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A136206 Triangle H(n,j) (n=1,2,3,..., j=2,3,4,...) read by rows: let X(k,l,n) := Stirling2(n,k)*Stirling2(k,l) for 1<=k<=n and 1<=l<=k. Then H(n,j)= sum_{k+l=j, 1<=k<=n and 1<=l<=k} X(k,l,n).

%I A136206 #10 Sep 21 2015 01:38:22
%S A136206 1,1,1,1,1,3,4,3,1,1,7,13,19,13,6,1,1,15,40,85,96,75,35,10,1,1,31,121,
%T A136206 335,560,616,471,240,80,15,1,1,63,364,1253,2891,4221,4502,3353,1806,
%U A136206 665,161,21,1,1,127,1093,4599,13923,26222,36225,36205,26895,14756,5887,1638,294,28,1
%N A136206 Triangle H(n,j) (n=1,2,3,..., j=2,3,4,...) read by rows: let X(k,l,n) := Stirling2(n,k)*Stirling2(k,l) for 1<=k<=n and 1<=l<=k. Then H(n,j)= sum_{k+l=j, 1<=k<=n and 1<=l<=k} X(k,l,n).
%C A136206 Row n has 2n-1 terms. The row sums are given by A000258.
%H A136206 Gottfried Helms, <a href="/A136206/a136206.txt">Comments on A136206 and A136248</a>
%e A136206 Triangle begins:
%e A136206 ..........................1
%e A136206 .....................1....1....1
%e A136206 ................1....3....4....3....1
%e A136206 ...........1....7...13...19...13....6...1
%e A136206 ......1...15...40...85...96...75...35..10..1
%e A136206 ..1..31..121..335..560..616..471..240..80..15..1
%e A136206 .................................................
%e A136206 Assume a matrix-function rowshift(M) which computes M1 = rowshift(M) in the following way: M =
%e A136206 [a,b,c,...]
%e A136206 [k,l,m,...]
%e A136206 [r,s,t,...]
%e A136206 [.........]
%e A136206 becomes M1 =
%e A136206 [a,b,c, ......]
%e A136206 [0,k,l,m, ....]
%e A136206 [0,0,r,s,t,...]
%e A136206 [ ............]
%e A136206 Define the lower-triangular matrix of Stirling-numbers of the second kind S =
%e A136206 [1 0 0 0 ...]
%e A136206 [1 1 0 0 ...]
%e A136206 [1 3 1 0 ...]
%e A136206 [1 7 6 1 ...]
%e A136206 [ ..........]
%e A136206 Then with H0 =
%e A136206 [1]
%e A136206 [1]
%e A136206 [1]
%e A136206 [1]
%e A136206 ...
%e A136206 we have
%e A136206 H1 = S * rowshift(H0) \\ = S
%e A136206 H2 = S * rowshift(H1)
%e A136206 H3 = S * rowshift(H2)
%e A136206 ...
%e A136206 H1 =
%e A136206 1 . . . .
%e A136206 1 1 . . .
%e A136206 1 3 1 . .
%e A136206 1 7 6 1 .
%e A136206 1 15 25 10 1
%e A136206 H2=
%e A136206 1 . . . . . . . .
%e A136206 1 1 1 . . . . . .
%e A136206 1 3 4 3 1 . . . .
%e A136206 1 7 13 19 13 6 1 . .
%e A136206 1 15 40 85 96 75 35 10 1
%e A136206 H3=
%e A136206 1 . . . . . . . . . . . .
%e A136206 1 1 1 1 . . . . . . . . .
%e A136206 1 3 4 6 4 3 1 . . . . . .
%e A136206 1 7 13 26 31 31 25 13 6 1 . . .
%e A136206 1 15 40 100 171 220 255 215 156 85 35 10 1
%e A136206 (based on the Maple implementation from _R. J. Mathar_)
%p A136206 # From _R. J. Mathar_: (Start)
%p A136206 X := proc(k,l,n)
%p A136206 if k >=1 and k <=n and l >=1 and l <= n then
%p A136206 combinat[stirling2](n,k)*combinat[stirling2](k,l) ;
%p A136206 else
%p A136206 0 ;
%p A136206 fi ;
%p A136206 end:
%p A136206 H := proc(n,j)
%p A136206 add( X(j-l,l,n),l=1..floor(j/2)) ;
%p A136206 end:
%p A136206 for n from 1 to 10 do
%p A136206 for j from 2 to 2*n do
%p A136206 printf("%d ",H(n,j)) ;
%p A136206 od:
%p A136206 printf("\n") ;
%p A136206 od:
%p A136206 # (End)
%Y A136206 Cf. A136248.
%K A136206 nonn,tabf
%O A136206 1,6
%A A136206 _Gottfried Helms_, Apr 15 2008
%E A136206 Definition in terms of Stirling2 numbers found by _R. J. Mathar_, Apr 15 2008