A246381 - OEIS

%I #13 Feb 25 2020 00:40:13

%S 1,1,1,3,2,1,12,13,3,1,73,80,34,4,1,584,701,296,70,5,1,5889,7680,3463,

%T 816,125,6,1,73184,100519,49432,12139,1876,203,7,1,1089057,1571040,

%U 810268,217728,34294,3808,308,8,1,19019632,28717865,15455072,4354260,751792,83406,7056,444,9,1

%N Triangular matrix T defined by T = exp(L) where L(n,k) = C(2*n, 2*k+1)/2, as read by rows n >= 0, k=0..n.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 12, 13, 3, 1;

%e 73, 80, 34, 4, 1;

%e 584, 701, 296, 70, 5, 1;

%e 5889, 7680, 3463, 816, 125, 6, 1;

%e 73184, 100519, 49432, 12139, 1876, 203, 7, 1;

%e 1089057, 1571040, 810268, 217728, 34294, 3808, 308, 8, 1;

%e 19019632, 28717865, 15455072, 4354260, 751792, 83406, 7056, 444, 9, 1;

%e 384301729, 603257920, 338772685, 99130208, 17974226, 2186368, 181602, 12192, 615, 10, 1; ...

%e The matrix logarithm, L, begins:

%e 0;

%e 1, 0;

%e 2, 2, 0;

%e 3, 10, 3, 0;

%e 4, 28, 28, 4, 0;

%e 5, 60, 126, 60, 5, 0;

%e 6, 110, 396, 396, 110, 6, 0;

%e 7, 182, 1001, 1716, 1001, 182, 7, 0;

%e 8, 280, 2184, 5720, 5720, 2184, 280, 8, 0;

%e 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 0;

%e 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10, 0; ...

%e where L(n,k) = C(2*n, 2*k+1)/2.

%e The matrix square begins:

%e 1;

%e 2, 1;

%e 8, 4, 1;

%e 46, 32, 6, 1;

%e 376, 280, 80, 8, 1;

%e 3962, 3304, 972, 160, 10, 1;

%e 52268, 47100, 15400, 2552, 280, 12, 1;

%e 837574, 803852, 283394, 51704, 5642, 448, 14, 1;

%e 15919312, 16175600, 6028944, 1187632, 141136, 11088, 672, 16, 1; ...

%o (PARI) {T(n,k)=local(L=matrix(n+1,n+1,r,c,if(r>=c,binomial(2*r-2,2*c-1)/2)),A);

%o A=sum(m=0,n,L^m/m!);A[n+1,k+1]}

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%Y Cf. A246382, A246383, A246384, A246385, A246386, A246387.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Aug 24 2014