A126457
Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 3, n-k) for n>=k>=0.
Original entry on oeis.org
1, 4, 1, 21, 6, 1, 286, 66, 9, 1, 8855, 1540, 171, 13, 1, 501942, 66045, 5984, 378, 18, 1, 45057474, 4582116, 341055, 18424, 741, 24, 1, 5843355957, 470155077, 29034396, 1353275, 47905, 1326, 31, 1, 1029873432159, 66983637864, 3470108187, 140364532
Offset: 0
Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 3, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 3, n-k) = C(22,3) = 1540;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 3, n-k) = C(19,2) = 171;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 3, n-k) = C(34,3) = 5984.
Triangle begins:
1;
4, 1;
21, 6, 1;
286, 66, 9, 1;
8855, 1540, 171, 13, 1;
501942, 66045, 5984, 378, 18, 1;
45057474, 4582116, 341055, 18424, 741, 24, 1;
5843355957, 470155077, 29034396, 1353275, 47905, 1326, 31, 1; ...
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 *)
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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 *)
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a(n)=binomial((n+1)*(n+2)*(n+3)/3!, n)
A126456
Column 1 of triangle A126454; a(n) = C( C(n+3,3) + 1, n).
Original entry on oeis.org
1, 5, 55, 1330, 58905, 4187106, 437353560, 63140314380, 12049276177620, 2938311614386005, 891655291709643461, 329600203128234828790, 145830232567505064233200, 76102715775896720790887700
Offset: 0
-
Table[Binomial[Binomial[n+3,3]+1,n],{n,0,20}] (* Harvey P. Dale, Feb 03 2015 *)
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a(n)=binomial((n+1)*(n+2)*(n+3)/3!+1, n)
A126458
Column 0 of triangle A126457; a(n) = C( C(n+2,3) + 3, n).
Original entry on oeis.org
1, 4, 21, 286, 8855, 501942, 45057474, 5843355957, 1029873432159, 236236542585120, 68292983465630781, 24268885951464043344, 10392619362579990298763, 5276256293478688846049120
Offset: 0
Showing 1-5 of 5 results.
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