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A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n. 17
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_{k=2..floor((n+2)/2)} a(n-k+1, k-1). - 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
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) / 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
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 *)
T[_, 1] = 1; T[n_, k_] /; 1<k<n := T[n, k] = T[n-k, k]+T[n-k, k-1]; T[_, _] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 26 2015 *)
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: A218786 A218787 A325336 * A117408 A228360 A303138
KEYWORD
nonn,tabl,nice,easy
AUTHOR
EXTENSIONS
More terms, recurrence, etc. from Henry Bottomley, Mar 26 2001
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)