1,5

The row sums form A098569: {1,2,5,14,43,143,510,1936,7775,32869,...}. How do the terms of row k tend to be distributed as k grows?

Remarkably, column k of the matrix inverse (A121434) equals signed column k of the triangular matrix power: A107876^(k*(k+1)/2) for k>=0. - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 25 2006

Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 30 2007

Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.

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

G.f. of columns are: 1/(1-x), 1/(1-x)^3, 1/(1-x)^10, 1/(1-x)^15, ...

Rows begin:

[1],

[1,1],

[1,3,1],

[1,6,6,1],

[1,10,21,10,1],

[1,15,56,55,15,1],

[1,21,126,220,120,21,1],

[1,28,252,715,680,231,28,1],

[1,36,462,2002,3060,1771,406,36,1],

[1,45,792,5005,11628,10626,4060,666,45,1],

[1,55,1287,11440,38760,53130,31465,8436,1035,55,1],

[1,66,2002,24310,116280,230230,201376,82251,16215,1540,66,1],...

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

Cf. A098569.

Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.

Cf. A131338.

Sequence in context: A107105 A088925 A100862 this_sequence A131235 A133713 A008278

Adjacent sequences: A098565 A098566 A098567 this_sequence A098569 A098570 A098571

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2004