A372321 -id:A372321

Search: a372321 -id:a372321

Displaying 1-3 of 3 results found.

page 1

A372315

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..17.

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

FORMULA

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

PROG

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

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

CROSSREFS

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

Cf. A362694.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 27 2024

STATUS

approved

A372316

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..17.

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

FORMULA

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

PROG

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

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

CROSSREFS

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

Cf. A362734.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 27 2024

STATUS

approved

A372320

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

+10
3

-1, 0, 4, 36, 464, 8560, 206112, 6104896, 214376192, 8701657344, 400748710400, 20642974511104, 1175888936749056, 73389707156586496, 4980134850525986816, 365062349226075463680, 28747688571714736160768, 2420266280392895064506368

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

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

FORMULA

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 06 2024

PROG

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

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

CROSSREFS

Cf. A088957, A360193, A372315, A372316, A372321.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Apr 27 2024

STATUS

approved

page 1

Search completed in 0.010 seconds