

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
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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

Table of n, a(n) for n=1..89.


FORMULA

G.f.: G(t,x) = Sum_{j>=1} (x^j * Product_{i=1..j1} (1 + t*x^i))/(1x^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..j1)/(1x^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



