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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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

# From R. J. Mathar: (Start)

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:

# (End)

CROSSREFS

Cf. A136248.

Sequence in context: A201162 A096646 A249790 * A262979 A011190 A201935

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

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Last modified December 6 14:49 EST 2016. Contains 278781 sequences.