A367941 -id:A367941 - cp's OEIS frontend

A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

1, 6, 27216, 1344924798336, 3605580335899213007486976, 1648055031941075082958467426002632704000000, 312704667066499295437237787452750428210311485710262201221120000000

Offset: 0

Vaclav Kotesovec, Dec 05 2023

nonn

In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).

c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.

Cf. A324403 (d=1), A367941 (d=2), A367942 (d=3), A367943 (d=4).

Cf. A367542, A367543, A367958.

Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...

A367942 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*j^2).

Original entry on oeis.org

1, 4, 5824, 45861064704, 9751658280030585225216, 176005320076923781520069562958715289600, 656508955366282248103393001602851493819854909361664242483200

Offset: 0

Vaclav Kotesovec, Dec 05 2023

nonn

Cf. A324403, A367941, A367943, A367944.

Cf. A367542, A367543.

Table[Product[i^2+3*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

a(n) ~ c * n^(2*n^2 - 1/2) * 4^(n*(n+1)) * 3^(-n/2) * exp(5*Pi*n*(n+1)/(6*sqrt(3)) - 3*n^2), where c = 0.4612030005343304845802441101292774353695846313857765074861837886133930626...

A367943 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*j^2).

Original entry on oeis.org

1, 5, 13600, 294372000000, 252880261890048000000000, 27099784799070466617992871936000000000000, 882065676199020188908312950703217787436793856000000000000000000

Offset: 0

Vaclav Kotesovec, Dec 05 2023

nonn

Cf. A324403, A367941, A367942, A367944.

Cf. A367542, A367543.

Table[Product[i^2+4*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

a(n) ~ c * n^(2*n^2 - 1/2) * 5^(n*(n+1)) * 2^(-n) * exp(n*(n+1)*(2*Pi - 3*arctan(2))/2 - 3*n^2) , where c = 0.4523180383519335764034720087114905921141637339852374451758854101884791581...

cp's OEIS Frontend

A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

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A367942 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*j^2).

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Programs

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A367943 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*j^2).

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Author

Keywords

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Programs

Mathematica

Formula