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A185412 Triangle T(n,m) read by rows: the matrix product A130595 * A156919.

%I A185412 #10 Jan 21 2019 20:04:37
%S A185412 1,1,1,1,8,1,1,33,33,1,1,112,378,112,1,1,353,2938,2938,353,1,1,1080,
%T A185412 18987,44912,18987,1080,1,1,3265,111051,520523,520523,111051,3265,1,1,
%U A185412 9824,612820,5131040,9998182,5131040,612820,9824,1
%N A185412 Triangle T(n,m) read by rows: the matrix product A130595 * A156919.
%C A185412 Row sums are A165968(n+1).
%C A185412 Because A130595 is the inverse of the Pascal triangle A007318, we have A007318 *(this lower triangular matrix) = A156919.
%F A185412 G.f.: 1/(1+x-xy-2x/(1-3xy/(1+x-4x/(1-5xy/(1+x-6x/(1-7xy/(1+x-8x/(1-9xy/(1+ ... (continued fraction).
%e A185412 Triangle begins in row n=0 with columns 0 <= m <= n:
%e A185412   1;
%e A185412   1,    1;
%e A185412   1,    8,      1;
%e A185412   1,   33,     33,       1;
%e A185412   1,  112,    378,     112,       1;
%e A185412   1,  353,   2938,    2938,     353,       1;
%e A185412   1, 1080,  18987,   44912,   18987,    1080,      1;
%e A185412   1, 3265, 111051,  520523,  520523,  111051,   3265,    1;
%e A185412   1, 9824, 612820, 5131040, 9998182, 5131040, 612820, 9824, 1;
%p A185412 A156919 := proc(n,m) if n=m then 1; elif m=0 then 2^n ; elif m<0 or m>n then 0; else 2*(m+1)*procname(n-1,m)+(2*n-2*m+1)*procname(n-1,m-1) ; end if; end proc:
%p A185412 A130595 := proc(n,m) (-1)^(n+m)*binomial(n,m) ; end proc:
%p A185412 A185412 := proc(n,m) local a,j; a := 0 ; for j from m to n do a := a+A130595(n,j)*A156919(j,m) ; end do: a ; end proc: # _R. J. Mathar_, Feb 03 2011
%K A185412 nonn,easy,tabl
%O A185412 0,5
%A A185412 _Paul Barry_, Jan 26 2011