login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A026082 Irregular triangular array T read by rows: T(n,k) = C(n,k) for k=0..n for n = 0,1,2,3. For n >= 4, T(n,0) = T(n,2n)=1, T(n,1) = T(n,2n-1) = n - 3, T(4,2) = 4, T(4,3) = 3, T(4,4) = 6; T(4,5) = 3, T(4,6)=4; for n >= 5, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k=2..2n-2. 21
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 2, 6, 8, 13, 12, 13, 8, 6, 2, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278, 300, 278, 226, 156, 93, 45, 18, 5, 1, 1, 6, 24, 68, 156, 294, 475, 660, 804 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

For n >= 4, T(n,k) = number of strings s(0)..s(n) such that s(n) = n - k, s(0) = 0, |s(i)-s(i-1)| = 1 for i=1,2,3 and |s(i)-s(i-1)| <= 1 for i >= 4.

LINKS

Clark Kimberling, Rows 0..100, flattened

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

FORMULA

G.f.: (1-y*z)^3 / (1-z*(1+y+y^2)).

EXAMPLE

First 6 rows:

1

1 1

1 2 1

1 3 3 1

1 1 4 3 6 3 4 1 1

1 2 6 8 12 12 13 8 6 2 1

MAPLE

A026082 := proc(n, k)

option remember;

if n < 0 or k < 0 or k > 2*n then

0 ;

elif n <= 3 then

binomial(n, k) ;

elif n = 4 then

op(k+1, [1, 1, 4, 3, 6, 3, 4, 1, 1]) ;

elif k =0 or k=2*n then

1 ;

else

procname(n-1, k-2)+procname(n-1, k-1)+procname(n-1, k) ;

end if;

end proc: # R. J. Mathar, Jun 23 2013

MATHEMATICA

z = 15; t[n_, 0] := 1 /; n >= 4; t[n_, 1] := n - 3 /; n >= 4;

t[4, 2] = 4; t[4, 3] = 3; t[4, 4] = 6; t[4, 5] = 3; t[4, 6] = 4;

t[n_, k_] := t[n, k] = Which[0 <= k <= n && 0 <= n <= 3, Binomial[n, k], n

>= 4 && k == 2 n, 1, k == 2 n - 1, n - 3, 2 <= k <= 2 n - 2, t[n - 1, k -

2] + t[n - 1, k - 1] + t[n - 1, k]]; s = Table[Binomial[n, k], {n, 0, 3},

{k, 0, n}]; u = Join[s, Table[t[n, k], {n, 4, z}, {k, 0, 2 n}]];

TableForm[u] (* A026082 array *)

Flatten[u] (* A026082 sequence *)

CROSSREFS

First differences of A024996.

Sequence in context: A307116 A212626 A090402 * A117185 A129181 A157694

Adjacent sequences: A026079 A026080 A026081 * A026083 A026084 A026085

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling

EXTENSIONS

Updated by Clark Kimberling, Aug 28 2014

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 17:25 EST 2022. Contains 358668 sequences. (Running on oeis4.)