A126454
Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) for n>=k>=0.
Original entry on oeis.org
1, 3, 1, 15, 5, 1, 220, 55, 8, 1, 7315, 1330, 153, 12, 1, 435897, 58905, 5456, 351, 17, 1, 40475358, 4187106, 316251, 17296, 703, 23, 1, 5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1, 962889794295, 63140314380, 3295144749, 134153712
Offset: 0
Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 2, n-k) = C(21,3) = 1330;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 2, n-k) = C(18,2) = 153;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 2, n-k) = C(33,3) = 5456.
Triangle begins:
1;
3, 1;
15, 5, 1;
220, 55, 8, 1;
7315, 1330, 153, 12, 1;
435897, 58905, 5456, 351, 17, 1;
40475358, 4187106, 316251, 17296, 703, 23, 1;
5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1; ...
-
Table[Binomial[Binomial[n+2,3]-Binomial[k+2,3]+2,n-k],{n,0,10},{k,0,n}]// Flatten (* Harvey P. Dale, Dec 17 2020 *)
-
T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+2, n-k)
A126447
Column 1 of triangle A126445; a(n) = C( C(n+3,3) - 1, n).
Original entry on oeis.org
1, 3, 36, 969, 46376, 3478761, 377447148, 56017460733, 10912535409348, 2703343379981793, 830496702831140346, 310006778438284515093, 138247735223480364826280, 72613463426660610635960445
Offset: 0
-
Table[Binomial[Binomial[n+3,3]-1,n],{n,0,20}] (* Harvey P. Dale, Apr 22 2022 *)
-
a(n)=binomial((n+1)*(n+2)*(n+3)/3!-1, n)
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
-
Table[Binomial[Binomial[n+3,3],n],{n,0,20}] (* Harvey P. Dale, Oct 09 2014 *)
-
a(n)=binomial((n+1)*(n+2)*(n+3)/3!, n)
A126455
Column 0 of triangle A126454; a(n) = C( C(n+2,3) + 2, n).
Original entry on oeis.org
1, 3, 15, 220, 7315, 435897, 40475358, 5373200880, 962889794295, 223581013518060, 65230517839369311, 23345156728397937888, 10052806195411162278095, 5126493560257678027274800
Offset: 0
-
Table[Binomial[Binomial[n+2,3]+2,n],{n,0,20}] (* Harvey P. Dale, Dec 08 2018 *)
-
a(n)=binomial(n*(n+1)*(n+2)/3!+2, n)
A126459
Column 1 of triangle A126457; a(n) = C( C(n+3,3) + 2, n).
Original entry on oeis.org
1, 6, 66, 1540, 66045, 4582116, 470155077, 66983637864, 12655529067060, 3062465626261470, 923729223066105456, 339813167168828020668, 149762733221010818774320, 77904783726874238769542600
Offset: 0
Showing 1-5 of 5 results.
Comments