A051675 -id:A051675 - cp's OEIS frontend

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

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

A317747 Numerator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Original entry on oeis.org

1, -1, 1, 259193, -1036793, -201551328007, 9137074752049, 9142431862033871923, -11105299580705049589, -11003865617473929216508154207, 114467620015003245418244743007, 32505236416490926096399421788847363, -254505521478572052318535393350091231, -1828472168539763642032546635313363411876021

Offset: 0

Seiichi Manyama, Sep 01 2018

1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.

a(n) is the numerator of b(n).

			1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..174
Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2).

Cf. A051675, A243262 (A_2).

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
a(n) is the numerator of c_n.

A317796 Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Original entry on oeis.org

1, 360, 259200, 1959552000, 2821754880000, 5079158784000000, 76796880814080000000, 304115648023756800000000, 125121866615488512000000000, 258236518070374430146560000000000, 929651465053347948527616000000000000, 334674527419205261469941760000000000000, 920050700832433373350094438400000000000000

Offset: 0

Seiichi Manyama, Sep 01 2018

1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.

a(n) is the denominator of b(n).

			1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..206
Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2).

Cf. A051675, A243262 (A_2).

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
a(n) is the denominator of c_n.

A330716 n-th Gosper hyperfactorial of n.

Original entry on oeis.org

1, 1, 16, 1952152956156672

Offset: 0

Greg Huber, Dec 27 2019

Gosper's m-th hyperfactorial of n is the product 1^(1^m)*2^(2^m)*3^(3^m)*...*n^(n^m).

The 0th hyperfactorial is the factorial function.

			n=3: a(3) = 1^(1^3)*2^(2^3)*3^(3^3) = 2^8 * 3^27.
a(4) has 198 decimal digits and a(5) has 2927 digits.

R. W. Gosper, "Fac Fun" (ca. 1979).

R. W. Gosper, Gosper's facfun document

Cf. A000142, A002109, A051675, A255321, A255323, A255344 (0th through 5th Gosper hyperfactorials, respectively).

nmax:=3; Table[Product[i^(i^n),{i,1,n}],{n,0,nmax}] (* Stefano Spezia, Dec 29 2019 *)

A290770 a(n) = Product_{k=1..n} k^(2*k).

Original entry on oeis.org

1, 1, 16, 11664, 764411904, 7464960000000000, 16249593066946560000000000, 11020848942410302096869949440000000000, 3102093199396597590886754340698424229232640000000000, 465607547420733489126893933985879279492195953053596584509440000000000

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Hyperfactorial
Index entries for sequences related to factorial numbers

Cf. A000178, A001044, A001818, A002109, A051675, A055209, A061742, A062206, A074962, A184877, A185141, A260122.

[1] cat [(&*[k^(2*k): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[k^(2 k), {k, 1, n}], {n, 0, 9}]
Table[Hyperfactorial[n]^2, {n, 0, 9}]
Table[n!^(2 n)/BarnesG[n + 1]^2, {n, 0, 9}]

a(n) = prod(k=1, n, k^(2*k)) \\ Felix Fröhlich, Aug 10 2017

a(n) = A002109(n)^2.
a(n) = A185141(n)/A000178(n-1)^2 for n > 0.
a(n) = (n!)^(2*n)/G(n+1)^2, where G() is the Barnes G-function.
a(n) ~ A^2*exp(-n^2/2)*n^(n*(n+1))*n^(1/6), where A is the Glaisher-Kinkelin constant (A074962).

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

Original entry on oeis.org

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}]

A372150 a(n) = Product_{k=1..n} k!^(k^2).

Original entry on oeis.org

1, 1, 16, 161243136, 1953714516870533385423459188736, 18637697331204402735774894643901575833450808531469488619520000000000000000000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2024

nonn

Cf. A051675, A255269.

Table[Product[k!^(k^2), {k, 1, n}], {n, 0, 6}]

a(n) ~ (2*Pi)^(n^3/6 + n^2/4 + n/12) * n^(n^4/4 + 2*n^3/3 + n^2/2 + n/12 - 1/90) / (A^(1/6) * exp(5*n^4/16 + 5*n^3/9 + n^2/8 - n/12 - zeta(3)/(8*Pi^2) - zeta'(-3)/3 - 13/720)), where A is the Glaisher-Kinkelin constant A074962, zeta(3) = A002117, zeta'(-3) = A259068.

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

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A317747 Numerator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

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A317796 Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

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A330716 n-th Gosper hyperfactorial of n.

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A290770 a(n) = Product_{k=1..n} k^(2*k).

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

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A372150 a(n) = Product_{k=1..n} k!^(k^2).

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