A372177 -id:A372177 - cp's OEIS frontend

A372232 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * (1 + A(x)^(1/2)) ).

1, 4, 40, 688, 17152, 564864, 23212288, 1145627648, 66082594816, 4365282304000, 325074868781056, 26950224851927040, 2462208223872286720, 245811899064585814016, 26626175172644096180224, 3110339882223194198769664, 389786352057654976473726976

Offset: 0

Seiichi Manyama, Apr 23 2024

nonn

Cf. A138764, A372177.

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

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A138764.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
a(n) = 2^n * A372177(n).

A372246 E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 14, 182, 3528, 91572, 2988124, 117646664, 5429848160, 287596190960, 17197966810224, 1146212005029456, 84257333026857472, 6772618660901287040, 590968891266018673664, 55635634440230961625088, 5621016808791883758841344, 606656453852999167732922112

Offset: 0

Seiichi Manyama, Apr 24 2024

nonn

Cf. A372177, A372236, A372251.

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

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A372251.

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).

cp's OEIS Frontend

A372232 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * (1 + A(x)^(1/2)) ).

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