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A090011 T(n,k) = number of partitions of binomial(n,k), 0<=k<=n, triangular array read by rows.

%I A090011 #14 Feb 16 2025 08:32:51
%S A090011 1,1,1,1,2,1,1,3,3,1,1,5,11,5,1,1,7,42,42,7,1,1,11,176,627,176,11,1,1,
%T A090011 15,792,14883,14883,792,15,1,1,22,3718,526823,4087968,526823,3718,22,
%U A090011 1,1,30,17977,26543660,3519222692,3519222692,26543660,17977,30,1,1,42,89134
%N A090011 T(n,k) = number of partitions of binomial(n,k), 0<=k<=n, triangular array read by rows.
%C A090011 a(n) = A000041(A007318(n));
%C A090011 T(n,0) = T(n,n) = 1; T(n,1) = T(n,n-1) = A000041(n), n>0.
%H A090011 Indranil Ghosh, <a href="/A090011/b090011.txt">Rows 0..20, flattened</a>
%H A090011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Partition.html">Partition</a>
%H A090011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient</a>
%e A090011 Triangle begins:
%e A090011 1;
%e A090011 1, 1;
%e A090011 1, 2, 1;
%e A090011 1, 3, 3, 1;
%e A090011 1, 5, 11, 5, 1;
%e A090011 1, 7, 42, 42, 7, 1;
%e A090011 1, 11, 176, 627, 176, 11, 1;
%e A090011 1, 15, 792, 14883, 14883, 792, 15, 1;
%e A090011 1, 22, 3718, 526823, 4087968, 526823, 3718, 22, 1;
%e A090011 1, 30, 17977, 26543660, 3519222692, 3519222692, 26543660, 17977, 30, 1;
%e A090011 1, 42, 89134, 1844349560, 9275102575355, 269232701252579, 9275102575355, 1844349560, 89134, 42, 1; ...
%t A090011 Flatten[Table[PartitionsP[Binomial[n,k]],{n,0,10},{k,0,n}]] (* _Indranil Ghosh_, Feb 21 2017 *)
%o A090011 (PARI) T(n,k)=numbpart(binomial(n,k))
%o A090011 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Jun 14 2013
%Y A090011 Cf. A226659 (row sums), A128855 (central terms).
%K A090011 nonn,tabl
%O A090011 0,5
%A A090011 _Reinhard Zumkeller_, Jan 28 2004
%E A090011 Data section corrected by _Indranil Ghosh_, Feb 21 2017