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

1, 4, 1, 15, 5, 1, 84, 28, 6, 1, 715, 220, 45, 7, 1, 8568, 2380, 455, 66, 8, 1, 134596, 33649, 5985, 816, 91, 9, 1, 2629575, 593775, 98280, 12650, 1330, 120, 10, 1, 61523748, 12620256, 1947792, 237336, 23751, 2024, 153, 11, 1, 1677106640, 314457495, 45379620, 5245786, 501942, 40920, 2925, 190, 12, 1

Offset: 0

Author

Paul D. Hanna, Jun 04 2005

Keywords

• nonn
• tabl

Comments

Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.

Examples

			Triangle begins:
        1;
        4,      1;
       15,      5,     1;
       84,     28,     6,     1;
      715,    220,    45,     7,    1;
     8568,   2380,   455,    66,    8,   1;
   134596,  33649,  5985,   816,   91,   9,  1;
  2629575, 593775, 98280, 12650, 1330, 120, 10, 1; ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A107862, A107865, A107867, A107870, A107874, A107875, A107876.

Programs

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

Mathematica

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 *)

PARI

T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2 +n-k+3,n-k)

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

Formula

From G. C. Greubel, Feb 19 2022: (Start)

T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).

T(n, 0) = A107874(n).

T(n, 1) = A107875(n). (End)