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
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)
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
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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)
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
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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