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

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

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

Crossrefs

Cf. A081494, A081495, A081496, A000984.

Cf. A000108.

Programs

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);

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

Extensions

More terms from James Sellers