A128855 -id:A128855 - cp's OEIS frontend

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

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, 15, 792, 14883, 14883, 792, 15, 1, 1, 22, 3718, 526823, 4087968, 526823, 3718, 22, 1, 1, 30, 17977, 26543660, 3519222692, 3519222692, 26543660, 17977, 30, 1, 1, 42, 89134

Offset: 0

Reinhard Zumkeller, Jan 28 2004

a(n) = A000041(A007318(n));

T(n,0) = T(n,n) = 1; T(n,1) = T(n,n-1) = A000041(n), n>0.

			Triangle begins:
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, 15, 792, 14883, 14883, 792, 15, 1;
1, 22, 3718, 526823, 4087968, 526823, 3718, 22, 1;
1, 30, 17977, 26543660, 3519222692, 3519222692, 26543660, 17977, 30, 1;
1, 42, 89134, 1844349560, 9275102575355, 269232701252579, 9275102575355, 1844349560, 89134, 42, 1; ...

Indranil Ghosh, Rows 0..20, flattened
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Binomial Coefficient

Cf. A226659 (row sums), A128855 (central terms).

Flatten[Table[PartitionsP[Binomial[n,k]],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Feb 21 2017 *)

T(n,k)=numbpart(binomial(n,k))
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jun 14 2013

Data section corrected by Indranil Ghosh, Feb 21 2017

A226659 Sum_{k=0..n} A000041( binomial(n,k) ), where A000041(n) is the number of partitions of n.

Original entry on oeis.org

1, 2, 4, 8, 23, 100, 1003, 31382, 5149096, 7091568720, 287786595280763, 539018517346414192796, 1130813038175196801809538188145, 2336855300714703790840987155549462486654700, 7636154577344556445476348286247799105605643795614728449082014

Offset: 0

Paul D. Hanna, Jun 14 2013

nonn

Compare to the number of partitions of 2^n (A068413).

			Equals the row sums of triangle A090011, which begins:
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, 15, 792, 14883, 14883, 792, 15, 1;
1, 22, 3718, 526823, 4087968, 526823, 3718, 22, 1; ...

Indranil Ghosh, Table of n, a(n) for n = 0..20

Cf. A090011, A068413, A128855, A000041.

Table[Sum[PartitionsP[Binomial[n,k]],{k,0,n}],{n,0,20}] (* Indranil Ghosh, Feb 21 2017 *)

{a(n)=sum(k=0,n,numbpart(binomial(n,k)))}
for(n=0,15,print1(a(n),", "))

Row sums of triangle A090011.

cp's OEIS Frontend

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

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