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A168597 Squares of the central trinomial coefficients (A002426). 5
1, 1, 9, 49, 361, 2601, 19881, 154449, 1225449, 9853321, 80156209, 658076409, 5444816521, 45343869481, 379735715529, 3195538786449, 27004932177129, 229066136374761, 1949470542590481, 16640188083903609, 142415188146838161 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Ignoring initial term, a(n) equals the logarithmic derivative of A168598.

Partial sums of A007987. Hence, a(n) is the number of irreducible words of length at most 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero. - Max Alekseyev, Jun 05 2011

The number of ways a king, starting at the origin of an infinite chessboard, can return to the origin in n moves, where leaving the king where it is counts as a move. Cf. A094061. - Peter Bala, Feb 14 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = A002426(n)^2.

G.f.: hypergeom([1/12, 5/12],[1],1728*x^4*(x-1)*(9*x-1)*(3*x+1)^2/(81*x^4-36*x^3-26*x^2-4*x+1)^3)/(81*x^4-36*x^3-26*x^2-4*x+1)^(1/4).  - Mark van Hoeij, May 07 2013

G.f.: 1 / AGM(1+3*x, sqrt((1-x)*(1-9*x))), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Sep 04 2014

G.f.: 1 / AGM((1-x)*(1-3*x), (1+x)*(1+3*x)) = Sum_{n>=0} a(n)*x^(2*n). - Paul D. Hanna, Oct 04 2014

a(n) = (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([(1-n)/2,-n/2],[1],4). - Peter Luschny, Nov 10 2014

MAPLE

a := n -> (-1)^n*hypergeom([1/2, -n], [1], 4)*hypergeom([1/2-n/2, -n/2], [1], 4): seq(simplify(a(n)), n=0..20); # Peter Luschny, Nov 10 2014

MATHEMATICA

Table[(-1)^n*Hypergeometric2F1[1/2, -n, 1, 4] * Hypergeometric2F1[(1 - n)/2, -n/2, 1, 4], {n, 0, 50}] (* G. C. Greubel, Feb 26 2017 *)

PROG

(PARI) {a(n)=polcoeff((1+x+x^2 +x*O(x^n))^n, n)^2}

for(n=0, 20, print1(a(n), ", "))

(PARI) /* Using AGM: */

{a(n)=polcoeff( 1 / agm(1+3*x, sqrt((1+3*x)^2 - 16*x +x*O(x^n))), n)}

for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 04 2014

CROSSREFS

Cf. A002426, A133053, A168598, A243949, A094061.

Sequence in context: A188235 A286810 A066558 * A169724 A135625 A160603

Adjacent sequences:  A168594 A168595 A168596 * A168598 A168599 A168600

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Dec 01 2009

STATUS

approved

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Last modified August 24 00:29 EDT 2017. Contains 291052 sequences.