A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

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

Offset: 1

Author

Mircea Merca, Feb 26 2014

Keywords

• nonn
• tabl

Examples

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

Links

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

Crossrefs

Columns k=1..6 are A238215, A238217, A238218, A238219, A238220, A238221.

Row sums are A238132.

Cf. A015716, A067659, A067661, A238450.

Programs

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, 0)} \\ Andrew Howroyd, Apr 29 2020

Formula

T(n,k) = Sum_{j=1..round(n/(2*k))} A067659(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067661(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) - A238450(n,k). - Andrew Howroyd, Apr 29 2020