|
| |
|
|
A090181
|
|
Triangle of Narayana (A001263) with 0<=k<=n, read by rows.
|
|
15
| |
|
|
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,9
|
|
|
COMMENTS
| Row sums : A000108 (Catalan). Columns give : A000217, A002415, A006542, A006857 row n=0 : 1 row n=1 : 0, 1 row n=2 : 0, 1, 1, row n=3 : 0, 1, 3, 1 row n=4 : 0, 1, 6, 6, 1 row n=5 : 0, 1, 10, 20, 10, 1 row n=6 : 0, 1, 15, 50, 50, 15, 1
Coefficient array of the polynomials P(n,x)=x^n*2F1(-n,-n+1;2;1/x). [From Paul Barry (pbarry(AT)wit.ie), Nov 10 2008]
Mirror image of triangle A131198 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2008]
|
|
|
REFERENCES
| P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, Arxiv preprint arXiv:1107.5490, 2011.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
|
|
|
FORMULA
| Triangle T(n, k), read by rows, given by : [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938 . T(0, 0) = 1, T(n, 0) = 0 for n>0, T(n, k) = C(n-1, k-1)*C(n, k-1)/k for k>0.
Sum_{k, 0<=k<=n}T(n,k)*x^k=A000007(n),A000108(n),A006318(n),A047891(n+1),A082298(n),A082301(n),A082302(n),A082305(n),A082366(n),A082367(n)for x=0,1,2,3,4,5,6,7,8,9 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 10 2006
Sum_{k, 0<=k<=n}x^(n-k)*T(n,k)=A090192(n+1), A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 21 2006
Sum_[j, j>=0}T(n,j)*binomial(j,k) = A060693(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 04 2007
Sum_{k, 0<=k<=n} T(n,k)*x^k*(x-1)^(n-k) = A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2007
Sum_{k, 0<=k<=n}T(n,k)*10^k = A143749(n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 14 2008]
T(n,k)=sum{j=0..n, (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j)}. [From Paul Barry (pbarry(AT)wit.ie), Nov 10 2008]
Sum_{k, 0<=k<=n}T(n,k)*5^k*3^(n-k) = A152601(n) . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2008]
Sum{k=0..n, T(n,k)*(-2)^k} = A152681(n); sum{k=0..n, T(n,k)*(-1)^k} = A105523(n) . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 03 2009]
Sum_{k, 0<=k<=n} T(n,k)*2^(n+k) = A156017(n). - DELEHAM Philippe, Nov 27 2011
|
|
|
EXAMPLE
| Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 10 2008: (Start)
Triangle begins
1,
0, 1,
0, 1, 1,
0, 1, 3, 1,
0, 1, 6, 6, 1,
0, 1, 10, 20, 10, 1,
0, 1, 15, 50, 50, 15, 1 (End)
|
|
|
CROSSREFS
| Cf. A001263 A084938.
Sequence in context: A059045 A122935 A131198 * A144417 A085791 A144645
Adjacent sequences: A090178 A090179 A090180 * A090182 A090183 A090184
|
|
|
KEYWORD
| easy,nonn,tabl,changed
|
|
|
AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 19 2004
|
| |
|
|
