login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A105306 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having the top of the rightmost column at height k. 7
1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 16, 28, 25, 14, 5, 1, 32, 64, 66, 44, 20, 6, 1, 64, 144, 168, 129, 70, 27, 7, 1, 128, 320, 416, 360, 225, 104, 35, 8, 1, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 512, 1536, 2400, 2528, 1970, 1182, 553, 200, 54, 10, 1, 1024 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

From Gary W. Adamson, Apr 24 2005 (Start):

Let A be the array

1, 0, 0, 0, 0, 0,...

0, 1, 0, 0, 0, 0,...

1, 0, 1, 0, 0, 0,...

0, 2, 0, 1, 0, 0,...

1, 0, 3, 0, 1, 0,...

0, 3, 0, 4, 0, 1,...

...

where columns are bin(n,k) with alternating zeros. (Row sums = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...(Fibonacci numbers)) Let P = infinite lower triangular Pascal triangle matrix (A007318). Form P * A: this gives the rows of the present sequence.. (End)

T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of rightmost peak equal to k. Example: T(4,1)=4 because we have UDUDUDUD, UDUUDDUD, UUDDUDUD and UUUDDDUD, where U=(1,1) and D=(1,-1). Sum of row n = fibonacci(2n-1) (A001519). Basically the same as A062110.

T(n,k) is the number of permutations of [n] with length n-k that avoid the patterns 321 and 3412. - Bridget Tenner, Sep 28 2005

T(2*n-1,n)/n = A001003(n-1) (little Schroeder numbers). Proof with Lagrange inversion of inverse of g.f. of A001003.

Row sums = odd indexed Fibonacci numbers.

Diagonal sums : A077998. [Philippe Deléham, Nov 16 2008]

Central coefficients are A176479. Inverse is A125692. [Paul Barry, Apr 18 2010]

Riordan matrix ((1-x)/(1-2x),(x-x^2)/(1-2x)) [Emanuele Munarini, Mar 22 2011]

T(n,k) is the number of ideals in the fence Z(2n-1) with  n-k elements of rank 0. [Emanuele Munarini, Mar 22 2011]

Triangle T(n,k), 1<=k<=n, read by rows, given by (0,1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011

T(n,k) is the number of permutations of [n] for which k is equal to both the length and reflection length. - Bridget Tenner, Feb 22 2012

REFERENCES

V. E. Hoggatt, Jr. and Marjorie Bicknell, editors: "A Primer for the Fibonacci Numbers", 1970, p. 87.

LINKS

Table of n, a(n) for n=1..67.

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877, 2012, and J. Int. Seq. 15 (2012) #12.9.3

E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

T. K. Petersen and B. E. Tenner, The depth of a permutation, arXiv:1202.4765v1 [math.CO].

M. Pétréolle, Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups, arXiv preprint arXiv:1403.1130 [math.GR], 2014

S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

FORMULA

T(n,k) = sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) (0<=k<=n). (This appears to be incorrect, Emanuele Munarini, Mar 22 2011)

G.f.: t*z*(1-z)/[1-2*z-t*z*(1-z)].

From Emanuele Munarini, Mar 22 2011 (Start)

T(n,k) = sum(binomial(k+1,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i),i,0,n-k)

T(n,k) = sum(binomial(k+1,i)*M(i,n-k-i)*2^(n-k-i),i,0,n-k), where M(n,k)=n*(n+1)*(n+2)*...*(n+k-1)/k!. (End)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 30 2013

EXAMPLE

Triangle begins:

1;

1, 1;

2, 2, 1;

4, 5, 3, 1;

8, 12, 9, 4, 1;

16, 28 25, 14, 5, 1;

32, 64, 66, 44, 20, 6, 1;

64, 144, 168, 129, 70, 27, 7, 1;

...

From Paul Barry, Apr 18 2010: (Start)

Production matrix is

1, 1,

1, 1, 1,

0, 1, 1, 1,

-1, 0, 1, 1, 1,

0, -1, 0, 1, 1, 1,

2, 0, -1, 0, 1, 1, 1,

0, 2, 0, -1, 0, 1, 1, 1,

-5, 0, 2, 0, -1, 0, 1, 1, 1,

0, -5, 0, 2, 0, -1, 0, 1, 1, 1,

14, 0, -5, 0, 2, 0, -1, 0, 1, 1, 1 (End)

MAPLE

T:=proc(n, k) if k<n-1 then sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) elif k=n-1 then n-1 elif k=n then 1 else 0 fi end: for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

t[n_, k_] := 2^(n-2*k-1)*Binomial[n, k]*Hypergeometric2F1[-k-1, -k, -n, -1]; t[n_, n_] = 1; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

PROG

(Maxima) create_list(sum(binomial(k+1, i)*binomial(n-i, k)*(-1)^i*2^(n-k-i), i, 0, n-k), n, 0, 8, k, 0, n); /* Emanuele Munarini, Mar 22 2011 */

CROSSREFS

Cf. A001519. Essentially the same array as A062110.

Row sums = A001519(n-1), n>=1.

Sequence in context: A001404 A104580 A202193 * A183191 A273713 A064189

Adjacent sequences:  A105303 A105304 A105305 * A105307 A105308 A105309

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 25 2005

EXTENSIONS

Entry revised by N. J. A. Sloane, Apr 27 2007

Corrected comment. - Philippe Deléham, Dec 09 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 00:36 EDT 2018. Contains 316327 sequences. (Running on oeis4.)