A005717 - cp's OEIS frontend

A005717 Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.

1, 2, 6, 16, 45, 126, 357, 1016, 2907, 8350, 24068, 69576, 201643, 585690, 1704510, 4969152, 14508939, 42422022, 124191258, 363985680, 1067892399, 3136046298, 9217554129, 27114249960, 79818194925, 235128465026, 693085098852, 2044217638456, 6032675068061

Offset: 1

N. J. A. Sloane

Number of ordered trees with n+1 edges, having root of even degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch, Aug 02 2002
The connection to Motzkin numbers comes from the Lagrange inversion formula. - Michael Somos, Oct 10 2003
Number of horizontal steps in all Motzkin paths of length n. - Emeric Deutsch, Nov 09 2003
Number of UHD's in all Motzkin paths of length n+2 (here U=(1,1), H=(1,0) and D=(1,-1)). Example: a(2)=2 because in the nine Motzkin paths of length 4, HHHH, HHUD, HUDH, H(UHD), UDHH, UDUD, (UHD)H, UHHD and UUDD, we have altogether two UHD's (shown between parentheses). - Emeric Deutsch, Dec 26 2003
Number of ordered trees with n+1 edges, having exactly one leaf at even height. Number of Dyck path of semilength n+1, having exactly one peak at even height. Example: a(3)=6 because we have uuu(ud)ddd, u(ud)dudud, udu(ud)dud, ududu(ud)d, u(ud)uuddd and uuudd(ud)d (here u=(1,1),d=(1,-1) and the unique peak at even height is shown between parentheses). - Emeric Deutsch, Mar 10 2004
a(n) is the number of Dyck (n+1)-paths containing exactly one UDU. - David Callan, Jul 15 2004
Number of peaks in all Motzkin paths of length n+1. - Emeric Deutsch, Sep 01 2004
This is a kind of Motzkin transform of A059841 because the substitution x -> x*A001006(x) in the independent variable of the g.f. of A059841 generates 1,0,1,2,6,16,... that is 1,0 followed by this sequence here. - R. J. Mathar, Nov 08 2008
a(n) is the number of lattice paths avoiding N^(>=3) from (0,0) to (n,n). - Shanzhen Gao, Apr 20 2010
a(n+1) is the number of binary strings having n 0's and n 1's and no appearance of 000.  For example, for n = 1, there 2 strings: 01 and 10.  For n = 2, there are 6: 0011, 0101, 0110, 1001, 1010, 1100. - Toby Gottfried, Sep 12 2011
a(n) is the number of paths in the half-plane x>=0, from (0,0) to (n,1), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=3, we have the 6 paths HHU, HUH, UDU, UUD, UHH, DUU. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) is the number of ways to tile a strip of length 2*n+1 with squares, dominos, and trominos, where the number of trominos is always one more than the number of squares. - Greg Dresden and Anna Kalynchuk, Jul 30 2025

			G.f. = x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 126*x^6 + 357*x^7 + ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (Terms 1 to 200 computed by T. D. Noe; terms 201 to 1000 by G. C. Greubel, Jan 15 2017)
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.
Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, The Combinatorics of Motzkin Polyominoes, arXiv:2401.06228 [math.CO], 2024. See page 13.
Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees, and branch lengths, Discrete Mathematics 282 (2004), 89-94.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Richard K. Guy, Letter to N. J. A. Sloane, 1987.
Stanislav Krymski and Alexander Okhotin, Longer Shortest Strings in Two-Way Finite Automata, in: Jirásková G., Pighizzini G. (eds) Descriptional Complexity of Formal Systems. DCFS 2020. Lecture Notes in Computer Science, vol 12442. Springer, Cham.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.
Chenying Wang, Piotr Miska, and István Mező, The r-derangement numbers, Discrete Mathematics 340(7) (2017), 1681-1692.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.

A diagonal of A027907.

