A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+n*k+k, n*k+k).

1, 1, 1, 1, 3, 1, 1, 10, 5, 1, 1, 35, 28, 7, 1, 1, 126, 165, 55, 9, 1, 1, 462, 1001, 455, 91, 11, 1, 1, 1716, 6188, 3876, 969, 136, 13, 1, 1, 6435, 38760, 33649, 10626, 1771, 190, 15, 1, 1, 24310, 245157, 296010, 118755, 23751, 2925, 253, 17, 1, 1, 92378, 1562275

Offset

0,5

Comments

Main diagonal is A108288. Antidiagonal sums is A108289. Inverse binomial transforms of each row give triangle A108290. G.f. of row n multiplied by (1-x)^(n+1) equals g.f. of row n of triangle A108267 (rows sums of A108267 equal (n+1)^n).

Links

Harry J. Smith, Table of n, a(n) for n=0..252

Formula

a(n) = A060539(n, k)/n = A007318(nk, k)/n = A060540(n, k)/A060540(n-1, k).

Example

row 1: (2*n+1)/1!
row 2: (3*n+1)*(3*n+2)/2!
row 3: (4*n+1)*(4*n+2)*(4*n+3)/3!
row 4: (5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/4!
row 5: (6*n+1)*(6*n+2)*(6*n+3)*(6*n+4)*(6*n+5)/5!.
Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,5,7,9,11,13,15,17,19,21,23,25,27,...
1,10,28,55,91,136,190,253,325,406,496,...
1,35,165,455,969,1771,2925,4495,6545,...
1,126,1001,3876,10626,23751,46376,82251,...
1,462,6188,33649,118755,324632,749398,...
1,1716,38760,296010,1344904,4496388,...

Prog

(PARI) T(n, k)=binomial(n+n*k+k, n*k+k)
(PARI) { i=0; write("b060543.txt", "0 1"); for (m=0, 20, for (k=0, m + 1, n=m - k + 1; write("b060543.txt", i++, " ", binomial(n + n*k + k, n*k + k))); ) } \\ Harry J. Smith, Jul 06 2009

Crossrefs

Cf. A108290, A108267, A108288, A108289, A060544 (row 2), A015219 (row 3).

Rows include A000012, A001700, A025174. Columns include A000012, A005408, A060544. Main diagonal is A060545.

Sequence in context: A106268 A267655 A263864 * A267751 A185996 A261215

Adjacent sequences: A060540 A060541 A060542 * A060544 A060545 A060546

Keyword

nonn,tabl

Author

Henry Bottomley, Apr 02 2001

Extensions

Entry revised by Paul D. Hanna, May 31 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

Status

approved