%I #14 Sep 24 2018 16:53:13

%S 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,

%T 11,1,1,1716,6188,3876,969,136,13,1,1,6435,38760,33649,10626,1771,190,

%U 15,1,1,24310,245157,296010,118755,23751,2925,253,17,1,1,92378,1562275

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

%C 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).

%H Harry J. Smith, <a href="/A060543/b060543.txt">Table of n, a(n) for n=0..252</a>

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

%e row 1: (2*n+1)/1!

%e row 2: (3*n+1)*(3*n+2)/2!

%e row 3: (4*n+1)*(4*n+2)*(4*n+3)/3!

%e row 4: (5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/4!

%e row 5: (6*n+1)*(6*n+2)*(6*n+3)*(6*n+4)*(6*n+5)/5!.

%e Table begins:

%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

%e 1,3,5,7,9,11,13,15,17,19,21,23,25,27,...

%e 1,10,28,55,91,136,190,253,325,406,496,...

%e 1,35,165,455,969,1771,2925,4495,6545,...

%e 1,126,1001,3876,10626,23751,46376,82251,...

%e 1,462,6188,33649,118755,324632,749398,...

%e 1,1716,38760,296010,1344904,4496388,...

%o (PARI) T(n,k)=binomial(n+n*k+k,n*k+k)

%o (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

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

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

%K nonn,tabl

%O 0,5

%A _Henry Bottomley_, Apr 02 2001

%E Entry revised by _Paul D. Hanna_, May 31 2005

%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 17 2007