A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

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

Offset

1,29

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Formula

T(n,k) = Sum_{j=1..round(n/(2*k))} A067661(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067659(n-2*j*k).

G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q)_{inf} + (1/2)*(q^k/(1-q^k))*(q;q)_{inf}.

T(n,k) = A015716(n,k) - A238451(n,k). - Andrew Howroyd, Apr 29 2020

Example

n\k | 1 2 3 4 5 6 7 8 9 10
   1: 1
   2: 0 1
   3: 0 0 1
   4: 0 0 0 1
   5: 0 0 0 0 1
   6: 1 1 1 0 0 1
   7: 1 1 0 1 0 0 1
   8: 2 1 1 1 1 0 0 1
   9: 2 2 2 1 1 1 0 0 1
  10: 3 2 2 1 2 1 1 0 0 1

Prog

(PARI) T(n, k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) + prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, m==0)} \\ Andrew Howroyd, Apr 29 2020

Crossrefs

Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.

Row sums are A238131.

Cf. A015716, A067659, A067661, A238451.

Sequence in context: A229745 A339366 A016397 * A251926 A335504 A037908

Adjacent sequences: A238447 A238448 A238449 * A238451 A238452 A238453

Keyword

nonn,tabl

Author

Mircea Merca, Feb 26 2014

Extensions

Terms a(79) and beyond from Andrew Howroyd, Apr 29 2020

Status

approved