cp's OEIS Frontend

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.

%I A306625 #10 Mar 06 2019 08:10:38
%S A306625 2,6,12,24,36,80,100,150,240,560,420,660,1020,1680,4032,1764,2940,
%T A306625 4620,7224,12096,29568,7392,13104,21280,33320,52416,88704,219648,
%U A306625 30888,58212,98280,156870,244800,386496,658944,1647360,128700,257400,452760,742140,1170540,1821600,2882880,4942080,12446720
%N 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.
%H A306625 R. T. Eakin, <a href="https://doi.org/10.1016/j.jnt.2012.04.009">A combinatorial partition of Mersenne numbers arising from spectroscopy</a>, Journal of Number Theory, Volume 132, Issue 10, October 2012, Pages 2166-2183.
%e A306625 Triangle begins
%e A306625      2,
%e A306625      6,   12,
%e A306625     24,   36,   80,
%e A306625    100,  150,  240,  560,
%e A306625    420,  660, 1020, 1680,  4032,
%e A306625   1764, 2940, 4620, 7224, 12096, 29568,
%e A306625   ...
%o A306625 (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));
%o A306625 tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")););
%Y A306625 Sum of n-th row equals A000984(n)*A000225(n).
%Y A306625 Right diagonal is A069723 starting at index 2.
%K A306625 nonn,tabl
%O A306625 1,1
%A A306625 _Michel Marcus_, Mar 01 2019