0,3

This sequence counts the "almost triangular" partitions of n. A partition is triangular if it is of the form 0+1+2+...+k. Examples 3=0+1+2, 6=0+1+2+3. An "almost triangular" partition is a triangular partition with at most 1 added to each of the parts. Examples: 7=1+1+2+3=0+2+2+3=0+1+3+3=0+1+2+4. Thus a(7)=4. 8=1+2+2+3=1+1+3+3=1+1+2+4=0+2+3+3=0+2+2+4=0+1+3+4 so a(8)=6. - Moshe Newman (moshneumann(AT)hotmail.com), Dec 19 2002

The "almost triangular" partitions are the ones cycled by the operation of "bulgarian solitaire", as defined by Martin Gardner.

Also the number of increasing acyclic functions from {1,...,n-k+1} to {1,2,...,n+2}. A function f is acyclic if for every subset B of the domain the image of B under f does not equal B. For example, T(3,1)=4 since there are exactly 4 increasing acyclic functions from {1,2,3} to {1,2,3,4,5}: f1={(1,2),(2,3),(3,4)}, f2={(1,2),(2,3),(3,5)}, f3={(1,2),(2,4),(3,5)} and f4={(1,3),(2,4),(4,5)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Mar 14 2008

J. R. Griggs, , bulgarian solitaire partitions

D. P. Walsh, A short note on increasing acyclic functions.

T(n, k)=Sum_{i=0..n} C(i, k) = C(n+1, k).

Row k has g.f. (1+x)^(k+1)-x^(k+1).

T(n, k)=T(n-1, k-1)+T(n-1, k) for k: 0<k<n, T(n, 0)=1, T(n, n)=n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2005

Start with A007318 - I, then delete right border of zeros; (I = Identity matrix). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2007

T(4,2)=10 = 0+0+1+3+6.

{1}, {1,2}, {1,3,3}, {1,4,6,4}, {1,5,10,10,5},...

Triangle begins:

1

1, 2

1, 3, 3

1, 4, 6, 4

1, 5, 10, 10, 5

1, 6, 15, 20, 15, 6

1, 7, 21, 35, 35, 21, 7

1, 8, 28, 56, 70, 56, 28, 8

1, 9, 36, 84, 126, 126, 84, 36, 9

1, 10, 45, 120, 210, 252, 210, 120, 45, 10

1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11

1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12

for n from 0 to 11 do seq(binomial(n+1, k), k =0..n); od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 09 2006

Table[Sum[Binomial[k, m], {k, 0, n}], {n, 0, 12}, {m, 0, n}] or Table[ Binomial[n, m], {n, 1, 12}, {m, 1, n}]

Cf. A007318. Row sums are A000225, diagonal sums are A052952.

Cf. A007318.

The number of acyclic functions is A058127.

Sequence in context: A131251 A057145 A134394 this_sequence A135278 A034356 A075195

Adjacent sequences: A074906 A074907 A074908 this_sequence A074910 A074911 A074912

easy,nonn,tabl

Wouter Meeussen (wouter.meeussen(AT)pandora.be), Oct 01 2002

I added an initial 1 at the suggestion of Paul Barry, which makes the triangle a little nicer but may mean that some of the formulae will now need adjusting. - N. J. A. Sloane (njas(AT)research.att.com), Feb 11 2003