A268190 Triangle read by rows: T(n,k) (n, k>=1) is the number of partitions of n such that the difference between the two largest distinct parts is k; T(n,0) is the number of partitions of n in which all parts are equal.

1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 7, 5, 2, 2, 1, 1, 3, 11, 5, 5, 2, 2, 1, 1, 4, 13, 10, 5, 4, 2, 2, 1, 1, 2, 20, 11, 8, 5, 4, 2, 2, 1, 1, 6, 23, 16, 10, 8, 4, 4, 2, 2, 1, 1, 2, 33, 20, 15, 9, 8, 4, 4

Offset: 1

Author

Emeric Deutsch, Feb 10 2016

Keywords

• nonn
• tabf

Comments

Sum of entries in row n is A000041(n) (the partition numbers).

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

T(n,1) = A083751(n+1) for n>=3.

Sum(k*T(n,k),k>=1) = A268191(n).

T(2n,n) = A002865(n) for n>=2. - Alois P. Heinz, Feb 11 2016

Examples

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

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Cf. A000005, A000041, A002865, A083751, A268191.

Programs

Maple

G := add(x^k/(1-x^k), k = 1 .. 80)+ add(add(t^(i-j)*x^(i+j)/((1-x^i)*mul(1-x^k,k = 1 .. j)), j = 1 .. i-1), i = 2 .. 80): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: 1; for n from 2 to 25 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, l, i) option remember; `if`(irem(n, i)=0, x^
      `if`(l=0, 0, i-l), 0) +`if`(i>n, 0, add(b(n-i*j,
      `if`(j=0, l, i), i+1), j=0..(n-1)/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=1..30);  # Alois P. Heinz, Feb 11 2016

Mathematica

b[n_, l_, i_] := b[n, l, i] = If[Mod[n, i] == 0, x^If[l == 0, 0, i-l], 0] + If[i>n, 0, Sum[b[n-i*j, If[j == 0, l, i], i+1], {j, 0, (n-1)/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

Formula

G.f.: G(t,x) = Sum_{k>0} (x^k/(1-x^k)) + Sum_{k>1} (Sum_{j=1..i-1} t^{i-j}*x^{i+j}/((1 - x^i)*Product_{k=1..j} (1 - x^k))).