A027914 T(n,0) + T(n,1) + ... + T(n,n), T given by A027907.

1, 2, 6, 17, 50, 147, 435, 1290, 3834, 11411, 34001, 101400, 302615, 903632, 2699598, 8068257, 24121674, 72137547, 215786649, 645629160, 1932081885, 5782851966, 17311097568, 51828203475, 155188936431, 464732722872

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Let b(n)=a(n) mod 2; then b(n)=1/2+(-1)^n*(1/2-A010060(floor(n/2))). - Benoit Cloitre, Mar 23 2004

Binomial transform of A027306. Inverse binomial transform of = A032443. Hankel transform is {1, 2, 3, 4, ..., n, ...}. - Philippe Deléham, Jul 20 2005

Sums of rows of the triangle in A111808. - Reinhard Zumkeller, Aug 17 2005

Number of 3-ary words of length n in which the number of 1's does not exceed the number of 0's. - David Scambler, Aug 14 2012

The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Moa Apagodu and Ji-Cai Liu, Congruence properties for the trinomial coefficients, arXiv:1907.13547 [math.NT], 2019.
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 24.

Crossrefs

Cf. A025191, A027915, A081673, A092255, A055217, A005773.

Programs

Haskell

a027914 n = sum $ take (n + 1) $ a027907_row n
-- Reinhard Zumkeller, Jan 22 2013

Maple

a := n -> simplify((3^n + GegenbauerC(n,-n,-1/2))/2):
seq(a(n), n=0..25); # Peter Luschny, May 12 2016

Mathematica

CoefficientList[ Series[ (1 + x + Sqrt[1 - 2x - 3x^2])/(2 - 4x - 6x^2), {x, 0, 26}], x] (* Robert G. Wilson v, Jul 21 2015 *)
Table[(3^n + Hypergeometric2F1[1/2 - n/2, -n/2, 1, 4])/2, {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
f[n_] := Plus @@ Take[ CoefficientList[ Sum[x^k, {k, 0, 2}]^n, x], n +1]; Array[f, 26, 0] (* Robert G. Wilson v, Jan 30 2017 *)

PARI

a(n)=sum(i=0,n,polcoeff((1+x+x^2)^n,i,x))

PARI

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,j,if(i+j+k-n,0,(n!/i!/j!/k!)))))

PARI

x='x+O('x^99); Vec((1+x+(1-2*x-3*x^2)^(1/2))/(2*(1-2*x-3*x^2))) \\ Altug Alkan, May 12 2016

Formula

a(n) = ( 3^n + A002426(n) )/2; lim n -> infinity a(n+1)/a(n) = 3; 3^n < 2*a(n) < 3^(n+1). - Benoit Cloitre, Sep 28 2002

From Benoit Cloitre, Jan 26 2003: (Start)

a(n) = (1/2) *( Sum{k = 0..n} binomial(n,k)*binomial(n-k,k) + 3^n );

a(n) = Sum_{k = 0..n} Sum_{i = 0..k} binomial(n,i)*binomial(n-i,k);

a(n) = 3^n/2*(1+c/sqrt(n)+O(n^-1/2)) where c=0.5... (End)

c = sqrt(3/Pi)/2 = 0.4886025119... - Vaclav Kotesovec, May 07 2016

a(n) = n!*Sum(i+j+k=n, 1/(i!*j!*k!)) 0<=i<=n, 0<=k<=j<=n. - Benoit Cloitre, Mar 23 2004

G.f.: (1+x+sqrt(1-2x-3x^2))/(2(1-2x-3x^2)); a(n) = Sum_{k = 0..n} floor((k+2)/2)*Sum_{i = 0..floor((n-k)/2)} C(n,i)*C(n-i,i+k)* ((k+1)/(i+k+1)). - Paul Barry, Sep 23 2005; corrected Jan 20 2008

D-finite with recurrence: n*a(n) +(-5*n+4)*a(n-1) +3*(n-2)*a(n-2) +9*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012

G.f.: (1+x+1/G(0))/(2*(1-2*x-3*x^2)), where G(k)= 1 + x*(2+3*x)*(4*k+1)/(4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2013

From Peter Bala, Jul 21 2015: (Start)

a(n) = [x^n]( 3*x - 1/(1 - x) )^n.

1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + ... is the o.g.f. for A005773. (End)

a(n) = (3^n + GegenbauerC(n,-n,-1/2))/2. - Peter Luschny, May 12 2016