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A003169 Number of 2-line arrays; or number of P-graphs with 2n edges.
(Formerly M2973)
14
1, 3, 14, 79, 494, 3294, 22952, 165127, 1217270, 9146746, 69799476, 539464358, 4214095612, 33218794236, 263908187100, 2110912146295, 16985386737830, 137394914285538, 1116622717709012, 9113225693455362, 74659999210200292 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First column of triangle A100326. - Paul D. Hanna, Nov 16 2004

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

M. Bicknell and V. E. Hoggatt, Jr., Sequences of matrix inverses from Pascal, Catalan and related convolution arrays, Fib. Quart., 14 (1976), 224-232.

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 416

R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) no 1, 47-63.

Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

FORMULA

For formula see Read reference.

a(n) = ( (324*n^2-708*n+360)*a(n-1) - (371*n^2-1831*n+2250)*a(n-2) + (20*n^2-130*n+210)*a(n-3) )/(16*n*(2*n-1)) for n>2, with a(0)=0, a(1)=1, a(2)=3. - Paul D. Hanna, Nov 16 2004

G.f. satisfies: A(x) = x*(1+A(x))/(1-A(x))^2 where A(0)=0. G.f. satisfies: (1+A(x))/(1-A(x)) = 2*G003168(x)-1, where G003168 is the g.f. of A003168. - Paul D. Hanna, Nov 16 2004

a(n) = (1/n)*Sum_{i=0..n-1} binomial(n,i)*binomial(3*n-i-2,n-i-1). - Vladeta Jovovic, Sep 13 2006

Appears to be 1/n*Jacobi_P(n-1,1,n-1,3). If so then a(n) = 1/(2*n-1)*sum {k = 0..n-1} binomial(n-1,k)*binomial(2*n+k-1,k+1) = 1/n*sum {k = 0..n} binomial(n,k)*binomial(2*n-2,n+k-1)*2^k. - Peter Bala, Aug 01 2012

a(n) ~ sqrt(33/sqrt(17)-7) * ((71+17*sqrt(17))/16)^n / (4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2013

The o.g.f. A(x) = 1 + 3*x + 14*x^2 + ... taken with offset 0, satisfies 1 + x*A'(x)/A(x) = 1 + 3*x + 19*x^2 + 138*x^3 + ..., the o.g.f. for A156894. - Peter Bala, Oct 05 2015

MAPLE

a[0]:=0:a[1]:=1:a[2]:=3:for n from 3 to 30 do a[n]:=((324*n^2-708*n+360)*a[n-1] -(371*n^2-1831*n+2250)*a[n-2]+(20*n^2-130*n+210)*a[n-3])/(16*n*(2*n-1)) od:seq(a[n], n=1..25); # Emeric Deutsch, Jan 31 2005

MATHEMATICA

lim = 21; t[0, 0] = 1; t[n_, 0] := t[n, 0] = Sum[(k + 1)*t[n - 1, k], {k, 0, n - 1}]; t[n_, k_] := t[n, k] = Sum[t[j + 1, 0]*t[n - j - 1, k - 1], {j, 0, n - k}]; Table[ t[n, 0], {n, lim}] (* Jean-François Alcover, Sep 20 2011, after Paul D. Hanna's comment *)

PROG

(PARI) {a(n)=if(n==0, 0, if(n==1, 1, if(n==2, 3, ( (324*n^2-708*n+360)*a(n-1) -(371*n^2-1831*n+2250)*a(n-2)+(20*n^2-130*n+210)*a(n-3))/(16*n*(2*n-1)) )))} \\ Paul D. Hanna, Nov 16 2004

(PARI) {a(n)=local(A=x+x*O(x^n)); if(n==1, 1, for(i=1, n, A=x*(1+A)/(1-A)^2); polcoeff(A, n))}

(Haskell)

a003169 = flip a100326 0  -- Reinhard Zumkeller, Nov 21 2015

CROSSREFS

Cf. A003168, A100324, A100326, A156894.

Sequence in context: A001564 A277132 A059276 * A086621 A020089 A218677

Adjacent sequences:  A003166 A003167 A003168 * A003170 A003171 A003172

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Emeric Deutsch, Jan 31 2005

STATUS

approved

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Last modified March 19 19:30 EDT 2019. Contains 321330 sequences. (Running on oeis4.)