A138774 Triangle read by rows: T(n,k) is the number of partitions of k that fit into a 2n by n box (n>=0; 0<=k<=2n^2).

1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 21, 23, 27, 28, 31, 31, 33, 31, 31, 28, 27, 23, 21, 17, 15, 11, 9, 6, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 36, 45, 53, 63, 72, 83, 92, 103, 111, 121

Offset: 0

Author

Emeric Deutsch, May 03 2008

Keywords

• nonn
• tabf

Comments

Row n contains 1+2n^2 terms.

Sum of entries in row n is binomial(3n,n) (=A005809(n)).

Also, number of n-multisets in [0..2n] whose elements sum up to k. M. F. Hasler, Apr 15 2012

Examples

			T(2,4)=3 because we have 4, 31 and 22.
T(3,13)=5 because we have 661,652,643,553 and 544.
Triangle starts:
1;
1,1,1;
1,1,2,2,3,2,2,1,1;
1,1,2,3,4,5,7,7,8,8,8,7,7,5,4,3,2,1,1;
		

References

• G. E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, 2004, pp. 67-69.

Links

• Alois P. Heinz, Rows n = 0..25, flattened

Crossrefs

Cf. A005809, A063746.

Programs

Maple

br:=proc(n) options operator, arrow: sum(q^i,i=0..n-1) end proc: f:= proc(n) options operator, arrow: mul(br(j),j=1..n) end proc: cbr:=proc(n,k) options operator, arrow: simplify(f(n)/(f(k)*f(n-k))) end proc: for n from 0 to 5 do P[n]:=sort(expand(cbr(3*n,n))) end do: for n from 0 to 5 do seq(coeff(P[n],q, j),j=0..2*n^2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1 or k<1, 0, b(n, i-1, k)+
      `if`(i>n, 0, b(n-i, i, k-1))))
    end:
T:= n-> seq(b(k, min(n, k), 2*n), k=0..2*n^2):
seq(T(n), n=0..6);  # Alois P. Heinz, Apr 05 2012

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1 || k<1, 0, b[n, i-1, k] + If[i>n, 0, b[n-i, i, k-1]]]]; T[n_] := Table[b[k, Min[n, k], 2*n], {k, 0, 2 n^2}]; Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

PARI

T138774(n,k)=polcoeff(prod(i=0,2*n,sum(j=0,n,x^(j*(i*(2*n^2+n+1)+1)),O(x^(k*(2*n^2+n+1)+n+1)))),k*(2*n^2+n+1)+n) /* replacing the inner sum by the expression for the geometric series seems less efficient */
for(n=0,5,for(k=0,2*n^2,print1(T138774(n,k)","))) \\ M. F. Hasler, Apr 15 2012

Formula

G.f. of row n = the q-binomial coefficient [3n,n].