A381989 -id:A381989 - cp's OEIS frontend

A382001 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^6.

1, 1, 16, 462, 20672, 1261400, 97728672, 9190016416, 1016963389696, 129485497897728, 18648682990461440, 2997567408967391744, 531985786683988512768, 103321584851593487961088, 21798243872991807130685440, 4964302861788729054456729600, 1213816740632458735310221672448

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

In general, if k>1 and e.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^k, then a(n) ~ sqrt(k) * sqrt(1 + LambertW(2*(k-1)^(k-1)/k^k)) * 2^n * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW(2*(k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Mar 22 2025

Cf. A336950, A381997, A381998, A381999, A382000.

Cf. A002295, A381989.

a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002295(k)/(n-k)!.

a(n) ~ sqrt(3*(1 + LambertW(3125/23328))) * 2^(n + 1/2) * n^(n-1) / (5^(3/2) * exp(n) * LambertW(3125/23328)^n). - Vaclav Kotesovec, Mar 22 2025

A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 11, 160, 3941, 134486, 5851327, 309520436, 19283504585, 1382980764106, 112223497464371, 10165461405056552, 1016801830348902061, 111312715288354681310, 13237965546409421546471, 1699516550894276788156156, 234263144339070269872076177, 34507561203827621878485498386

Offset: 0

Seiichi Manyama, Mar 12 2025

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 5 the sequence becomes [1, 2, 1, 0, 1, 1, 2, 1, 0, 1, ...] with period 5. In particular, a(5*n+3) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Cf. A381984, A381988, A381989.

Cf. A002293, A365340.

seq(simplify(hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27)), n = 0..17); # Peter Bala, Mar 13 2025

Table[HypergeometricPFQ[{-n, 1/2, 1/4, 3/4}, {2/3, 4/3}, -256/27], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)

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

a(n) = n! * Sum_{k=0..n} A002293(k)/(n-k)!.
From Peter Bala, Mar 13 2025: (Start)
a(n) = hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27).
3*(3*n - 1)*(3*n + 1)*a(n) = n*(256*n^2 - 303*n + 95)*a(n-1) - 3*(n - 1)*(256*n^2 - 485*n + 245)*a(n-2) + 3*(256*n - 375)*(n - 1)*(n - 2)*a(n-3) - 256*(n - 1)*(n - 2)*(n - 3)*a(n-4) with a(0) = 1, a(1) = 2, a(2) = 11 and a(3) = 160. (End)
a(n) ~ 2^(8*n+1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 27/256)). - Vaclav Kotesovec, Mar 14 2025

A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 15, 313, 10773, 510981, 30876463, 2267990159, 196204786025, 19539828320905, 2201822913234771, 276969947671828995, 38473403439454795837, 5849221857618942870029, 966078641687956464576119, 172251173569831561500070711, 32975613823747758363130520529, 6746227557293225645352382744593

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A381987, A381989.

Cf. A002293, A002294, A346647, A377526.

a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A377526. F(x) = B(x*A(x)) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002294(k)/(n-k)!.

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A382001 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^6.

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A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

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A382001 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^6.

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A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382001 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^6.

A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.