A134375 -id:A134375 - cp's OEIS frontend

A134370 a(n) = ((2n+1)!)^(n+2).

1, 216, 207360000, 3252016064102400000, 2283380023591730815784976384000000, 161469323688736156802100136913438716723200000000000000, 2260697901194635682690248130915498742378267539496871953338204160000000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134371, A134374, A134375.

Mathematica
```
Table[((2n+1)!)^(n + 2), {n, 0, 10}]
```

a(n) ~ 2^(2*(n+1)*(n+2)) * exp(13/24 - 2*n*(n+2)) * n^((n+2)*(4*n+3)/2) * Pi^(n/2 + 1). - Vaclav Kotesovec, Oct 26 2017

Typo in a(6) corrected by Georg Fischer, Apr 10 2024

A134373 a(n) = ((2n)!)^3.

Original entry on oeis.org

1, 8, 13824, 373248000, 65548320768000, 47784725839872000000, 109903340320478724096000000, 662559760549147780765974528000000, 9159226129831418921308831875072000000000, 262435789155225791087396177124997988352000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Christian Elsholtz, Golomb’s Conjecture on Prime Gaps, arXiv:1604.06903 [math.NT], 2016.
Christian Elsholtz, Golomb’s Conjecture on Prime Gaps, The American Mathematical Monthly 124.4 (2017): 365-368.

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134373, A134374, A134375.

Table[((2n)!)^(3), {n, 0, 10}]
((2*Range[0, 10])!)^3 (* Harvey P. Dale, Jul 25 2016 *)

[factorial(2*n)**3 for n in range(0,9)] # Stefano Spezia, Apr 22 2025

Definition corrected by Harvey P. Dale, Jul 25 2016

A381162 a(n) = (8n)!/((n!)^4(4*n)!).

Original entry on oeis.org

1, 1680, 32432400, 999456057600, 37905932634570000, 1617318175088527591680, 74451445170005824874553600, 3614146643656788883257309696000, 182458061523203642337177421198794000, 9493111901274733909567003010522405280000, 505860213332178847817809654781948251947782400

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 7.

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See pp. 14-15.

Cf. A100733, A134375, A195392.

Cf. A381161, A381163, A381164, A381165.

a[n_]:=(8n)!/((n!)^4*(4n)!); Array[a,11,0]

G.f.: hypergeom([1/8, 3/8, 5/8, 7/8], [1, 1, 1], 2^16*x).

a(n) ~ 2^(16*n - 3/2) / (Pi^2*n^2). - Vaclav Kotesovec, May 29 2025

A381166 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+4, j) with 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 46, 23216, 70437736, 911400637082, 39931366088759328, 5015203546888139970264, 1592320463242701429692077472, 1158339311156769223634640734447744, 1783702957209729441902140461938160455424, 5447268928199100257603373050876725987854119216, 31237114830378466799129128930824084710690680271414364

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

			a(3) = 23216:
  [1, 15,  25]
  [1, 31,  90]
  [1, 63, 301]

Cf. A008277, A134375 (determinant), A381160.

a[n_]:=Permanent[Table[StirlingS2[i+4, j], {i, n}, {j, n}]]; Join[{1}, Array[a, 12]]

a(n) = matpermanent(matrix(n, n, i, j, stirling(i+4,j,2))); \\ Michel Marcus, Feb 16 2025

A381267 a(n) = numerator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).

Original entry on oeis.org

1, 15, 31185, 6381375, 409933148625, 115551955934415, 561860686475913825, 179982394552964750175, 245527483089290688069980625, 84259935283701238220954169375, 473788223464393905637179153328785, 169752647693877043154936308907932575, 15821279983229628402902553309640505635425

Offset: 0

Stefano Spezia, Feb 18 2025

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 24.

Cf. A381268 (denominator).

Cf. A009971, A134375.

a[n_]:=Numerator[SeriesCoefficient[1/Sqrt[1-(x+y+z+u(y+z))],{x,0,n},{y,0,n},{z,0,n},{u,0,n}]]; Array[a,13,0]

a(n) = numerator( [x^n] hypergeom( [1/2, 1/6, 1/2, 5/6], [1, 1, 1], 108*x) ).

a(n) = numerator( 2^(2*n-1) * 27^n * Gamma(n+1/6) * Gamma(n+1/2)^2 * Gamma(n+5/6)/(Pi^2 * (n!)^4) ).

A381268 a(n) = denominator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).

Original entry on oeis.org

1, 4, 256, 1024, 1048576, 4194304, 268435456, 1073741824, 17592186044416, 70368744177664, 4503599627370496, 18014398509481984, 18446744073709551616, 73786976294838206464, 4722366482869645213696, 18889465931478580854784, 4951760157141521099596496896, 19807040628566084398385987584

Offset: 0

Stefano Spezia, Feb 18 2025

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 24.

Cf. A381267 (numerator).

Cf. A009971, A134375, A278142.

a[n_]:=Denominator[SeriesCoefficient[1/Sqrt[1-(x+y+z+u(y+z))],{x,0,n},{y,0,n},{z,0,n},{u,0,n}]]; Array[a,13,0]

a(n) = denominator( [x^n] hypergeom( [1/2, 1/6, 1/2, 5/6], [1, 1, 1], 108*x) ).
a(n) = denominator( 2^(2*n-1) * 27^n * Gamma(n+1/6) * Gamma(n+1/2)^2 * Gamma(n+5/6)/(Pi^2 * (n!)^4) ).
a(2*n) = A278142(n).

cp's OEIS Frontend

A134370 a(n) = ((2n+1)!)^(n+2).

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A134373 a(n) = ((2n)!)^3.

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A381162 a(n) = (8*n)!/((n!)^4*(4*n)!).

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A381166 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+4, j) with 1 <= i,j <= n.

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A381267 a(n) = numerator( [(x*y*z*u)^n] 1/sqrt(1 - (x + y + z + u*(y + z))) ).

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Mathematica

Formula

A381268 a(n) = denominator( [(x*y*z*u)^n] 1/sqrt(1 - (x + y + z + u*(y + z))) ).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A381162 a(n) = (8n)!/((n!)^4(4*n)!).

A381267 a(n) = numerator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).

A381268 a(n) = denominator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).