A306625 Regular triangle T(n,k) = binomial(2*n-2*k,n-k)*((n+1)/k)*Sum_{k=0..floor((k-1)/2)} (-1)^k*binomial(2*k,k)*binomial(n+3*k-2*k,k-2*k-1), read by rows.

2, 6, 12, 24, 36, 80, 100, 150, 240, 560, 420, 660, 1020, 1680, 4032, 1764, 2940, 4620, 7224, 12096, 29568, 7392, 13104, 21280, 33320, 52416, 88704, 219648, 30888, 58212, 98280, 156870, 244800, 386496, 658944, 1647360, 128700, 257400, 452760, 742140, 1170540, 1821600, 2882880, 4942080, 12446720

Offset: 1

Author

Michel Marcus, Mar 01 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins
     2,
     6,   12,
    24,   36,   80,
   100,  150,  240,  560,
   420,  660, 1020, 1680,  4032,
  1764, 2940, 4620, 7224, 12096, 29568,
  ...
		

Links

• R. T. Eakin, A combinatorial partition of Mersenne numbers arising from spectroscopy, Journal of Number Theory, Volume 132, Issue 10, October 2012, Pages 2166-2183.

Crossrefs

Sum of n-th row equals A000984(n)*A000225(n).

Right diagonal is A069723 starting at index 2.

Programs

PARI

T(n,r) = binomial(2*n-2*r,n-r)*((n+1)/r)*sum(k=0, (r-1)\2, (-1)^k*binomial(2*r,k)*binomial(n+3*r-2*k,r-2*k-1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")););