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 A117456 Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part to the largest part occurs and the number of parts is k (1<=k<=n). 1
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 2, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,18 COMMENTS Row sums yield A034296. sum(k*T(n,k),k=1..n)=A117457(n). LINKS FORMULA G.f.=G(t,x)=sum(t^j*x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..infinity). EXAMPLE T(10,5)=3 because we have [3,3,2,1,1],[3,2,2,2,1] and [2,2,2,2,2]. Triangle starts: 1; 1,1; 1,1,1; 1,1,1,1; 1,1,1,1,1; 1,1,2,1,1,1; 1,1,1,2,1,1,1; 1,1,1,2,2,1,1,1; MAPLE g:=sum(t^j*x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..60): gser:=simplify(series(g, x=0, 55)): for n from 1 to 15 do P[n]:=coeff(gser, x^n) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form CROSSREFS Cf. A034296, A117457. Sequence in context: A030379 A030392 A165633 * A030621 A120336 A039738 Adjacent sequences:  A117453 A117454 A117455 * A117457 A117458 A117459 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Mar 18 2006 STATUS approved

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