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; ...
A126446
Column 0 of triangle A126445; a(n) = binomial( binomial(n+2,3), n).
Original entry on oeis.org
1, 1, 6, 120, 4845, 324632, 32468436, 4529365776, 840261910995, 200063149171380, 59473554359599446, 21592914273609648996, 9403538945961296957821, 4838670732821812768919800, 2904538537066424425438417800
Offset: 0
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[(Binomial(Binomial(n+3, n), n+1)): n in [-1..20]]; // Vincenzo Librandi, Mar 04 2018
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Table[Binomial[n (n + 1) (n + 2) / 3!, n], {n, 0, 20}] (* Vincenzo Librandi, Mar 04 2018 *)
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a(n)=binomial(n*(n+1)*(n+2)/3!, n)
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[(binomial(binomial(n+3,n),n+1)) for n in range(-1, 12)] # Zerinvary Lajos, Nov 30 2009
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
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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)
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
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
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a(n)=sum(k=0,n,binomial(binomial(n+2,3)-binomial(k+2,3)+1, n-k))
Showing 1-6 of 6 results.
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