A115604 - OEIS

A115604

Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n).

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 2, 1, 0, 1, 0, 0, 1, 3, 2, 1, 1, 1, 0, 1, 0, 0, 1, 3, 3, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4, 2, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 5, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1

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OFFSET

1,22

COMMENTS

Row sums yield A000009. T(n,1)=A087897(n+2). Sum(k*T(n,k),k=1..n)=A092268(n).

LINKS

Table of n, a(n) for n=1..105.

FORMULA

G.f.=G(t,x)=sum(tx^(2k-1)/[(1-tx^(2k-1))product(1-x^(2i-1), i=k+1..infinity)], k=1..infinity).

EXAMPLE

T(14,2)=4 because we have [9,3,1,1],[7,7],[7,5,1,1] and [3,3,3,3,1,1].

Triangle starts:

0,1;

1,0,1;

1,0,0,1;

1,1,0,0,1;

1,1,1,0,0,1;

2,1,0,1,0,0,1;

MAPLE

g:=sum(t*x^(2*k-1)/(1-t*x^(2*k-1))/product(1-x^(2*i-1), i=k+1..40), k=1..40): gser:=simplify(series(g, x=0, 55)): for n from 1 to 15 do P[n]:=expand(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. A000009, A087897, A092268, A117408.

Sequence in context: A333815 A308103 A368987 * A128617 A116488 A356242

Adjacent sequences: A115601 A115602 A115603 * A115605 A115606 A115607

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Mar 13 2006

STATUS

approved