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A097304 Triangle of numbers of partitions of n with m parts which are all odd. 3
1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 3, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 4, 0, 6, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,17

COMMENTS

Row sums: A000009 (number of partitions of n into odd parts).

LINKS

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

W. Lang, First 10 rows.

FORMULA

T(n, m) := 0 if 1<= n <m, else T(n, m)= number of partitions of n with m parts which are all odd. Hence T(2*k, 2*j-1)=0, k>=1, k>=j>=1; T(2*k-1, 2*j)=0, k>=1, k-1>=j>=1.

G.f.=1/product(1-tx^(2j-1),j=1..infinity). - Emeric Deutsch, Feb 24 2006

EXAMPLE

[1];[0,1]; [1,0,1]; [0,1,0,1]; [1,0,1,0,1]; [0,2,0,1,0,1]...

T(6,2)=2 because 6= 1+5 = 3+3; T(6,1)= 0 = T(6,3): there are no partitions of 6 into either one or three parts with only odd numbers;

T(6,4)=1 from 6 = 1+1+1+3; T(6,6)=1 from 6= 1+1+1+1+1+1.

MAPLE

g:=1/product(1-t*x^(2*j-1), j=1..30)-1: gser:=simplify(series(g, x=0, 17)): for n from 1 to 15 do P[n]:=sort(coeff(gser, x^n)) od: seq(seq(coeff(P[n], t^j), j=1..n), n=1..15); - Emeric Deutsch, Feb 24 2006

CROSSREFS

Cf. A008284 (partitions of n into k parts).

Sequence in context: A082886 A236511 A235924 * A136745 A214157 A246720

Adjacent sequences:  A097301 A097302 A097303 * A097305 A097306 A097307

KEYWORD

nonn,tabl,easy

AUTHOR

Wolfdieter Lang, Aug 13 2004

STATUS

approved

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Last modified April 27 00:46 EDT 2017. Contains 285506 sequences.