A186414 -id:A186414 - cp's OEIS frontend

A186420 a(n) = binomial(2n,n)^4.

1, 16, 1296, 160000, 24010000, 4032758016, 728933458176, 138735983333376, 27435582641610000, 5588044012339360000, 1165183173971324375296, 247639903129149250277376, 53472066459540320483696896, 11701285507234585729600000000, 2589980371199606611713600000000

Offset: 0

Emanuele Munarini, Feb 21 2011

			G.f.: 4F3({1/2,1/2,1/2,1/2},{1,1,1},256x) where 4F3 is a hypergeometric series.

Seiichi Manyama, Table of n, a(n) for n = 0..417

Cf. binomial(2n,n)^k: A000984 (k=1), A002894 (k=2), A002897 (k=3), this sequence (k=4).

Cf. A000108, A000888, A186414, A186415, A186416, A186418, A186419, A000897, A006480, A008977, A188662.

Table[Binomial[2n,n]^4,{n,0,20}]
Table[Coefficient[Series[HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {1, 1, 1}, 256 x], {x, 0, n}], x, n], {n, 0, 14}] (* Michael De Vlieger, Jul 13 2016 *)

Maxima
```
makelist(binomial(2*n,n)^4,n,0,40);
```

a(n) = A000984(n)^4 = A002894(n)^2.
a(n) = binomial(2*n,n)^4 = ( [x^n](1 + x)^(2*n) )^4 = [x^n](F(x)^(16*n)), where F(x) = 1 + x + 25*x^2 + 1798*x^3 + 183442*x^4 + 22623769*x^5 + 3142959012*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977 and A188662. - Peter Bala, Jul 14 2016
a(n) ~ 256^n/(Pi*n)^2. - Ilya Gutkovskiy, Jul 13 2016

A186415 a(n) = binomial(2n,n)^3/(n+1).

Original entry on oeis.org

1, 4, 72, 2000, 68600, 2667168, 112698432, 5053029696, 236860767000, 11493303192800, 573327757086656, 29253930349198464, 1521079361361956032, 80361335659444000000, 4304087536829486400000, 233271979857187430688000, 12774642558686527109607000, 706008965215713532853436000, 39337406606398593529683000000

Offset: 0

Emanuele Munarini, Feb 21 2011

Seiichi Manyama, Table of n, a(n) for n = 0..557

Cf. A000108, A002894, A000888, A002897, A186414.

A186415 := proc(n) binomial(2*n,n)^3/(n+1) ; end proc: # R. J. Mathar, Feb 23 2011

Mathematica
```
Table[Binomial[2n,n]^3/(n+1),{n,0,40}]
```

makelist(binomial(2*n,n)^3/(n+1),n,0,40);

G.f.: 3F2(1/2,1/2,1/2;1,2;64x), where 3F2(.,.,.;.,.;.) is a generalized hypergeometric series.
a(n) = A000888(n)*A000984(n). - R. J. Mathar, Feb 23 2011
a(n) ~ 64^n/(Pi^(3/2)*n^(5/2)). - Ilya Gutkovskiy, Nov 01 2016

A186416 a(n) = binomial(2n,n)^4/(n+1)^3.

Original entry on oeis.org

1, 2, 48, 2500, 192080, 18670176, 2125170432, 270968717448, 37634544090000, 5588044012339360, 875419364366134016, 143310129125665075392, 24338673855047938317568, 4264316875814353400000000, 767401591466550107174400000, 141345980472409642279275210000, 26569505644587874058090478570000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415.

A186416 := proc(n) binomial(2*n,n)^4/(n+1)^3 ; end proc: # R. J. Mathar, Feb 23 2011

Table[Binomial[2n,n]^4/(n+1)^3,{n,0,40}]

makelist(binomial(2*n,n)^4/(n+1)^3,n,0,40);

G.f.: 4F3(1/2,1/2,1/2,1/2;2,2,2;256*x), where nFm(...;..;.) denotes a generalized hypergeometric series.

a(n) = (A000108(n))^3*A000984(n). - R. J. Mathar, Feb 23 2011

A186418 a(n) = binomial(2*n,n)^4/(n + 1)^2.

Original entry on oeis.org

1, 4, 144, 10000, 960400, 112021056, 14876193024, 2167749739584, 338710896810000, 55880440123393600, 9629613008027474176, 1719721549507980904704, 316402760115623198128384, 59700436261400947600000000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415, A186416.

Table[Binomial[2n,n]^4/(n+1)^2,{n,0,40}]

makelist(binomial(2*n,n)^4/(n+1)^2,n,0,40);

G.f.: 4F3({1/2,1/2,1/2,1/2},{1,2,2},256x), where 4F3 is a hypergeometric series.

A186419 a(n) = binomial(2*n,n)^4/(n + 1).

Original entry on oeis.org

1, 8, 432, 40000, 4802000, 672126336, 104133351168, 17341997916672, 3048398071290000, 558804401233936000, 105925743088302215936, 20636658594095770856448, 4113235881503101575668992, 835806107659613266400000000, 172665358079973774114240000000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415, A186416, A186418.

Mathematica
```
Table[Binomial[2n,n]^4/(n+1),{n,0,40}]
```

makelist(binomial(2*n,n)^4/(n+1),n,0,12);

G.f.: 4F3({1/2,1/2,1/2,1/2},{1,1,2},256x), where 4F3 is a hypergeometric series.

cp's OEIS Frontend

A186420 a(n) = binomial(2n,n)^4.

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A186415 a(n) = binomial(2n,n)^3/(n+1).

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A186416 a(n) = binomial(2n,n)^4/(n+1)^3.

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A186418 a(n) = binomial(2*n,n)^4/(n + 1)^2.

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A186419 a(n) = binomial(2*n,n)^4/(n + 1).

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