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

1,1

Links

Table of n, a(n) for n=1..45.

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.

Example

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

Prog

(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), ", ")); );

Crossrefs

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

Right diagonal is A069723 starting at index 2.

Sequence in context: A067718 A210594 A307252 * A262986 A307559 A211978

Adjacent sequences: A306622 A306623 A306624 * A306626 A306627 A306628

Keyword

nonn,tabl

Author

Michel Marcus, Mar 01 2019

Status

approved