A295238 -id:A295238 - cp's OEIS frontend

A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

1, 2, 14, 150, 2264, 44370, 1073772, 30998954, 1041094448, 39909978594, 1720526113460, 82422717484602, 4345035540566184, 250012958308399442, 15594180423126432428, 1048169467357831893930, 75535629221800163853152, 5810132660615400890909634, 475146028302302130377698404

Offset: 0

Seiichi Manyama, Nov 02 2024

nonn

Cf. A006153, A377575, A377576.

Cf. A002999, A295238, A377553.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A295238.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+2,k)/( (k+1)*(n-k)! ).
a(n) ~ 2^(5/2) * sqrt(1 + LambertW(1/4)) * n^(n-1) / (LambertW(1/4)^n * exp(n)). - Vaclav Kotesovec, Nov 02 2024

A377740 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^2.

Original entry on oeis.org

1, 3, 19, 199, 2957, 57341, 1377175, 39531927, 1321803705, 50491876825, 2170432191491, 103726081148339, 5456983990544773, 313449393386822421, 19521567325327386831, 1310428405901227674511, 94325931842372734994417, 7248016420075574268626225, 592190617414334419733622139

Offset: 0

Seiichi Manyama, Nov 05 2024

nonn

Cf. A072597, A377741.

Cf. A295238, A377574.

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

E.g.f.: 4*exp(x)/(1 + sqrt(1 - 4*x*exp(x)))^2.
a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(2*k+2,k)/(n-k)!.
a(n) = A295238(n+1)/(n+1).

A380044 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2xexp(x)*A(x) ).

Original entry on oeis.org

1, 1, 7, 81, 1393, 32025, 924831, 32208337, 1314511297, 61553580849, 3253663709335, 191661481308561, 12451241630689137, 884434637282286025, 68195094329460133231, 5672843158404577658385, 506413381554227338302721, 48290505275596520116029537, 4899034372132659112326787239

Offset: 0

Seiichi Manyama, Jan 10 2025

nonn

Cf. A295238, A380045.

a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(3*k/2+1/2, k)/((3*k+1)*(n-k)!));

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

A380045 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3xexp(x)*A(x) )^(1/3).

Original entry on oeis.org

1, 1, 8, 109, 2220, 60585, 2079166, 86098929, 4179685560, 232849349425, 14645304783450, 1026614846280441, 79371261554884036, 6709919722961129337, 615776691767279304822, 60968162469515187248545, 6478143744223567852425456, 735290556968263062361451745, 88790542940636437330983140146

Offset: 0

Seiichi Manyama, Jan 10 2025

nonn

Cf. A295238, A380044.

a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(4*k/3+1/3, k)/((4*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} 3^k * k^(n-k) * binomial(4*k/3+1/3,k)/( (4*k+1)*(n-k)! ).

cp's OEIS Frontend

A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A377740 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380044 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2xexp(x)*A(x) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380045 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3xexp(x)*A(x) )^(1/3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A377740 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380044 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*exp(x)*A(x) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380045 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x)*A(x) )^(1/3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380044 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2xexp(x)*A(x) ).

A380045 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3xexp(x)*A(x) )^(1/3).