A097893 Partial sums of the central trinomial coefficients (A002426).

1, 2, 5, 12, 31, 82, 223, 616, 1723, 4862, 13815, 39468, 113257, 326198, 942425, 2730032, 7926659, 23061590, 67214399, 196211252, 573590621, 1678941350, 4920076877, 14433305000, 42381641381, 124558477682, 366371703833

Offset: 0

Author

Emeric Deutsch, Sep 03 2004

Keywords

• nonn
• easy

Comments

a(n) is the number of peaks at odd height in all Motzkin paths of length n+2. Example: a(2)=5 counts the peaks shown between parentheses in the 9 Motzkin paths of length 4: HHHH, HH(UD), H(UD)H, HUHD, (UD)HH, (UD)(UD), UHDH, UHHD and UUDD.

Binomial transform of 1,1,2,2,6,6,20,20,70,70...... (A000984 doubled). It would appear that the Hankel transform of this sequence is a signed version of A128055, with sign pattern given by s(n)=(2/3-sqrt(3)/3)cos(5*Pi*n/6)-sin(5*Pi*n/6)/3+(sqrt(3)/3+2/3)*cos(Pi*n/6)-sin(Pi*n/6)/3-cos(Pi*n/2)/3+sin(Pi*n/2)/3. - Paul Barry, Jan 03 2008

Define triangle T(n,1) = T(n,n) = 1 and T(r,c) = T(r,c-1) + T(r-1,c-1) + T(r-2,c-1). Then the sum of the terms in row(n) is a(n+1). - J. M. Bergot, Apr 30 2013

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
• Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
• Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 14.
• László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
• 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. 25.

Crossrefs

Cf. A002426, A025178.

Programs

Haskell

a097893 n = a097893_list !! n
a097893_list = scanl1 (+) a002426_list
-- Reinhard Zumkeller, Jan 22 2013

Maple

ser:=series(1/(1-z)/sqrt(1-2*z-3*z^2),z=0,32): 1,seq(coeff(ser,z^n),n=1..31);
a := n -> (n+1)*hypergeom([1/2,(1-n)/2,-n/2],[1,3/2],4):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Oct 29 2015

Mathematica

Table[ Sum[ Binomial[n, k]*Binomial[k, n-k], {k, 0, n}], {n, 0, 26}] // Accumulate (* Jean-François Alcover, Jul 10 2013 *)
CoefficientList[Series[1/((1-x)*Sqrt[1-2*x-3*x^2]), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)

PARI

a(n)=sum(i=0,n,sum(j=0,i,binomial(i,i-j)*binomial(j,i-j)))

PARI

vector(30, n, n--; sum(k=0, n\2, binomial(n+1, 2*k+1)* binomial(2*k, k))) \\ Altug Alkan, Oct 29 2015

PARI

x='x+O('x^30); Vec(1/((1-x)*sqrt(1-2*x-3*x^2))) \\ G. C. Greubel, Dec 21 2017

Python

from math import comb
def A097893(n): return sum(comb(n+1,(k<<1)|1)*comb(k<<1,k) for k in range((n>>1)+1)) # Chai Wah Wu, Aug 14 2025

Formula

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

a(n) = Sum_{0<=j<=i<=n} C(i, i-j)*C(j, i-j). - Benoit Cloitre, Oct 23 2004

a(n) = sum_{k=0..n} Sum_{j=0..n-k} C(k,j)C(n-k,j)C(2j,j). - Paul Barry, Jan 03 2008

Logarithm g.f. atan(x*M(x)), M(x) - o.g.f. for Motzkin numbers (A001006). - Vladimir Kruchinin_, Aug 11 2010

D-finite with recurrence -n*a(n) +(3*n-1)*a(n-1) +(n-2)*a(n-2) +3*(1-n)*a(n-3)=0. - R. J. Mathar, Nov 09 2012 [Since A002426(n) = a(n) - a(n-1), this third-order recurrence follows easily from the second-order recurrence given in A002426. - Peter Bala, Oct 28 2015]

G.f.: G(0)/(1-x), 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, Jun 29 2013

a(n) ~ 3^(n+3/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 23 2013

a(n) = Sum_{k = 0..floor(n/2)} binomial(n + 1,2*k + 1) *binomial(2*k,k). Cf. A025178. - Peter Bala, Oct 28 2015

a(n) = (n+1)*hypergeom([1/2,(1-n)/2,-n/2],[1,3/2],4). - Peter Luschny, Oct 29 2015

a(n) = (n+1)*Sum_{k=0..floor(n/2)} multinomial(n;n-2*k,k,k)/(2*k+1). - Chai Wah Wu, Aug 14 2025