) - 1)/3). - Vaclav Kotesovec, May 06 2024

cp's OEIS Frontend

A372315 Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A372316 Expansion of e.g.f. exp( x - LambertW(-3*x)/3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A372321 Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula