A255322 -id:A255322 - cp's OEIS frontend

A255403 Product_{k=1..n} (k^k)!.

1, 24, 261332866810040451858432000000

Offset: 1

Vaclav Kotesovec, Feb 22 2015

nonn

The next term (a(4)) has 537 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 1..4

Cf. A000178, A255322, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269.

Table[Product[(k^k)!, {k, 1, n}], {n, 1, 4}]

A272096 a(n) = Product_{k=0..n} (k*n)!.

Original entry on oeis.org

1, 1, 48, 1567641600, 9698137182219213471744000000, 21488900044302744250061179567064173417691432878080000000000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

The next term has 126 digits.

Cf. A000178, A098694, A268504, A268505, A268506, A271946.

Cf. A255322, A272093.

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

a(n) ~ A^n * n^(1/4 + 13*n/12 + n^2 + n^3) * (2*Pi)^(1/4 + n/2) / exp(n*(2 + 2*n + 3*n^2)/4), where A = A074962 is the Glaisher-Kinkelin constant.

A371499 Decimal expansion of Sum_{k>=0} 1/(k^2)!.

Original entry on oeis.org

2, 0, 4, 1, 6, 6, 9, 4, 2, 2, 3, 9, 8, 6, 3, 6, 8, 6, 0, 0, 2, 9, 0, 5, 5, 8, 6, 0, 7, 2, 1, 0, 6, 6, 2, 9, 1, 1, 7, 5, 3, 7, 9, 7, 4, 8, 0, 0, 7, 6, 6, 3, 5, 1, 3, 3, 1, 5, 4, 8, 9, 1, 4, 5, 6, 7, 4, 8, 4, 6, 8, 0, 2, 0, 0, 4, 7, 6, 1, 8, 9, 8, 3, 9, 9, 4

Offset: 0

Clark Kimberling, Mar 30 2024

			2.04166942239863686002905586072...

Cf. A255322, A371498.

s[x_] := s[x] = Sum[x^n/((n^2)!), {n, 0, Infinity}]
First[RealDigits[N[s[1], 100]]]

suminf(k=0,1/(k^2)!) \\ Hugo Pfoertner, Mar 30 2024

(This constant) - (constant in A371498) = 2.00000551146384...

A371645 a(n) = Product_{k=0..n} (2*k^2)!.

Original entry on oeis.org

1, 2, 80640, 516287415629905920000, 135851139773030831011345707357554182190215987200000000000

Offset: 0

Vaclav Kotesovec, Mar 31 2024

nonn

The next term has 121 digits.

Cf. A255322.

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

a(n) ~ c * 2^(2*n^3/3 + n^2 + 4*n/3) * Pi^(n/2) * n^(4*n^3/3 + 2*n^2 + 5*n/3 + 1/2) / exp(10*n^3/9 + n^2 + n), where c = 3.03114159524498232897099596239043047625469924617878376047699145874714076...

A324442 a(n) = Product_{i=1..n, j=1..n} (i^2 + j).

Original entry on oeis.org

1, 2, 180, 6652800, 402265543680000, 109211487076824381849600000, 295382703175843424854047228769075200000000, 15385012566245626089929288743828190926813939944652800000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A101686, A255322, A324403.

a:= n-> mul(mul(i^2+j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^2 + j, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
Table[Product[Pochhammer[1 + i^2, n], {i, 1, n}], {n, 1, 10}]

From Vaclav Kotesovec, Dec 27 2023: (Start)
a(n) ~ c * n^(2*n^2 + n/2 - 1/4) / exp(2*n^2 - 2*Pi*n^(3/2)/3 - Pi*sqrt(n)/2), where c = 0.31906...
For n>1, a(n) = a(n-1) * Gamma(n - i*sqrt(n)) * Gamma(n + i*sqrt(n)) * Gamma(n^2 + n + 1) * sinh(Pi*sqrt(n)) / (Pi * n^(5/2) * Gamma(n^2)), where i is the imaginary unit. (End)

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A370483 a(n) = Product_{k=0..n} binomial(n^2 + k^2, k^2).

Original entry on oeis.org

1, 2, 350, 347633000, 101143578356902991250, 422044560230008480282938965899488406272, 1208807563912714402070105775158111317516306396248661153276031151000

Offset: 0

Vaclav Kotesovec, Mar 31 2024

nonn

Cf. A255322, A371643.

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

a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2).
a(n) = A371643(n) / ((n^2)!^(n+1) * A255322(n)).
a(n) ~ 2^(4*n^3/3 + n^2 + n/6 + 1/4) * exp((Pi-4)*n^3/3 + Pi*n/4) / (A255504 * n^(n + 1/2) * Pi^(n/2)).

cp's OEIS Frontend

A255403 Product_{k=1..n} (k^k)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A272096 a(n) = Product_{k=0..n} (k*n)!.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A371499 Decimal expansion of Sum_{k>=0} 1/(k^2)!.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A371645 a(n) = Product_{k=0..n} (2*k^2)!.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A324442 a(n) = Product_{i=1..n, j=1..n} (i^2 + j).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A370483 a(n) = Product_{k=0..n} binomial(n^2 + k^2, k^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula