A186415 -id:A186415 - 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

A186414 a(n) = binomial(2n,n)^3/(n+1)^2.

Original entry on oeis.org

1, 2, 24, 500, 13720, 444528, 16099776, 631628712, 26317863000, 1149330319280, 52120705189696, 2437827529099872, 117006104720150464, 5740095404246000000, 286939169121965760000, 14579498741074214418000

Offset: 0

Emanuele Munarini, Feb 21 2011

Seiichi Manyama, Table of n, a(n) for n = 0..559

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

[Binomial(2*n,n)^3/(n+1)^2: n in [0..50]]; // Vincenzo Librandi, Mar 27 2011

Table[Binomial[2n, n]^3/(n + 1)^2, {n, 0, 20}]

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

G.f.: 3F2({1/2, 1/2, 1/2}, {2, 2}, 64x), where 3F2 is a hypergeometric function.

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.

A367178 Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 5, 1, 16, 72, 80, 14, 1, 25, 200, 500, 350, 42, 1, 36, 450, 2000, 3150, 1512, 132, 1, 49, 882, 6125, 17150, 18522, 6468, 429, 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430, 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862

Offset: 0

Peter Luschny, Nov 07 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,    2;
  [3] 1,  9,   18,     5;
  [4] 1, 16,   72,    80,     14;
  [5] 1, 25,  200,   500,    350,     42;
  [6] 1, 36,  450,  2000,   3150,   1512,    132;
  [7] 1, 49,  882,  6125,  17150,  18522,   6468,    429;
  [8] 1, 64, 1568, 15680,  68600, 131712, 103488,  27456,   1430;
  [9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;

Cf. A086618 (row sums), A186415 (central column), A000108 (main diagonal).

Cf. A098474, A367022, A367023, A387024, A387024, A367177.

T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (k + 1).

T(n, k) = [x^n] hypergeom([1/2, -n, -n], [1, 2], 4*x).

cp's OEIS Frontend

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

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A186414 a(n) = binomial(2n,n)^3/(n+1)^2.

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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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Maxima

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A367178 Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).

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Maple

Formula