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 A117470 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 difference between the largest part and the smallest part is k (n >= 0, k >= 0). 1
 1, 2, 2, 1, 3, 1, 2, 3, 4, 2, 1, 2, 5, 1, 4, 4, 2, 3, 6, 4, 4, 6, 4, 1, 2, 9, 6, 1, 6, 6, 9, 2, 2, 11, 10, 3, 4, 10, 11, 6, 4, 11, 17, 6, 1, 5, 11, 17, 10, 1, 2, 15, 21, 12, 2, 6, 12, 24, 18, 3, 2, 17, 28, 20, 5, 6, 14, 31, 26, 8, 4, 17, 38, 31, 10, 1, 4, 18, 37, 41, 14, 1, 2, 21, 45, 45, 19, 2, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row n contains ceiling((sqrt(9+8n)-3)/2) terms, i.e., 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ... Row sums yield A034296. LINKS FORMULA G.f.: G(t,x) = Sum_{j>=1} (x^j * Product_{i=1..j-1} (1 + t*x^i))/(1-x^j). T(n,0) = A000005(n) (number of divisors of n). Sum_{k>=0} k*T(n,k) = A117471(n). EXAMPLE T(9,2)=4 because we have [4,3,2],[3,3,2,1],[3,2,2,1,1] and [3,2,1,1,1,1]. Triangle starts:   1;   2;   2, 1;   3, 1;   2, 3;   4, 2, 1;   2, 5, 1; MAPLE g:=sum(x^j*product(1+t*x^i, i=1..j-1)/(1-x^j), j=1..30): gser:=simplify(series(g, x=0, 28)): for n from 1 to 28 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 25 do seq(coeff(P[n], t, j), j=0..ceil((sqrt(9+8*n)-5)/2)) od; # yields sequence in triangular form CROSSREFS Cf. A034296, A000005, A117471. Sequence in context: A227083 A327571 A166363 * A070786 A255542 A117500 Adjacent sequences:  A117467 A117468 A117469 * A117471 A117472 A117473 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Mar 20 2006 STATUS approved

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Last modified June 17 00:17 EDT 2021. Contains 345080 sequences. (Running on oeis4.)