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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 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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