A107862 Triangle, read by rows, where T(n,k) = C(n*(n-1)/2 - k*(k-1)/2 + n-k, n-k).

1, 1, 1, 3, 2, 1, 20, 10, 3, 1, 210, 84, 21, 4, 1, 3003, 1001, 220, 36, 5, 1, 54264, 15504, 3060, 455, 55, 6, 1, 1184040, 296010, 53130, 7315, 816, 78, 7, 1, 30260340, 6724520, 1107568, 142506, 14950, 1330, 105, 8, 1, 886163135, 177232627, 26978328, 3262623, 324632, 27405, 2024, 136, 9, 1

Offset

0,4

Comments

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n,k) = binomial( (n-k)*(n+k+1)/2, n-k). - G. C. Greubel, Feb 19 2022

Example

Triangle begins:
        1;
        1,      1;
        3,      2,     1;
       20,     10,     3,    1;
      210,     84,    21,    4,   1;
     3003,   1001,   220,   36,   5,  1;
    54264,  15504,  3060,  455,  55,  6, 1;
  1184040, 296010, 53130, 7315, 816, 78, 7, 1; ...

Mathematica

T[n_, k_]:= Binomial[(n-k)*(n+k+1)/2, n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 19 2022 *)

Prog

(PARI) T(n, k)=binomial(n*(n-1)/2-k*(k-1)/2+n-k, n-k)
(Magma) [Binomial(Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2022
(Sage) flatten([[binomial( (n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2022

Crossrefs

Cf. A014068 (column 0), A107863 (column 1), A099121 (column 2), A107865, A107867, A107870, A107876.

Sequence in context: A136733 A117269 A291080 * A117265 A107727 A346743

Adjacent sequences: A107859 A107860 A107861 * A107863 A107864 A107865

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 04 2005

Status

approved