A246437 Expansion of (1/2)*(1/(x+1)+1/(sqrt(-3*x^2-2*x+1))).

1, 0, 2, 3, 10, 25, 71, 196, 554, 1569, 4477, 12826, 36895, 106470, 308114, 893803, 2598314, 7567465, 22076405, 64498426, 188689685, 552675364, 1620567764, 4756614061, 13974168191, 41088418150, 120906613076, 356035078101, 1049120176954

Offset: 0

Author

Vladimir Kruchinin, Nov 14 2014

Keywords

• nonn
• easy

Comments

a(2101) has 1001 decimal digits. - Michael De Vlieger, Apr 25 2016

This is the analog for Coxeter type B of Motzkin sums (A005043) for Coxeter type A, see the article by Athanasiadis and Savvidou. - F. Chapoton, Jul 20 2017

Number of compositions of n into exactly n nonnegative parts avoiding part 1. a(4) = 10: 0004, 0022, 0040, 0202, 0220, 0400, 2002, 2020, 2200, 4000. - Alois P. Heinz, Aug 19 2018

From Peter Bala, Jan 07 2022: (Start)

The binary transform is A088218. The inverse binomial transform is a signed version of A027306 and the second inverse binomial transform is a signed version of A027914.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.

Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2100
• Christos A. Athanasiadis and Christina Savvidou, The Local h-Vector of the Cluster Subdivision of a Simplex, Séminaire Lotharingien de Combinatoire 66 (2012), Article B66c.
• Eric Marberg, On some actions of the 0-Hecke monoids of affine symmetric groups, arXiv:1709.07996 [math.CO], 2017.
• 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. A002426, A005043, A027306, A027914, A088218, A104507, A370616, A386548.

Programs

Mathematica

CoefficientList[Series[(1/2) (1 / (x + 1) + 1 / (Sqrt[-3 x^2 - 2 x + 1])), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
Table[(-1)^n (Hypergeometric2F1[1/2, -n, 1, 4] + 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)
Table[Sum[Binomial[n, k] Binomial[n - k - 1, n - 2 k], {k, 0, n/2}], {n, 0, 28}] (* Michael De Vlieger, Apr 25 2016 *)

Maxima

a(n):=sum(binomial(n,k)*binomial(n-k-1,n-2*k),k,0,n/2);

Sage

def a(n):
    if n < 3: return [1,0,2][n]
    return n*hypergeometric([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4)
[simplify(a(n)) for n in (0..28)] # Peter Luschny, Nov 14 2014

Formula

a(n) = Sum_{k = 0..n/2} binomial(n,k)*binomial(n-k-1,n-2*k).

A(x) = 1 + x*B'(x)/B(x), where B(x) = (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)) is the o.g.f. of A005043.

a(n) = n*hypergeom([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4) for n>=3. - Peter Luschny, Nov 14 2014

a(n) ~ 3^(n+1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 14 2014

a(n) = (-1)^n*(hypergeom([1/2, -n], [1], 4) + 1)/2. - Vladimir Reshetnikov, Apr 25 2016

D-finite with recurrence: n*(a(n) - a(n-1)) = (5*n-6)*a(n-2) + 3*(n-2)*a(n-3). - Vladimir Reshetnikov, Oct 13 2016

a(n) = [x^n]( (1 - x + x^2)/(1 - x) )^n. - Peter Bala, Jan 07 2022