A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n.

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy

Comments

Also number of partitions of n-k(k+1)/2 into at most k parts (not necessarily distinct).

A025147(n) = Sum_{k=2..floor((n+2)/2)} a(n-k+1, k-1). - Reinhard Zumkeller, Nov 04 2007

Examples

			Triangle starts
[ 1]  1,
[ 2]  1, 0,
[ 3]  1, 1, 0,
[ 4]  1, 1, 0, 0,
[ 5]  1, 2, 0, 0, 0,
[ 6]  1, 2, 1, 0, 0, 0,
[ 7]  1, 3, 1, 0, 0, 0, 0,
[ 8]  1, 3, 2, 0, 0, 0, 0, 0,
[ 9]  1, 4, 3, 0, 0, 0, 0, 0, 0,
[10]  1, 4, 4, 1, 0, 0, 0, 0, 0, 0,
[11]  1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0,
[12]  1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0,
[13]  1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
[14]  1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, ...
T(8,3)=2 since 8 can be written in 2 ways as the sum of 3 distinct positive integers: 5+2+1 and 4+3+1.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Columns (offset) include A057427, A004526, A001399, A001400, A001401, etc. Cf. A000009 (row sums), A008289 (without zeros), A030699 (row maximum), A008284 (partition triangle including duplications).

See A008289 for another version.

Programs

Maple

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
T:= proc(n) local l; l:= subsop(1=NULL, b(n, n));
      l[], 0$(n-nops(l))
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Dec 12 2012

Mathematica

Flatten[Table[Length[Select[IntegerPartitions[n,{k}],Max[Transpose[ Tally[#]][[2]]]==1&]],{n,20},{k,n}]] (* Harvey P. Dale, Feb 27 2012 *)
T[, 1] = 1; T[n, k_] /; 1, ] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 26 2015 *)

PARI

N=16;  q='q+O('q^N);
gf=sum(n=0,N, z^n * q^((n^2+n)/2) / prod(k=1,n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf,'q);
{ for (n=1,N-1,
    p = Pol(polcoeff(P, n),'z);
    p += 'z^(n+1);  /* preserve trailing zeros */
    v = Vec(polrecip(p));
    v = vector(n,k,v[k]); /* trim to size n */
    print(v);
); }
/* Joerg Arndt, Oct 20 2012 */

Formula

T(n, k) = T(n-k, k) + T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise].

G.f.: Sum_{n>=0} z^n * q^((n^2+n)/2) / Product_{k=1..n} (1-q^k), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A000009; cf. to g.f. for A072574. - Joerg Arndt, Oct 20 2012

Extensions

More terms, recurrence, etc. from Henry Bottomley, Mar 26 2001