cp's OEIS Frontend

A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

%I A168021 #16 Jan 24 2023 10:07:55
%S A168021 1,2,1,3,0,1,5,2,0,1,7,0,0,0,1,11,3,2,0,0,1,15,0,0,0,0,0,1,22,5,0,2,0,
%T A168021 0,0,1,30,0,3,0,0,0,0,0,1,42,7,0,0,2,0,0,0,0,1,56,0,0,0,0,0,0,0,0,0,1,
%U A168021 77,11,5,3,0,2,0,0,0,0,0,1
%N A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.
%C A168021 The row-reversed version is A168016.
%C A168021 Also see A168020.
%H A168021 G. C. Greubel, <a href="/A168021/b168021.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A168021 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D)</a>
%H A168021 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A168021 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A168021 T(n,k) = A000041(n/k) if k|n, else T(n,k)=0.
%F A168021 Sum_{k=1..n} T(n, k) = A047968(n).
%F A168021 From _G. C. Greubel_, Jan 12 2023: (Start)
%F A168021 T(2*n, n) = 2*A000012(n).
%F A168021 T(2*n-1, n+1) = A000007(n-2). (End)
%e A168021 Triangle begins:
%e A168021 ==============================================
%e A168021 ...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12
%e A168021 ==============================================
%e A168021 n=1 ..... 1,
%e A168021 n=2 ..... 2, 1,
%e A168021 n=3 ..... 3, 0, 1,
%e A168021 n=4 ..... 5, 2, 0, 1,
%e A168021 n=5 ..... 7, 0, 0, 0, 1,
%e A168021 n=6 .... 11, 3, 2, 0, 0, 1,
%e A168021 n=7 .... 15, 0, 0, 0, 0, 0, 1,
%e A168021 n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,
%e A168021 n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,
%e A168021 n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,
%e A168021 n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A168021 n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
%e A168021 ...
%t A168021 T[n_, k_]:= If[IntegerQ[n/k], PartitionsP[n/k], 0];
%t A168021 Table[T[n, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jan 12 2023 *)
%o A168021 (SageMath)
%o A168021 def A168021(n,k): return number_of_partitions(n/k) if (n%k)==0 else 0
%o A168021 flatten([[A168021(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jan 12 2023
%Y A168021 Cf. A000007, A000012, A000041, A035377, A035444, A047968 (row sums).
%Y A168021 Cf. A135010, A138121, A168016, A168017, A168018, A168019, A168020.
%K A168021 easy,nonn,tabl
%O A168021 1,2
%A A168021 _Omar E. Pol_, Nov 20 2009, Nov 21 2009
%E A168021 Edited by _Charles R Greathouse IV_, Mar 23 2010