A168021 - OEIS

%I #16 Jan 24 2023 10:07:55

%S 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 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 77,11,5,3,0,2,0,0,0,0,0,1

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

%C The row-reversed version is A168016.

%C Also see A168020.

%H G. C. Greubel, <a href="/A168021/b168021.txt">Rows n = 1..50 of the triangle, flattened</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>

%F T(n,k) = A000041(n/k) if k|n, else T(n,k)=0.

%F Sum_{k=1..n} T(n, k) = A047968(n).

%F From _G. C. Greubel_, Jan 12 2023: (Start)

%F T(2*n, n) = 2*A000012(n).

%F T(2*n-1, n+1) = A000007(n-2). (End)

%e Triangle begins:

%e ==============================================

%e ...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12

%e ==============================================

%e n=1 ..... 1,

%e n=2 ..... 2, 1,

%e n=3 ..... 3, 0, 1,

%e n=4 ..... 5, 2, 0, 1,

%e n=5 ..... 7, 0, 0, 0, 1,

%e n=6 .... 11, 3, 2, 0, 0, 1,

%e n=7 .... 15, 0, 0, 0, 0, 0, 1,

%e n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,

%e n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,

%e n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,

%e n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

%e n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,

%e ...

%t T[n_, k_]:= If[IntegerQ[n/k], PartitionsP[n/k], 0];

%t Table[T[n, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jan 12 2023 *)

%o (SageMath)

%o def A168021(n,k): return number_of_partitions(n/k) if (n%k)==0 else 0

%o flatten([[A168021(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jan 12 2023

%Y Cf. A000007, A000012, A000041, A035377, A035444, A047968 (row sums).

%Y Cf. A135010, A138121, A168016, A168017, A168018, A168019, A168020.

%K easy,nonn,tabl

%O 1,2

%A _Omar E. Pol_, Nov 20 2009, Nov 21 2009

%E Edited by _Charles R Greathouse IV_, Mar 23 2010