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A116683 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the even parts is k (n>=0, 0<=k<=n). 2
1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 1, 0, 1, 0, 2, 0, 2, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 3, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 3, 0, 3, 0, 2, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,25

COMMENTS

Row 2n-1 has 2n-1 terms; row 2n has 2n+1 terms. Row sums yield A000009. T(n,0)=A000700(n). Columns 2n-1 contain only 0's. Sum(k*T(n,k), k=0..n)=A116684(n).

LINKS

Table of n, a(n) for n=0..104.

FORMULA

G.f.=product((1+x^(2j-1))(1+(tx)^(2j)), j=1..infinity).

EXAMPLE

T(9,6)=2 because we have [6,3] and [4,3,2].

Triangle starts:

1;

1;

0,0,1;

1,0,1;

1,0,0,0,1;

1,0,1,0,1;

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

MAPLE

g:=product((1+x^(2*j-1))*(1+(t*x)^(2*j)), j=1..30): gser:=simplify(series(g, x=0, 20)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(gser, x^n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..2*floor(n/2)) od; # yields sequence in triangular form

CROSSREFS

Cf. A000009, A000700, A116684.

Sequence in context: A255327 A255391 A255396 * A079748 A073368 A321756

Adjacent sequences:  A116680 A116681 A116682 * A116684 A116685 A116686

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Feb 22 2006

STATUS

approved

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Last modified April 5 00:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)