A381162 -id:A381162 - cp's OEIS frontend

A381161 a(n) = (10n)!/((n!)^3(2n)!(5*n)!).

1, 15120, 3491888400, 1304290155168000, 601680868708529610000, 312696069714024464473125120, 175460887238127057573116837126400, 103865765423748548466734695459219968000, 63958974275578307119821712720619705931210000, 40596987692554701292235753375257230410967703200000

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 2.

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. 14.

Cf. A000442, A010050, A100734, A195394.

Cf. A381162, A381163, A381164, A381165.

a[n_]:=(10n)!/((n!)^3*(2n)!*(5n)!); Array[a,10,0]

G.f.: hypergeom([1/10, 3/10, 7/10, 9/10], [1, 1, 1], 2^8*5^5*x).

a(n) ~ 9*2^(3+8*n)*5^(1+5*n)/((1 + 24*n)*(1 + 60*n)*Pi^2).

A381163 a(n) = Sum_{k=0..n} binomial(n,k)(4k)!(2k)!/(k!)^6.

Original entry on oeis.org

1, 49, 15217, 7437505, 4444068913, 2978797867489, 2151085262277121, 1636678166183569873, 1294384621280668799665, 1054623536679756097536097, 879831837105310233485202337, 748258333337818719124808979313, 646586399881218539235007860940609, 566284969531710881501724274920081265

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 76.

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. 15-16.

Cf. A007318, A010050, A100733, A307618.

Cf. A381161, A381162, A381164, A381165.

a[n_]:=Sum[Binomial[n,k](4k)!(2k)!/k!^6,{k,0,n}]; Array[a,14,0]

G.f.: hypergeom([1/2, 1/2, 1/4, 3/4], [1, 1, 1], 2^10*x/(1-x))/(1-x).
a(n) = hypergeom([1/4, 1/2, 1/2, 3/4, -n], [1, 1, 1, 1], -2^10).
a(n) == 1 (mod 48).
a(n) ~ 5^(2*n+4) * 41^(n+2) / (2^(41/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

A381164 a(n) = Sum_{k=0..n} binomial(n,k)(5k)!/(k!)^5.

Original entry on oeis.org

1, 121, 113641, 168508561, 306213587881, 624890127114721, 1374618918516663841, 3187068298971939367561, 7682172545187676630759081, 19079663136489248380982551201, 48525227073661262262248690661841, 125818607409307965748858681991235961, 331488456546076036761442657285875590881

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 79.

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. 16.

Cf. A007318, A008978, A100734.

Cf. A381161, A381162, A381163, A381165.

a[n_]:=Sum[Binomial[n, k](5k)!/k!^5, {k, 0, n}]; Array[a, 13, 0]

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-x))/(1-x).
a(n) = hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1, 1, 1, 1], -5^5).
a(n) == 1 (mod 120).
a(n) ~ 2^n * 3^(n+2) * 521^(n+2) / (5^(19/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).

Original entry on oeis.org

1, 122, 114126, 169305620, 307902541870, 628881704226972, 1384648756554128604, 3213280613371692112392, 7752574653184355259506670, 19272593072633780827550508620, 49062146831202726778631520779476, 127331178560917294198014376933764792, 335791906923524740189894975371277920796

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 128.

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. 16.

Cf. A000442, A000984, A007318, A010050, A100734.

Cf. A381161, A381162, A381163, A381164.

a[n_]:=Sum[Binomial[2n,n]Binomial[n, k](5k)!/((k!)^3*(2k)!), {k, 0, n}]; Array[a, 13, 0]

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-4*x))/sqrt(1-4*x).
a(n) = binomial(2*n,n)*hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1/2, 1, 1, 1], -5^5/4).
a(n) ~ 3^(n + 3/2) * 7^(n + 3/2) * 149^(n +3/2) / (4 * 5^7 * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

cp's OEIS Frontend

A381161 a(n) = (10*n)!/((n!)^3*(2*n)!*(5*n)!).

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A381163 a(n) = Sum_{k=0..n} binomial(n,k)*(4*k)!*(2*k)!/(k!)^6.

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A381164 a(n) = Sum_{k=0..n} binomial(n,k)*(5*k)!/(k!)^5.

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Author

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Programs

Mathematica

Formula

A381165 a(n) = Sum_{k=0..n} binomial(2*n,n)*binomial(n, k)*(5*k)!/((k!)^3*(2*k)!).

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Author

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Comments

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Programs

Mathematica

Formula

A381161 a(n) = (10n)!/((n!)^3(2n)!(5*n)!).

A381163 a(n) = Sum_{k=0..n} binomial(n,k)(4k)!(2k)!/(k!)^6.

A381164 a(n) = Sum_{k=0..n} binomial(n,k)(5k)!/(k!)^5.

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).