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A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

%I A168016 #11 Jan 13 2023 03:28:36
%S A168016 1,1,2,1,0,3,1,0,2,5,1,0,0,0,7,1,0,0,2,3,11,1,0,0,0,0,0,15,1,0,0,0,2,
%T A168016 0,5,22,1,0,0,0,0,0,3,0,30,1,0,0,0,0,2,0,0,7,42,1,0,0,0,0,0,0,0,0,0,
%U A168016 56,1,0,0,0,0,0,2,0,3,5,11,77,1,0,0,0,0,0,0,0,0,0,0,0,101
%N A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.
%H A168016 G. C. Greubel, <a href="/A168016/b168016.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A168016 T(n, k) = A000041(n/k) if k|n; otherwise T(n,k) = 0.
%F A168016 T(n, n) = A000041(n).
%F A168016 From _G. C. Greubel_, Jan 12 2023: (Start)
%F A168016 T(2*n, n) = A000007(n-1).
%F A168016 Sum_{k=1..n} T(n, k) = A047968(n).
%F A168016 Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)
%e A168016 Triangle begins:
%e A168016 ==============================================
%e A168016 .... k: 12 11 10. 9. 8. 7. 6. 5. 4. 3.. 2.. 1.
%e A168016 ==============================================
%e A168016 n=1 ....................................... 1,
%e A168016 n=2 ................................... 1,  2,
%e A168016 n=3 ............................... 1,  0,  3,
%e A168016 n=4 ............................ 1, 0,  2,  5,
%e A168016 n=5 ......................... 1, 0, 0,  0,  7,
%e A168016 n=6 ...................... 1, 0, 0, 2,  3, 11,
%e A168016 n=7 ................... 1, 0, 0, 0, 0,  0, 15,
%e A168016 n=8 ................ 1, 0, 0, 0, 2, 0,  5, 22,
%e A168016 n=9 ............. 1, 0, 0, 0, 0, 0, 3,  0, 30,
%e A168016 n=10 ......... 1, 0, 0, 0, 0, 2, 0, 0,  7, 42,
%e A168016 n=11 ...... 1, 0, 0, 0, 0, 0, 0, 0, 0,  0, 56,
%e A168016 n=12 ... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77,
%e A168016 ...
%t A168016 T[n_, k_]:= If[IntegerQ[n/(n-k+1)], PartitionsP[n/(n-k+1)], 0];
%t A168016 Table[T[n, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jan 12 2023 *)
%o A168016 (SageMath)
%o A168016 def T(n,k): return number_of_partitions(n/(n-k+1)) if (n%(n-k+1))==0 else 0
%o A168016 flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jan 12 2023
%Y A168016 Cf. A000005, A000007, A000041, A035363, A035444, A047968, A135010.
%Y A168016 Cf. A138121, A168014, A168015, A168020, A168021, A168111.
%K A168016 easy,nonn,tabl
%O A168016 1,3
%A A168016 _Omar E. Pol_, Nov 21 2009
%E A168016 Edited and extended by _Max Alekseyev_, May 07 2010