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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A156197 T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows. 0
2, 2, 2, 3, 4, 3, 6, 8, 8, 6, 15, 18, 18, 18, 15, 43, 47, 42, 42, 47, 43, 133, 138, 110, 96, 110, 138, 133, 430, 436, 324, 240, 240, 324, 436, 430, 1431, 1438, 1036, 682, 550, 682, 1036, 1438, 1431, 4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388.
FORMULA
T(n,k) = -binomial(k + n, -1 + k) + binomial(k + n, n) + binomial(-k + 2*n, n] - binomial(-k + 2*n, -1 - k + n).
From Roger L. Bagula and Gary W. Adamson, Dec 03 2009: (Start)
T(n,k) = ((n - k + 1)*binomial(n + k, n) + (k + 1)*binomial(-k + 2*n, n))/(n + 1).
T(n,k) = A009766(n,k) + A033184(n,k). (End)
G.f.: (C(t*x) + C(x)*(1 - x*C(t*x) - t*x*C(t*x)))/((1 - t*x*C(x))*(1 - x*C(t*x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x). - Franck Maminirina Ramaharo, Dec 11 2018
EXAMPLE
Triangle begins:
2;
2, 2;
3, 4, 3;
6, 8, 8, 6;
15, 18, 18, 18, 15;
43, 47, 42, 42, 47, 43;
133, 138, 110, 96, 110, 138, 133;
430, 436, 324, 240, 240, 324, 436, 430;
1431, 1438, 1036, 682, 550, 682, 1036, 1438, 1431;
4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863;
...
MATHEMATICA
t0[n_, m_] = Binomial[n + m, n] - Binomial[n + m, m - 1];
T[n_, m_] = FullSimplify[t0[n, m] + t0[n, n - m]];
Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}] // Flatten
(* or *)
Table[Table[((1 - k + n)*Binomial[k + n, n] + (1 + k)*Binomial[-k + 2*n, n])/(1 + n), {k, 0, n}], {n, 0, 10}] // Flatten (* Roger L. Bagula and Gary W. Adamson, Dec 03 2009 *)
PROG
(Maxima) A009766(n, k) := binomial(n + k, n)*(n - k + 1)/(n + 1)$
create_list(A009766(n, k) + A009766(n, n - k), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Dec 11 2018 */
CROSSREFS
Row sums: 2*A000108 (without the first term).
Sequence in context: A098745 A029158 A241953 * A155162 A275444 A243322
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Feb 05 2009
EXTENSIONS
Edited by Franck Maminirina Ramaharo, Dec 11 2018
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 June 6 14:05 EDT 2023. Contains 363147 sequences. (Running on oeis4.)