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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.
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%I #11 Jan 13 2023 03:28:36

%S 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 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 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 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 G. C. Greubel, <a href="/A168016/b168016.txt">Rows n = 1..50 of the triangle, flattened</a>

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

%F T(n, n) = A000041(n).

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

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

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

%F Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)

%e Triangle begins:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%e ...

%t T[n_, k_]:= If[IntegerQ[n/(n-k+1)], PartitionsP[n/(n-k+1)], 0];

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

%o (SageMath)

%o def T(n,k): return number_of_partitions(n/(n-k+1)) if (n%(n-k+1))==0 else 0

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

%Y Cf. A000005, A000007, A000041, A035363, A035444, A047968, A135010.

%Y Cf. A138121, A168014, A168015, A168020, A168021, A168111.

%K easy,nonn,tabl

%O 1,3

%A _Omar E. Pol_, Nov 21 2009

%E Edited and extended by _Max Alekseyev_, May 07 2010