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

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059450 Triangle read by rows: T(n,k) = Sum_{j=0..k-1} T(n,j) + Sum_{j=1..n-k} T(n-j,k), with T(0,0)=1 and T(n,k) = 0 for k>n. 3
1, 1, 1, 2, 3, 5, 4, 8, 17, 29, 8, 20, 50, 107, 185, 16, 48, 136, 336, 721, 1257, 32, 112, 352, 968, 2370, 5091, 8925, 64, 256, 880, 2640, 7116, 17304, 37185, 65445, 128, 576, 2144, 6928, 20168, 53596, 129650, 278635, 491825, 256, 1280, 5120, 17664, 54880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

G.f. A(x,y) satisfies 0=-(1-x)^2+(1-x)(1-4x+3xy)A +2x(1-2x-2y+3xy)A^2. G.f.: (1-x)(-(1-4x+3xy)+sqrt((1-xy)(1-9xy)))/(4x(1-2x-2y+3xy))=2(1-x)/(1-4x+3xy+sqrt((1-xy)(1-9xy))). - Michael Somos, Mar 06 2004

T(n,k)=number of below-diagonal lattice paths from (0,0) to (n,k) consisting of steps (k,0) (k=1,2,...) and (0,k) (k=1,2,...). Example: T(2,1)=3 because we have (1,0)(1,0)(0,1), (2,0)(0,1) and (1,0)(0,1)(1,0). - Emeric Deutsch, Mar 19 2004

REFERENCES

C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

Wen-jin Woan, Diagonal lattice paths, Congressus Numerantium, 151, 2001, 173-178.

LINKS

Table of n, a(n) for n=0..49.

EXAMPLE

1; 1,2; 2,3,5; 4,8,17,29; 8,20,50,107,185; ...

MAPLE

l := 1:a[0, 0] := 1:b[l] := 1:T := (n, k)->sum(a[n, j], j=0..k-1)+sum(a[n-j, k], j=1..n-k): for n from 1 to 15 do for k from 0 to n do a[n, k] := T(n, k):l := l+1:b[l] := a[n, k]: od:od:seq(b[w], w=1..l); - Sascha Kurz

MATHEMATICA

t[0, 0] = 1; t[n_, k_] /; k > n = 0; t[n_, k_] := t[n, k] = Sum[t[n, j], {j, 0, k-1}] + Sum[t[n-j, k], {j, 1, n-k}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 08 2014 *)

PROG

(PARI) T(n, k)=if(k<0|k>n, 0, polcoeff(polcoeff(2*(1-x)/((1-4*x+3*x*y)+sqrt((1-x*y)*(1-9*x*y)+x^2*O(x^n))), n), k)) /* Michael Somos, Mar 06 2004 */

(PARI) T(n, k)=local(A, t); if(k<0|k>n, 0, A=matrix(n+1, n+1); A[1, 1]=1; for(m=1, n, t=0; for(j=0, m, t+=(A[m+1, j+1]=t+sum(i=1, m-j, A[m-i+1, j+1])))); A[n+1, k+1]) /* Michael Somos, Mar 06 2004 */

(PARI) T(n, k)=if(k<0|k>n, 0, (n==0)+sum(j=0, k-1, T(n, j))+sum(j=1, n-k, T(n-j, k))) /* Michael Somos, Mar 06 2004 */

CROSSREFS

Columns include A000079, A001792 (I guess), A086866, A059231. Rows sums give A086871.

A059231(n)=T(n, n).

Sequence in context: A244154 A182395 A244983 * A060000 A074050 A075301

Adjacent sequences:  A059447 A059448 A059449 * A059451 A059452 A059453

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, Sep 16 2003

EXTENSIONS

More terms from Ray Chandler, Sep 17 2003

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified November 28 20:22 EST 2014. Contains 250399 sequences.