A265247 Triangle read by rows: T(n,k) is the number of partitions of n in which the 2nd smallest part is k when the partition has at least 2 distinct parts and 0 otherwise; (n>=1, 0 <= k <= n).

1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 2, 2, 1, 4, 0, 3, 1, 2, 1, 2, 0, 5, 3, 2, 2, 1, 4, 0, 7, 4, 2, 2, 2, 1, 3, 0, 11, 6, 2, 3, 2, 2, 1, 4, 0, 15, 8, 6, 1, 3, 2, 2, 1, 2, 0, 22, 12, 6, 4, 2, 3, 2, 2, 1, 6, 0, 30, 15, 9, 4, 3, 2, 3, 2, 2, 1, 2, 0, 42, 22, 11, 8, 2, 4, 2, 3, 2, 2, 1, 4, 0, 56, 28, 16, 10, 6, 1, 4, 2, 3, 2, 2, 1, 4, 0, 77, 38, 19, 11, 7, 4, 2, 4, 2, 3, 2, 2, 1

Offset: 1

Author

Emeric Deutsch, Dec 24 2015

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n.

Sum of entries in row n is A000041(n) = number of partitions of n.

T(n,0) = A000005(n) = number of divisors of n.

T(n,1) = 0.

T(n,2) = A000041(n-3), i.e., for n>=3 the number of partitions of n having 2 as the 2nd smallest part is equal to the number of partitions of n-3 (follows from a simple bijection: delete a part 2 and a part 1).

Sum_{k>=0} k*T(n,k) = A265248(n).

Examples

			T(5,3) = 2 because of [3,2] and [3,1,1].
Triangle starts:
  1;
  2, 0;
  2, 0, 1;
  3, 0, 1, 1;
  2, 0, 2, 2, 1;
  4, 0, 3, 1, 2, 1.
		

Crossrefs

Cf. A000005, A000041, A265248.

Programs

Maple

g := add(x^i*(1+add(t^j*x^j/(mul(1-x^k, k=j..80)), j=i+1..80))/(1-x^i), i=1..80):
gser := simplify(series(g, x = 0, 25)):
for n to 20 do P[n] := sort(coeff(gser, x, n)) end do:
for n to 20 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

Formula

G.f.: G(t,x) = Sum_{i>=1} x^i/(1-x^i) (1 + Sum_{j>=i+1} t^j*x^j/Product_{k>=j}(1-x^k)).