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A005717 Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.
(Formerly M1612)
27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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) = 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

a(n) = A111808(n,n-1). - Reinhard Zumkeller, Aug 17 2005

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) = 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) = 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

REFERENCES

L. 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).

Wang, Chenying, Piotr Miska, and István Mező. "The r-derangement numbers." Discrete Mathematics 340.7 (2017): 1681-1692.

LINKS

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]

R. K. Guy, Letter to N. J. A. Sloane, 1987

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

Eric Weisstein's World of Mathematics, Trinomial Coefficient

FORMULA

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 Bessel function. - Michael Somos, Sep 09 2002.

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

(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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

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

(PARI)

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 */

(Sage)

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([A005717.next() for _ in range(29)]) # Peter Luschny, May 16 2016

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

CROSSREFS

A diagonal of A027907.

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

Sequence in context: A055544 A126285 A026163 * A025266 A074403 A290953

Adjacent sequences:  A005714 A005715 A005716 * A005718 A005719 A005720

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Erich Friedman, Jun 01 2001

STATUS

approved

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Last modified September 20 20:29 EDT 2018. Contains 315241 sequences. (Running on oeis4.)