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

1,2

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

Links

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

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))).

Example

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;

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
# Alternative:
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 *)

Crossrefs

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

Sequence in context: A236468 A116685 A355522 * A241150 A051135 A392627

Adjacent sequences: A268187 A268188 A268189 * A268191 A268192 A268193

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 10 2016

Status

approved