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A152140 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n into k odd parts. 5
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,24

COMMENTS

The number of partitions of n into k odd parts is equal to the number of partitions of (n+k)/2 into k parts; or equivalently the number of partitions of (n-k)/2 into at most k parts. - Franklin T. Adams-Watters, Sep 25 2009

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

EXAMPLE

n= 0, k= 0: [];

n= 1, k= 1: [1] ;

n= 2, k= 2: [1, 1] ;

n= 3, k= 1: [3] ;

n= 3, k= 3: [1, 1, 1] ;

n= 4, k= 2: [1, 3] ;

n= 4, k= 4: [1, 1, 1, 1];

n= 5, k= 1: [5];

n= 5, k= 3: [1, 1, 3];

n= 5, k= 5: [1, 1, 1, 1, 1];

n= 6, k= 2: [3, 3] or [1, 5];

n= 6, k= 4: [1, 1, 1, 3];

n= 6, k= 6: [1, 1, 1, 1, 1, 1];

Triangle begins:

1

0 1

0 0 1

0 1 0 1

0 0 1 0 1

0 1 0 1 0 1

0 0 2 0 1 0 1

0 1 0 2 0 1 0 1

0 0 2 0 2 0 1 0 1

0 1 0 3 0 2 0 1 0 1

0 0 3 0 3 0 2 0 1 0 1

0 1 0 4 0 3 0 2 0 1 0 1

0 0 3 0 5 0 3 0 2 0 1 0 1

0 1 0 5 0 5 0 3 0 2 0 1 0 1

0 0 4 0 6 0 5 0 3 0 2 0 1 0 1

0 1 0 7 0 7 0 5 0 3 0 2 0 1 0 1

0 0 4 0 9 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 8 0 10 0 7 0 5 0 3 0 2 0 1 0 1

0 0 5 0 11 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 10 0 13 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 5 0 15 0 14 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 12 0 18 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 6 0 18 0 20 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 14 0 23 0 21 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 6 0 23 0 26 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 16 0 30 0 28 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 7 0 27 0 35 0 29 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 19 0 37 0 38 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 7 0 34 0 44 0 40 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 21 0 47 0 49 0 41 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 8 0 39 0 58 0 52 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 24 0 57 0 65 0 54 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 8 0 47 0 71 0 70 0 55 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 27 0 70 0 82 0 73 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 0 9 0 54 0 90 0 89 0 75 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

0 1 0 30 0 84 0 105 0 94 0 76 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

MAPLE

b:= proc(n, i) option remember; local j; if n=0 then 1

      elif i<1 then 0 elif irem(i, 2)=0 then b(n, i-1)

      else []; for j from 0 to n/i do zip((x, y)->x+y, %,

      [0$j, b(n-i*j, i-2)], 0) od; %[] fi

    end:

T:= n-> b(n$2):

seq(T(n), n=0..13);  # Alois P. Heinz, May 31 2013

MATHEMATICA

nn = 10; CoefficientList[

Series[Product[1/(1 - y x^i), {i, 1, nn, 2}], {x, 0, nn}], {x, y}] (* Geoffrey Critzer, May 31 2013 *)

CROSSREFS

Cf. A000009 (row sums), A107379, A152146, A152157.

Sequence in context: A083912 A256003 A157187 * A244415 A104975 A191254

Adjacent sequences:  A152137 A152138 A152139 * A152141 A152142 A152143

KEYWORD

nonn,tabl

AUTHOR

R. J. Mathar, Sep 25 2009, offset corrected Jul 09 2012

STATUS

approved

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Last modified May 25 10:24 EDT 2017. Contains 287026 sequences.