A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

1, 1, 1, 1, 3, 4, 1, 6, 1, 8, 3, 12, 3, 18, 3, 1, 24, 6, 32, 10, 45, 10, 1, 59, 17, 1, 79, 21, 1, 104, 28, 3, 137, 37, 2, 177, 50, 4, 229, 64, 4, 295, 82, 8, 377, 105, 8, 477, 139, 10, 1, 605, 174, 13, 761, 220, 21, 956, 275, 24, 1193, 350, 31, 1

Offset

0,5

Comments

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

T(n,0) = A277099(n).

Sum(k*T(n,k), k>=0) = A276434(n).

Links

Alois P. Heinz, Rows n = 0..1000, flattened

Formula

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

Example

The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.
T(5,1) = 1, counting [1,1,3].
T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].
T(8,2) = 1, counting [1,1,2,2,2].
Triangle starts:
1;
1;
1,1;
3;
4,1;
6,1;
8,3.

Maple

G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 30 2016

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)

Crossrefs

Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102.

Sequence in context: A354718 A339964 A342915 * A343226 A030707 A349629

Adjacent sequences: A276430 A276431 A276432 * A276434 A276435 A276436

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 30 2016

Status

approved