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A060016
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Triangle T(n,k) = number of partitions of n into k distinct parts, 1<=k<=n.
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10
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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
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OFFSET
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1,12
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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].
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FORMULA
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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]
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Flatten[Table[Length[Select[IntegerPartitions[n, {k}], Max[Transpose[ Tally[#]][[2]]]==1&]], {n, 20}, {k, n}]] (* From Harvey P. Dale, Feb 27 2012 *)
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PROG
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(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 */
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CROSSREFS
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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 A079100 A123262
Adjacent sequences: A060013 A060014 A060015 * A060017 A060018 A060019
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KEYWORD
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nonn,tabl,nice,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms, recurrence, etc. from Henry Bottomley, Mar 26 2001
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STATUS
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approved
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