A264402 Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

1, 2, 2, 1, 3, 2, 2, 4, 1, 4, 5, 2, 2, 8, 4, 1, 4, 9, 7, 2, 3, 12, 10, 4, 1, 4, 14, 15, 7, 2, 2, 17, 20, 12, 4, 1, 6, 18, 27, 17, 7, 2, 2, 23, 33, 26, 12, 4, 1, 4, 24, 44, 35, 19, 7, 2, 4, 27, 51, 49, 28, 12, 4, 1, 5, 30, 64, 63, 41, 19, 7, 2, 2

Offset: 1

Author

Emeric Deutsch, Nov 21 2015

Keywords

• nonn
• tabf

Comments

T(n,k) = number of partitions of n in which the 2nd largest part is k (0 if all parts are equal). Example: T(7,2) = 4 because we have [3,2,1,1], [3,2,2], [4,2,1], and [5,2].

The fact that the above two statistics (in Name and in 1st Comment) have the same distribution follows at once by conjugation. - Emeric Deutsch, Dec 11 2015

Row sums yield the partition numbers (A000041).

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

Sum_{k>=0} k*T(n,k) = A182984(n).

Examples

			T(7,2) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,2,1].
Triangle starts:
1;
2;
2,1;
3,2;
2,4,1;
4,5,2;
2,8,4,1;
		

Links

• Alois P. Heinz, Rows n = 1..350, flattened

Crossrefs

Cf. A000005, A000041, A116685, A182984, A002541.

Programs

Maple

g := sum(x^i/((1-x^i)*(product(1-t*x^j, j = i+1 .. 100))), i = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n to 27 do P[n] := sort(coeff(gser, x, n)) end do: for n to 27 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, b(n, i-1) +add((p->[0, p[1]+
       expand(p[2]*x^j)])(b(n-i*j, i-1)) , j=1..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):
seq(T(n), n=1..20);  # Alois P. Heinz, Nov 29 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Sum[ Function[p, {0, p[[1]] + Expand[p[[2]]*x^j]}][b[n-i*j, i-1]], {j, 1, n/i} ]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n][[2]]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

Formula

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