A126445
Triangle, read by rows, where T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) for n >= k >= 0.
Original entry on oeis.org
1, 1, 1, 6, 3, 1, 120, 36, 6, 1, 4845, 969, 120, 10, 1, 324632, 46376, 4495, 300, 15, 1, 32468436, 3478761, 270725, 15180, 630, 21, 1, 4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1, 840261910995, 56017460733, 2967205528, 122391522, 3921225, 98770, 2016, 36, 1
Offset: 0
Formula: T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) is illustrated by:
T(n=4,k=1) = C(C(6,3) - C(3,3), n-k) = C(19,3) = 969;
T(n=4,k=2) = C(C(6,3) - C(4,3), n-k) = C(16,2) = 120;
T(n=5,k=2) = C(C(7,3) - C(4,3), n-k) = C(31,3) = 4495.
Triangle begins:
1;
1, 1;
6, 3, 1;
120, 36, 6, 1;
4845, 969, 120, 10, 1;
324632, 46376, 4495, 300, 15, 1;
32468436, 3478761, 270725, 15180, 630, 21, 1;
4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1;
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T[n_, k_]:= Binomial[Binomial[n+2,3] - Binomial[k+2,3], n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2022 *)
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T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!, n-k)
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def A126445(n,k): return binomial(binomial(n+2,3) - binomial(k+2,3), n-k)
flatten([[A126445(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 18 2022
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
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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)
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
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[Binomial(Binomial(n+2,3)+1,n): n in [0..20]]; // Vincenzo Librandi, Mar 10 2014
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Table[Binomial[Binomial[n+2,3]+1,n],{n,0,20}] (* Harvey P. Dale, Mar 08 2014 *)
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a(n)=binomial(n*(n+1)*(n+2)/3!+1, 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
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Table[Binomial[Binomial[n+2,3]+2,n],{n,0,20}] (* Harvey P. Dale, Dec 08 2018 *)
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a(n)=binomial(n*(n+1)*(n+2)/3!+2, 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
A126448
Column 2 of triangle A126445; a(n) = C( C(n+4,3) - 4, n).
Original entry on oeis.org
1, 6, 120, 4495, 270725, 24040016, 2967205528, 487444845680, 103073959989495, 27319423696620550, 8881600973913295056, 3478625214672347911080, 1616770762998304775695925, 880246034121663208464847200
Offset: 0
A126449
Row sums of triangle A126445; a(n) = Sum_{k=0..n} C( C(n+2,3) - C(k+2,3), n-k).
Original entry on oeis.org
1, 2, 10, 163, 5945, 375819, 36233754, 4932046435, 899372990826, 211481102358562, 62283285977509563, 22451501854089680715, 9722649026348549481236, 4980474318644453218716459
Offset: 0
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a(n)=sum(k=0,n,binomial(binomial(n+2,3)-binomial(k+2,3), n-k))
Showing 1-7 of 7 results.
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