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.
Original entry on oeis.org
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
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; ...
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)
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
-
[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 *)
-
a(n)=binomial(n*(n+1)*(n+2)/3!+1, 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 *)
-
a(n)=binomial((n+1)*(n+2)*(n+3)/3!+1, 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
A126453
Row sums of triangle A126450: a(n) = Sum_{k=0..n} C( C(n+2,3) - C(k+2,3) + 1, n-k).
Original entry on oeis.org
1, 3, 15, 218, 7273, 434654, 40417797, 5369210845, 962496995941, 223528473482380, 65221305164439085, 23343099723197369886, 10052235133879615066675, 5126300310101866339983229
Offset: 0
-
a(n)=sum(k=0,n,binomial(binomial(n+2,3)-binomial(k+2,3)+1, n-k))
Showing 1-6 of 6 results.
Comments