cp's OEIS Frontend

A107873 Triangle, read by rows, where T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).

%I A107873 #13 Feb 20 2022 02:12:06
%S A107873 1,4,1,15,5,1,84,28,6,1,715,220,45,7,1,8568,2380,455,66,8,1,134596,
%T A107873 33649,5985,816,91,9,1,2629575,593775,98280,12650,1330,120,10,1,
%U A107873 61523748,12620256,1947792,237336,23751,2024,153,11,1,1677106640,314457495,45379620,5245786,501942,40920,2925,190,12,1
%N A107873 Triangle, read by rows, where T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).
%C A107873 Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.
%H A107873 Harvey P. Dale, <a href="/A107873/b107873.txt">Table of n, a(n) for n = 0..5000</a>
%F A107873 From _G. C. Greubel_, Feb 19 2022: (Start)
%F A107873 T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).
%F A107873 T(n, 0) = A107874(n).
%F A107873 T(n, 1) = A107875(n). (End)
%e A107873 Triangle begins:
%e A107873         1;
%e A107873         4,      1;
%e A107873        15,      5,     1;
%e A107873        84,     28,     6,     1;
%e A107873       715,    220,    45,     7,    1;
%e A107873      8568,   2380,   455,    66,    8,   1;
%e A107873    134596,  33649,  5985,   816,   91,   9,  1;
%e A107873   2629575, 593775, 98280, 12650, 1330, 120, 10, 1; ...
%t A107873 Flatten[Table[Binomial[(n(n-1))/2-(k(k-1))/2+n-k+3,n-k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Oct 03 2015 *)
%o A107873 (PARI) T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2 +n-k+3,n-k)
%o A107873 (Magma) [Binomial(3+Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 19 2022
%o A107873 (Sage) flatten([[binomial(3+(n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 19 2022
%Y A107873 Cf. A107862, A107865, A107867, A107870, A107874, A107875, A107876.
%K A107873 nonn,tabl
%O A107873 0,2
%A A107873 _Paul D. Hanna_, Jun 04 2005