A362288 -id:A362288 - cp's OEIS frontend

A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584

Offset: 0

Paul D. Hanna, Nov 17 2009

Row sums of A219206.

Reinhard Zumkeller, Table of n, a(n) for n = 0..75
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A000169, A014062, A167009, A167010, A184731, A219206, A220359, A362288.

a167008 = sum . a219206_row  -- Reinhard Zumkeller, Feb 27 2015

[(&+[Binomial(n,j)^j: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Flatten[{1,Table[Sum[Binomial[n, k]^k, {k,0,n}], {n,20}]}]
(* Program for numerical value of the limit a(n)^(1/n^2) *) (1-r)^(-r/2)/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2},WorkingPrecision->100] (* Vaclav Kotesovec, Dec 12 2012 *)
Total/@Table[Binomial[n,k]^k,{n,0,20},{k,0,n}] (* Harvey P. Dale, Oct 19 2021 *)

PARI
```
a(n)=sum(k=0,n,binomial(n,k)^k)
```

[sum(binomial(n,j)^j for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Limit_{n->oo} a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Dec 12 2012

A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

Original entry on oeis.org

1, 1, 4, 1134, 333132800, 1319947441510156250, 876533819183888230348458418944000, 1185269534290897564185384010731432113450477770983533184

Offset: 0

Vaclav Kotesovec, Mar 30 2024

nonn

Cf. A001142, A255322, A272095, A296591, A362288, A371624, A371646.

Table[Product[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 8}]

a(n) = (n^2)!^(n+1) / (A255322(n) * A371624(n)).

a(n) ~ c * exp(2*n*(2*n^2/3 + 1)) / (A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(7*n/6 - 1/4)), where c = 0.6367427... and A is the Glaisher-Kinkelin constant A074962.

A371624 a(n) = Product_{k=0..n} (n^2 - k^2)!.

Original entry on oeis.org

1, 1, 144, 1755758592000, 66052111513207347990207922176000000000

Offset: 0

Vaclav Kotesovec, Mar 30 2024

nonn

The next term has 88 digits.

Cf. A001142, A255322, A296591, A362288, A371603.

Table[Product[((2*n-k)*k)!, {k, 0, n}], {n, 0, 6}]
Table[Product[(n^2 - k^2)!, {k, 0, n}], {n, 0, 6}]

a(n) = (n^2)!^(n+1) / (A255322(n) * A371603(n)).

a(n) ~ c * A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(4*n^3/3 + n^2 + 5*n/6 + 1/4) / exp(16*n^3/9 + n^2/2 + n), where c = 1.291409... = sqrt(2*Pi) / (A255504 * c from A371603) and A is the Glaisher-Kinkelin constant A074962.

A372117 a(n) = Product_{k=0..n} binomial(n+k, k)^k.

Original entry on oeis.org

1, 2, 108, 3200000, 1158107343750000, 119025168578031262646195453952, 82864944710388642300699757862681018776776867840000, 9481019710293786574190900386319772308050021208649248212215823364196925440000000

Offset: 0

Vaclav Kotesovec, Apr 19 2024

nonn

Cf. A067055, A255269, A306789, A362288, A372116.

Table[Product[Binomial[n+k,k]^k, {k,0,n}], {n,0,10}]

a(n) = Product_{k=0..n} binomial(n + k, n)^k.
a(n) = A372116(n) / (A255269(n) * A067055(n)).
a(n) ~ 2^(2*n^3/3 + 3*n^2/4 + n/6 + 1/24) * exp(n^3/12 + n^2/4 - n/24 + zeta(3)/(8*Pi^2) - 1/24) / (sqrt(A) * Pi^(n^2/4 + n/4) * n^(n^2/4 + n/4 + 1/24)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

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A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

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A371624 a(n) = Product_{k=0..n} (n^2 - k^2)!.

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A372117 a(n) = Product_{k=0..n} binomial(n+k, k)^k.

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