A386514 -id:A386514 - cp's OEIS frontend

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800

Offset: 0

Vaclav Kotesovec, Aug 24 2025

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293012, A000262, A082579, A091695, A361283.

Cf. A052845, A052887, A386514.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025

nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

A387264 Expansion of e.g.f. exp(x^3/(1-x)^4).

Original entry on oeis.org

1, 0, 0, 6, 96, 1200, 14760, 196560, 2983680, 52315200, 1041465600, 22912243200, 545443113600, 13887294220800, 376188856243200, 10816657377926400, 329526966472704000, 10612556870243328000, 360307460991724646400, 12857257599818926694400, 480829913352068087808000

Offset: 0

Enrique Navarrete, Aug 24 2025

For n > 0, a(n) is the number of ways to seat n people on benches and select 3 people from each bench.

A001805 is the number of ways if only 1 bench is used.

			a(6)=14760 since there are 14400 ways using one bench and 360 ways with 2 benches of 3 people each.

Cf. A001804, A001805, A386514, A387244.

nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 4*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 25 2025 *)

From Vaclav Kotesovec, Aug 25 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 4*k-1)/k!.
a(n) = 5*(n-1)*a(n-1) - 10*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(10*n-27)*a(n-3) - (n-3)*(n-2)*(n-1)*(5*n-21)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(-27/1280 + 13*2^(-22/5)*n^(1/5)/25 + 13*2^(-19/5)*n^(2/5)/15 - 2^(-6/5)*n^(3/5) + 5*2^(-8/5)*n^(4/5) - n) * n^(n-1/10). (End)

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A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

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