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A078718 Let f(i,j) = Sum_{k=0..2*i} (binomial(2*i, k)*binomial(2*j, i+j-k)*(-1)^(i+j-k) (this is essentially the same as the triangle in A068555); then a(n) = f(n, n-2)/2. 2
0, 1, 3, -5, 14, -45, 154, -546, 1980, -7293, 27170, -102102, 386308, -1469650, 5616324, -21544100, 82907640, -319929885, 1237518450, -4796857230, 18627909300, -72457790790, 282257178060, -1100982015900, 4299680491080, -16809921068850, 65785111513524, -257683159276956 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

FORMULA

G.f.: x^2*A'(x)*G(A(x))/A(x), where A(x) = x*(1+sqrt(1+4*x))/2, G(x) =(2*cosh(asinh((3^(3/2)*sqrt(x))/2)/3)*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)*sqrt(x))/(sqrt(3)*sqrt((27*x)/4+1))+(4*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)^2)/3+1. - Vladimir Kruchinin, Dec 16 2016

G.f.: (sqrt(4*x+1)*(4*x^2+x)+6*x^2+x)/(sqrt(4*x+1)+4*x+1). - Vladimir Kruchinin, Dec 17 2016

a(n) ~ (-1)^n * 2^(2*n-3) / sqrt(Pi*n). - Vaclav Kotesovec, Dec 17 2016

a(n) = (1/2)*Sum_{k=0,..,2*n} ( binomial(2*n, k)*binomial(2*(n - 2), 2*n - 2 - k)*(-1)^(2*n - 2 - k) ), with a(0)=0, a(1)=1. - G. C. Greubel, Feb 16 2017

MATHEMATICA

s = (Sqrt[4*x+1]*(4*x^2+x) + 6*x^2 + x)/(Sqrt[4*x+1] + 4*x+1) + O[x]^28; CoefficientList[s, x] (* Jean-Fran├žois Alcover, Dec 17 2016, after Vladimir Kruchinin *)

Table[(1/2)*Sum[ Binomial[2*n, k]*Binomial[2*(n - 2), 2*n - 2 - k]*(-1)^(2*n - 2 - k), {k, 0, 2*n}], {n, 0, 50}] (* G. C. Greubel, Feb 16 2017 *)

PROG

(Maxima)

A(x):=x*(1+sqrt(1+4*x))/2;

G(x):=(2*cosh(asinh((3^(3/2)*sqrt(x))/2)/3)*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)*sqrt(x))/(sqrt(3)*sqrt((27*x)/4+1))+(4*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)^2)/3+1;

taylor(x^2*diff(A(x), x)*G(A(x))/A(x), x, 0, 20); /* Vladimir Kruchinin, Dec 16 2016 */

(PARI) concat([0, 1], for(n=2, 25, print1(sum(k=0, 2*n, (1/2)* binomial(2*n, k)* binomial(2*( n-2), 2*n-k-2)*(-1)^(2*n-k-2)), ", "))) \\ G. C. Greubel, Feb 16 2017

CROSSREFS

Cf. A068555. Apart perhaps from signs, f(n, 0) and f(n, n) give A000984, f(n, 1) gives A002420, f(n, n-1) gives 2*A000984.

Sequence in context: A318227 A230585 A006395 * A081393 A129326 A167553

Adjacent sequences:  A078715 A078716 A078717 * A078719 A078720 A078721

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Dec 20 2002

STATUS

approved

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Last modified June 16 13:37 EDT 2019. Contains 324152 sequences. (Running on oeis4.)