A126453 -id:A126453 - cp's OEIS frontend

A126450 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) for n>=k>=0.

1, 2, 1, 10, 4, 1, 165, 45, 7, 1, 5985, 1140, 136, 11, 1, 376992, 52360, 4960, 325, 16, 1, 36288252, 3819816, 292825, 16215, 666, 22, 1, 4935847320, 406481544, 25621596, 1215450, 43680, 1225, 29, 1, 899749479915, 59487568920, 3127595016, 128164707

Offset: 0

Paul D. Hanna, Dec 27 2006

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.

			Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 1, n-k) = C(20,3) = 1140;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 1, n-k) = C(17,2) = 136;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 1, n-k) = C(32,3) = 4960.
Triangle begins:
1;
2, 1;
10, 4, 1;
165, 45, 7, 1;
5985, 1140, 136, 11, 1;
376992, 52360, 4960, 325, 16, 1;
36288252, 3819816, 292825, 16215, 666, 22, 1;
4935847320, 406481544, 25621596, 1215450, 43680, 1225, 29, 1; ...

Columns: A126451, A126452; A126453 (row sums); variants: A126445, A126454, A126457, A107867.

T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+1, n-k)

T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 1, n-k) for n>=k>=0.

A126451 Column 0 of triangle A126450; a(n) = C( C(n+2,3) + 1, n).

Original entry on oeis.org

1, 2, 10, 165, 5985, 376992, 36288252, 4935847320, 899749479915, 211531737340440, 62292206224983306, 22453501436688294427, 9723205992282927449305, 4980663327690172963041600, 2978877731799385928100461400, 2057145404864429538334152506640

Offset: 0

Paul D. Hanna, Dec 27 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A126450; A126452, A126453; A126446, A126455, A126458.

[Binomial(Binomial(n+2,3)+1,n): n in [0..20]]; // Vincenzo Librandi, Mar 10 2014

Table[Binomial[Binomial[n+2,3]+1,n],{n,0,20}] (* Harvey P. Dale, Mar 08 2014 *)

PARI
```
a(n)=binomial(n*(n+1)*(n+2)/3!+1, n)
```

More terms from Harvey P. Dale, Mar 08 2014

A126452 Column 1 of triangle A126450; a(n) = C( C(n+3,3), n).

Original entry on oeis.org

1, 4, 45, 1140, 52360, 3819816, 406481544, 59487568920, 11468588169060, 2818651865383860, 860587163078645431, 319667046321630491484, 141992594868360194121800, 74339194732961502662043600

Offset: 0

Paul D. Hanna, Dec 27 2006

nonn

Cf. A126450; A126451, A126453; A126447, A126456, A126459.

Table[Binomial[Binomial[n+3,3],n],{n,0,20}] (* Harvey P. Dale, Oct 09 2014 *)

PARI
```
a(n)=binomial((n+1)*(n+2)*(n+3)/3!, n)
```

cp's OEIS Frontend

A126450 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) for n>=k>=0.

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Formula

A126451 Column 0 of triangle A126450; a(n) = C( C(n+2,3) + 1, n).

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A126452 Column 1 of triangle A126450; a(n) = C( C(n+3,3), n).

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