Cf. A001006, A002426, A005043, A005773, A076540 (binomial transform).

seq(add(binomial(i, k) *binomial(i-k, k+1), k=0..floor(i/2)), i=1..30); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
M:= proc(n) option remember; `if` (n<2, 1, (3*(n-1)*M(n-2) +(2*n+1) *M(n-1))/ (n+2)) end: A005717 := n -> n*M(n-1):
seq(A005717(i), i=1..27); # Peter Luschny, Sep 12 2011
a := n -> simplify(GegenbauerC(n,-n-1,-1/2)):
seq(a(n), n=0..28); # Peter Luschny, May 07 2016

Table[Coefficient[Expand[(1+x+x^2)^n], x, n-1], {n, 1, 40}]
Table[n*Hypergeometric2F1[(1 - n)/2, 1 - n/2, 2, 4], {n, 29}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[GegenbauerC[n,-n-1,-1/2],{n,0,100}] (* Emanuele Munarini, Oct 20 2016 *)

makelist(ultraspherical(n,-n-1,-1/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, n-1))}; /* Michael Somos, Sep 09 2002 */

{a(n) = if( n<0, 0, n * polcoeff( serreverse( x / (1 + x + x^2) + x * O(x^n)), n))}; /* Michael Somos, Oct 10 2003 */

N=10^3;  x='x+'x*O('x^N);
gf = 2*x/(1-2*x-3*x^2+(1-x)*sqrt(1-2*x-3*x^2));
v005717 = Vec(gf);
/* Joerg Arndt, Aug 16 2012 */

def A():
    a, b, n = 0, 1, 1
    while True:
        yield b
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
A005717 = A()
print([next(A005717) for  in range(29)]) # _Peter Luschny, May 16 2016

a(n) = Sum_{k=1..n} T(k, k-1), where T is the array defined in A025177.
G.f.: 2*x/(1-2*x-3*x^2+(1-x)*sqrt(1-2*x-3*x^2)). - Emeric Deutsch, Aug 14 2002
E.g.f.: exp(x) * I_1(2x), where I_1 is the Bessel function. - Michael Somos, Sep 09 2002
a(n) = A111808(n,n-1). - Reinhard Zumkeller, Aug 17 2005
a(n) = Sum_{k=0..floor((n-1)/3)} (-1)^k * binomial(n,k) * binomial(2n-2-3k, n-1). - David Callan, Jul 03 2006
From Paul Barry, Feb 05 2007: (Start)
a(n) = n*Sum_{k=0..floor((n-1)/2), C(n-1,2k)*C(k)}, C(n) = A000108(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2k+1)*C(n,2k+1)*C(k).
a(n) = Sum_{k=0..n-1} ( Sum_{j=0..floor(k/2)} C(k,2j)*C(2j+1,j) ). (End)
a(n) = (A002426(n+1) - A002426(n))/2. - Paul Barry, May 22 2008
a(n) = n*A001006(n-1). - Paul Barry, Oct 05 2009
a(n) = Sum_{i=0..floor(n/2)} C(n+1,n-i) * C(n-i,i). - Shanzhen Gao, Apr 20 2010
D-finite with recurrence: (n+1)*a(n) - 3*n*a(n-1) - (n+3)*a(n-2) + 3*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 28 2011
a(n) ~ 3^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2013
0 = a(n) * 3*(n+1)*(n+2) + a(n+1) * (n+2)*(2*n+3) - a(n+2) * (n+1)*(n+3) for all n in Z. - Michael Somos, Apr 03 2014
G.f.: z*M(z)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
Working with an offset of 0, a(n) = [x^n](1 + x + x^2)^(n+1); binomial transform is A076540. - Peter Bala, Jun 15 2015
a(n) = GegenbauerC(n,-n-1,-1/2). - Peter Luschny, May 07 2016
a(n) = (-1)^(n+1) * n * hypergeom([3/2, 1-n], [3], 4). - Vladimir Reshetnikov, Sep 28 2016
a(n) = Sum_{k=0..n-1} binomial(n,k)*binomial(n-k, k+1) [Krymski and Okhotin]. - Michel Marcus, Dec 04 2020
a(n) = (1/2)*(A005773(n+1) - A005043(n)). - Peter Bala, Feb 11 2022
a(n) = A002426(n) - A005043(n). - Amiram Eldar, May 17 2024

More terms from Erich Friedman, Jun 01 2001

cp's OEIS Frontend

A005717 Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.

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