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A365676 Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct part sizes, for 0 <= k <= n.

%I A365676 #37 Mar 28 2025 02:10:45
%S A365676 1,0,1,0,2,0,0,2,1,0,0,3,2,0,0,0,2,5,0,0,0,0,4,6,1,0,0,0,0,2,11,2,0,0,
%T A365676 0,0,0,4,13,5,0,0,0,0,0,0,3,17,10,0,0,0,0,0,0,0,4,22,15,1,0,0,0,0,0,0,
%U A365676 0,2,27,25,2,0,0,0,0,0,0,0,0,6,29,37,5,0,0,0,0,0,0,0,0
%N A365676 Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct part sizes, for 0 <= k <= n.
%H A365676 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
%F A365676 T(n, k) = [t^k][x^n] Product_{j>=1} (1 + t*x^j / (1 - x^j)).
%F A365676 T(n, k) = 0 for n>0 and k=0. T(n,k) = 0 for k > floor([sqrt(1+8n)-1]/2). - _Chai Wah Wu_, Sep 15 2023
%e A365676 Triangle T(n, k) starts:
%e A365676   [0] 1;
%e A365676   [1] 0, 1;
%e A365676   [2] 0, 2,  0;
%e A365676   [3] 0, 2,  1,  0;
%e A365676   [4] 0, 3,  2,  0, 0;
%e A365676   [5] 0, 2,  5,  0, 0, 0;
%e A365676   [6] 0, 4,  6,  1, 0, 0, 0;
%e A365676   [7] 0, 2, 11,  2, 0, 0, 0, 0;
%e A365676   [8] 0, 4, 13,  5, 0, 0, 0, 0, 0;
%e A365676   [9] 0, 3, 17, 10, 0, 0, 0, 0, 0, 0;
%p A365676 P := proc(n, k, r) option remember; local j;  # after Amir Livne Bar-on
%p A365676   if n = 0 then return ifelse(k = 0, 1, 0) fi;
%p A365676   if k = 0 or r = 0 then return 0 fi;
%p A365676   add(P(n - r * j, k - 1, r - 1), j = 1..iquo(n, r)) + P(n, k, r - 1) end:
%p A365676 A365676row := n -> local k; seq(P(n, k, n), k = 0..n):
%p A365676 seq(print(A365676row(n)), n = 0..9);
%p A365676 # Using the generating function:
%p A365676 p := product(1 + t*x^j/(1 - x^j), j = 1..20):
%p A365676 ser := series(p, x, 20):
%p A365676 seq(seq(coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..9);
%t A365676 P[n_, k_, r_] := P[n, k, r] = Which[n == 0, If[k == 0, 1, 0], k == 0 || r == 0, 0, True, Sum[P[n-r*j, k-1, r-1], {j, 1, Quotient[n, r]}]+P[n, k, r-1]]; A365676row[n_] := Table[P[n, k, n], {k, 0, n}]; Table[A365676row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 21 2023, from 1st Maple program *)
%o A365676 (Python) # after Amir Livne Bar-on
%o A365676 from functools import cache
%o A365676 @cache
%o A365676 def P(n: int, k: int, r: int) -> int:
%o A365676     if n == 0: return 1 if k == 0 else 0
%o A365676     if k == 0 or r == 0: return 0
%o A365676     return sum(P(n - r * j, k - 1, r - 1)
%o A365676                for j in range(1, n // r + 1)) + P(n, k, r - 1)
%o A365676 def A365676Row(n) -> list[int]:
%o A365676     return [P(n, k, n) for k in range(n + 1)]
%o A365676 for n in range(10): print(A365676Row(n))
%o A365676 (PARI) T(n,k) = my(nb=0); forpart(p=n, if (#Set(p) == k, nb++)); nb; \\ _Michel Marcus_, Sep 17 2023
%Y A365676 Variants: A116608 (nonzero terms), A060177.
%Y A365676 Cf. A000041 (row sums), A000005 (T(n,1)), A002133 (T(n,2)), A002134 (T(n,3)), A365630 (T(n,4)), A365631 (T(n,5)).
%K A365676 nonn,tabl
%O A365676 0,5
%A A365676 _Peter Luschny_, Sep 15 2023