A255439 -id:A255439 - cp's OEIS frontend

A255504 Decimal expansion of a constant related to A255322.

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

Offset: 1

Vaclav Kotesovec, Feb 24 2015

			3.048330306522348566911920417337613015885313475689049184525483697684834...

Cf. A255322, A255511, A255438, A255439.

Cf. A074962, A243262, A243263, A243264, A243265.

Equals limit n->infinity (Product_{k=0..n} (k^2)!) / (n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n)).

Equals sqrt(2*Pi) * exp(Zeta(3)/(2*Pi^2)) * Product_{n>=1} ((n^2)!/stirling(n^2)), where stirling(n^2) = sqrt(2*Pi) * n^(2*n^2+1) / exp(n^2) is the Stirling approximation of (n^2)!. - Vaclav Kotesovec, Apr 20 2016

A255360 Product_{k=0..n} (k^5)!.

Original entry on oeis.org

1, 1, 263130836933693530167218012160000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

The next term a(3) has 512 digits.

In general (for m>1), product_{k=0..n} (k^m)! ~ c(m) * (2*Pi)^(n/2) * n^(m*(1/4 + n/2 + B(m+1)/(m+1) + (sum_{j=1..n} j^m) )) * exp(-m*n/2 - m*n^(m+1)/(m+1)^2 - (sum_{j=1..n} j^m) + m * (sum_{j=1..m-1} 1/(j+1) * B(j+1) * binomial(m, j) * n^(m-j) * (sum_{i=0..j-1} 1/(m-i)) )), where c(m) is a constant and B(n) is the Bernoulli number A027641(n)/A027642(n).

Vaclav Kotesovec, Table of n, a(n) for n = 0..3

Cf. A000178, A255322, A255358, A255359.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255439.

Table[Product[(k^5)!, {k, 0, n}], {n, 0, 4}]
Table[Product[j^(n - Ceiling[j^(1/5)] + 1), {j, 1, n^5}], {n, 0, 4}] (* Vaclav Kotesovec, Apr 25 2024 *)

a(n) ~ c * n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36), where c = A255439 = 11.354954749729782312106... .

a(n) = Product_{j=1..n^5} j^(n - ceiling(j^(1/5)) + 1). - Vaclav Kotesovec, Apr 25 2024

A255438 Decimal expansion of a constant related to A255359.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 24 2015

			6.644987918706354049483118316737842660075362652005201561326290428710371...

Cf. A255359, A255504, A255511, A255439.

Cf. A074962, A243262, A243263, A243264, A243265.

Equals limit n->infinity (Product_{k=0..n} (k^4)!) / (n^(1 + 28*n/15 + 4*n^3/3 + 2*n^4 + 4*n^5/5) * (2*Pi)^(n/2) / exp(19*n/9 + n^4/2 + 9*n^5/25)).

Equals 2*Pi*exp(-3*Zeta(5)/Pi^4) * Product_{n>=1} ((n^4)!/stirling(n^4)), where stirling(n^4) = sqrt(2*Pi) * n^(4*n^4 + 2) / exp(n^4) is the Stirling approximation of (n^4)! and Zeta(5) = A013663. - Vaclav Kotesovec, Apr 20 2016

A255511 Decimal expansion of a constant related to A255358.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 24 2015

			4.113740552015338123052453340090368136395763815194771589658140463089224...

Cf. A255358, A255504, A255438, A255439.

Cf. A074962, A243262, A243263, A243264, A243265.

Equals limit n->infinity (Product_{k=0..n} (k^3)!) / (n^(29/40 + 3*n/2 + 3*n^2/4 + 3*n^3/2 + 3*n^4/4) * (2*Pi)^(n/2) / exp(n*(n+2)*(12 - 6*n + 7*n^2)/16)).

Equals (2*Pi)^(3/4) * exp(-11/240 - 3*Zeta'(-3)) * Product_{n>=1} ((n^3)!/stirling(n^3)), where stirling(n^3) = sqrt(2*Pi) * n^(3*n^3 + 3/2) / exp(n^3) is the Stirling approximation of (n^3)! and Zeta'(-3) = A259068. - Vaclav Kotesovec, Apr 20 2016

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A255504 Decimal expansion of a constant related to A255322.

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A255360 Product_{k=0..n} (k^5)!.

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A255438 Decimal expansion of a constant related to A255359.

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A255511 Decimal expansion of a constant related to A255358.

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