A276427 Irregular triangle read by rows: T(n,k) = number of partitions of n having k distinct parts i of multiplicity i; 0 <= k <= A328806(n)-1 = largest index of a nonzero value; n >= 0.

1, 0, 1, 2, 2, 1, 3, 2, 5, 1, 1, 8, 3, 9, 6, 16, 5, 1, 19, 10, 1, 29, 11, 2, 36, 18, 2, 53, 21, 3, 65, 32, 4, 92, 38, 4, 1, 115, 54, 7, 154, 67, 10, 195, 88, 14, 257, 112, 15, 1, 318, 152, 19, 1, 419, 178, 29, 1, 516, 243, 31, 2, 663, 293, 44, 2, 821, 376, 56, 2, 1039, 465, 67, 4, 1277, 589, 89, 3, 1606, 715, 108, 7

Offset: 0

Author

Emeric Deutsch, Sep 19 2016

Keywords

• nonn
• tabf

Comments

The sum of entries in row n is A000041(n): the partition numbers. [This allows us to know the row length, i.e., when the largest value of k is reached for which T(n,k) is nonzero. The row lengths are now listed as A328806. - M. F. Hasler, Oct 28 2019]

Examples

			Triangle starts:
1;       (n=0: partition [] has k=0 parts i of multiplicity i: T(0,0) = 1.)
0, 1;    (n=1: partition [1] has k=1 part i of multiplicity i: T(1,1) = 1.)
2;       (n=2: partitions [1,1] and [2] have k=0 parts i occurring i times.)
2, 1;    (n=3: [1,1,1] and [3] have 0, [1,2] has 1 part i occurring i times)
3, 2;    (n=4: [4], [1,1,2] and [1,1,1,1] for k=0; [1,3] & [2,2] for k=1.)
5, 1, 1; (n=5: [1,4] has i=1, [1,2,2] has i=1 and i=2 occurring i times.)
(...)
The partition [1,2,3,3,3,4] has 2 parts i of multiplicity i: i=1 and i=3.
T(14,3) = 1, since [1,2,2,3,3,3] is the only partition of 14 having k=3 parts i with multiplicity i, namely i = 1, 2 and 3.
T(14,2) = 4, counting [1,2,2,3,6], [1,2,2,4,5], [1,2,2,9] (with i=1 and i=2), and [1,3,3,3,4] (with i=1 and i=3).
		

Links

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

Crossrefs

Cf. A000041 (row sums), A276428, A276429, A328806 (row lengths).

Programs

Maple

G := mul((t-1)*x^(i^2)+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=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 19 2016

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i==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, Oct 20 2016, after Alois P. Heinz *)

PARI

apply( A276427_row(n, r=List(0))={forpart(p=n, my(s, c=1); for(i=1, #p, p[i]==if(i<#p, p[i+1]) && c++ && next; c==p[i] && s++; c=1); while(#r<=s, listput(r,0)); r[s+1]++);Vec(r)}, [0..20]) \\ M. F. Hasler, Oct 27 2019

Formula

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^{i^2} + 1/(1-x^i)).

T(n,0) = A276429(n).

Sum(k*T(n,k), k>=0) = A276428(n).

Extensions

Name edited by M. F. Hasler, Oct 27 2019