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).

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, 335, 560, 616, 471, 240, 80, 15, 1, 1, 63, 364, 1253, 2891, 4221, 4502, 3353, 1806, 665, 161, 21, 1, 1, 127, 1093, 4599, 13923, 26222, 36225, 36205, 26895, 14756, 5887, 1638, 294, 28, 1

Offset

1,6

Comments

Row n has 2n-1 terms. The row sums are given by A000258.

Links

Table of n, a(n) for n=1..64.

Gottfried Helms, Comments on A136206 and A136248

Example

Triangle begins:
..........................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..335..560..616..471..240..80..15..1
.................................................
Assume a matrix-function rowshift(M) which computes M1 = rowshift(M) in the following way: M =
[a,b,c,...]
[k,l,m,...]
[r,s,t,...]
[.........]
becomes M1 =
[a,b,c, ......]
[0,k,l,m, ....]
[0,0,r,s,t,...]
[ ............]
Define the lower-triangular matrix of Stirling-numbers of the second kind S =
[1 0 0 0 ...]
[1 1 0 0 ...]
[1 3 1 0 ...]
[1 7 6 1 ...]
[ ..........]
Then with H0 =
[1]
[1]
[1]
[1]
...
we have
H1 = S * rowshift(H0) \\ = S
H2 = S * rowshift(H1)
H3 = S * rowshift(H2)
...
H1 =
1 . . . .
1 1 . . .
1 3 1 . .
1 7 6 1 .
1 15 25 10 1
H2=
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
H3=
1 . . . . . . . . . . . .
1 1 1 1 . . . . . . . . .
1 3 4 6 4 3 1 . . . . . .
1 7 13 26 31 31 25 13 6 1 . . .
1 15 40 100 171 220 255 215 156 85 35 10 1
(based on the Maple implementation from R. J. Mathar)

Maple

X := proc(k, l, n)
if k >=1 and k <=n and l >=1 and l <= n then
combinat[stirling2](n, k)*combinat[stirling2](k, l) ;
else
0 ;
fi ;
end:
H := proc(n, j)
add( X(j-l, l, n), l=1..floor(j/2)) ;
end:
for n from 1 to 10 do
for j from 2 to 2*n do
printf("%d ", H(n, j)) ;
od:
printf("\n") ;
od:
# R. J. Mathar

Crossrefs

Cf. A136248.

Sequence in context: A384118 A249790 A302713 * A262979 A011190 A279676

Adjacent sequences: A136203 A136204 A136205 * A136207 A136208 A136209

Keyword

nonn,tabf

Author

Gottfried Helms, Apr 15 2008

Extensions

Definition in terms of Stirling2 numbers found by R. J. Mathar, Apr 15 2008

Status

approved