0,5

Also square array T(n,k) (n >= 0, k >= 0) read by antidiagonals: T(n,k) = Sum_{i=0..k} C(n,i).

As a number triangle read by rows, this is T(n,k)=sum{i=n-2k..n-k, binomial(n-k,i)}, with T(n,k)=T(n-1,k)+T(n-2,k-1). Row sums are A000071(n+2). Diagonal sums are A023435(n+1). It is the reverse of the Whitney triangle A004070. - Paul Barry (pbarry(AT)wit.ie), Sep 04 2005

Index entries for triangles and arrays related to Pascal's triangle

T(n, m)= Sum( k=0..n, C(n-m, m-k) ) - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Oct 03 2002

Rows: {1}, {1,1}, {1,2,1}, {1,3,2,1}, {1,4,4,2,1}, ...

Triangle begins:

1

1,1

1,2,1

1,3,2,1

1,4,4,2,1

1,5,7,4,2,1

1,6,11,8,4,2,1

As a square array, this begins:

1 1 1 1 1 1 ...

1 2 2 2 2 2 ...

1 3 4 4 4 4 ...

1 4 7 8 8 8 ...

1 5 11 15 16 ...

1 6 16 26 31 32 ...

a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) fi:

if k=n then RETURN(1) fi: a(n-1, k)+a(n-2, k-1) end:for n from 0 to 20 do

for k from 0 to n do printf(`%d, `, a(n, k)) od:od:

with(combinat): for s from 0 to 20 do for n from s to 0 by -1 do if n=0 or s-n=0 then printf(`%d, `, 1) else printf(`%d, `, sum(binomial(n, i), i=0..s-n)) fi; od:od: - James A. Sellers (sellersj(AT)math.psu.edu), Mar 17 2000

Table[Sum[Binomial[n-m, m-k], {k, 0, n}], {n, 0, 10}, {m, 0, n}]

Cf. A054123, A054124, A007318, A008949.

Row sums = Fibonacci numbers - 1.

Columns give A000027, A000124, A000125, A000129, A006261, ...

Cf. A052509, A054123, A054124, A007318, A008949, A052553.

Partial sums across rows of (extended) Pascal's triangle A052553.

Sequence in context: A055794 A092905 A052511 this_sequence A093628 A114282 A112739

Adjacent sequences: A052506 A052507 A052508 this_sequence A052510 A052511 A052512

nonn,tabl,easy,nice

njas, Mar 17, 2000

More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Mar 17 2000

Entry formed by merging two earlier entries. Formulae probably need editing. - njas, Jun 17 2007