A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

1, 1, 1, 1, 2, 1, 4, 1, 5, 2, 7, 4, 10, 5, 15, 6, 1, 21, 8, 1, 28, 13, 1, 37, 18, 1, 50, 25, 2, 67, 31, 3, 88, 42, 5, 115, 55, 6, 150, 73, 8, 193, 93, 11, 248, 122, 15, 317, 154, 19, 402, 200, 24, 1, 508, 253, 30, 1, 640, 320, 41, 1, 802, 399, 53, 1, 1002, 503, 69, 1

Offset

0,5

Comments

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

T(n,0) = A277102(n).

Sum(k*T(n,k), k>=0) = A277101(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+1))).

Example

The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4.
T(5,1) = 2, counting [1,1,1,2] and [2,3].
T(8,2) = 1, counting [2,3,3].
Triangle starts:
1;
1;
1, 1;
2, 1;
4, 1;
5, 2;
7, 4;
...

Maple

g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): 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, Oct 10 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, Dec 08 2016 after Alois P. Heinz *)

Crossrefs

Cf. A000041, A116608, A276427, A276428, A276429, A276433, A276434, A277099, A277101, A277102.

Sequence in context: A325609 A006306 A322100 * A337363 A339243 A214579

Adjacent sequences: A277097 A277098 A277099 * A277101 A277102 A277103

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 10 2016

Status

approved