A028270 Central elements in 3-Pascal triangle A028262 (by row).

1, 3, 8, 26, 90, 322, 1176, 4356, 16302, 61490, 233376, 890188, 3409588, 13104756, 50517200, 195234120, 756197910, 2934686610, 11408741520, 44420399100, 173191792620, 676104403260, 2642356838160, 10337529691320, 40481034410700

Offset

0,2

Comments

Or, start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central triangle containing exactly 3 numbers. The first three triangles are

...1...........2.........6

.1...1.......3...3.....10..10

and the corresponding sums are 3, 8 and 26. - Amarnath Murthy, Mar 25 2003

This sequence starting at a(n+2) has Hankel transform A000032(2n+1)*2^n (empirical observation). - Tony Foster III, May 20 2016

Links

Table of n, a(n) for n=0..24.

Formula

G.f.: (x+1)/sqrt(1-4*x). - Vladeta Jovovic, Jan 08 2004

a(n) = binomial(2n, n)+binomial(2n-2, n-1)=A000984(n)+A000984(n-1). - Emeric Deutsch, Apr 20 2004

a(n) = 2binomial(2n-1, n-1)+binomial(2n-2, n-1). - Emeric Deutsch, Apr 20 2004

a(n) = (n+1)*C(n) + n*C(n-1), C = Catalan number (A000108). - Gary W. Adamson, Dec 28 2007

G.f.: G(0) where G(k)= 1 + x/(1 - (4*k+2)/((4*k+2) + (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012

D-finite with recurrence n*a(n) -3*n*a(n-1) +2*(-2*n+5)*a(n-2)=0. - R. J. Mathar, May 01 2024

Maple

seq(binomial(2*n, n)+binomial(2*n-2, n-1), n=0..24);
seq(2*binomial(2*n-1, n-1)+binomial(2*n-2, n-1), n=1..24);

Crossrefs

Cf. A081494, A081495, A081496, A000984.

Cf. A000108.

Sequence in context: A148818 A242903 A081497 * A124383 A148819 A148820

Adjacent sequences: A028267 A028268 A028269 * A028271 A028272 A028273

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from James A. Sellers

Status

approved