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A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1<=k<=n. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

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

A025147(n) = Sum(a(n-k+1,k-1): 1<k<=floor((n+2)/2). - Reinhard Zumkeller, Nov 04 2007

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].

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) / prod(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]

EXAMPLE

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.

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 *)

PROG

(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 */

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.

Sequence in context: A089605 A218786 A218787 * A117408 A228360 A244966

Adjacent sequences:  A060013 A060014 A060015 * A060017 A060018 A060019

KEYWORD

nonn,tabl,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

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Last modified December 22 22:37 EST 2014. Contains 252372 sequences.