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A110582 Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2.

%I A110582 #11 Sep 01 2017 03:04:28
%S A110582 1,1,2,1,2,3,1,4,3,4,1,4,7,4,5,1,6,10,10,5,6,1,6,14,16,13,6,7,1,8,18,
%T A110582 26,22,16,7,8,1,8,25,34,38,28,19,8,9,1,10,29,52,55,50,34,22,9,10,1,10,
%U A110582 37,66,84,76,62,40,25,10,11,1,12,44,90,116,122,97,74,46,28,11,12
%N A110582 Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2.
%C A110582 Related to planar partitions.
%H A110582 G. C. Greubel, <a href="/A110582/b110582.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A110582 T(n, k) = Sum_{j=0..k} T(n-k-1, j)*(k-j+1) with T(n, n) = n+1.
%F A110582 G.f.: A(x, y) = Sum_{j=0..n} x^j/Product_{i=1..j+1} (1-y*x^i)^2.
%e A110582 Triangle begins:
%e A110582 1;
%e A110582 1,2;
%e A110582 1,2,3;
%e A110582 1,4,3,4;
%e A110582 1,4,7,4,5;
%e A110582 1,6,10,10,5,6;
%e A110582 1,6,14,16,13,6,7;
%e A110582 1,8,18,26,22,16,7,8;
%e A110582 1,8,25,34,38,28,19,8,9;
%e A110582 1,10,29,52,55,50,34,22,9,10; ...
%e A110582 Row sums form A006330 (offset 1):
%e A110582 {1,3,6,12,21,38,63,106,170,272,422,653,...},
%e A110582 (planar partitions with only one row and one column).
%e A110582 G.f. of diagonal n, D_n(x), is generated from g.f. of
%e A110582 row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2:
%e A110582 D_3(x) = 1 + 4*x + 10*x^2 + 16*x^3 + 22*x^4 + ...
%e A110582 = (1 + 2*x + 3*x^2)/(1-x)^2 = R_2(x)/(1-x)^2;
%e A110582 D_4(x) = 1 + 6*x + 14*x^2 + 26*x^3 + 38*x^4 + ...
%e A110582 = (1+ 4*x+ 3*x^2+ 4*x^3)/(1-x)^2 = R_3(x)/(1-x)^2.
%t A110582 T[n_, k_] := T[n, k] = If[n < k || k < 0, 0, If[k == 0, 1, If[k == n, n + 1, Sum[T[n - k - 1, j]*(k - j + 1), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Sep 01 2017 *)
%o A110582 (PARI) T(n,k)=if(n<k || k<0,0,if(k==0,1,if(k==n,n+1, sum(j=0,k,T(n-k-1,j)*(k-j+1)));))
%o A110582 (PARI) T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));if(n<k || k<0,0, polcoeff(polcoeff(sum(j=0,n,X^j/prod(i=1,j+1,1-Y*X^i)^2),n,x),k,y))
%Y A110582 Cf. A006330 (row sums).
%K A110582 nonn,tabl
%O A110582 0,3
%A A110582 _Paul D. Hanna_, Jul 29 2005