This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A038112 a(n) = T(2n,n), where T(n,k) is in A037027. 6
 1, 2, 9, 40, 190, 924, 4578, 22968, 116325, 593450, 3045185, 15699840, 81260816, 421993040, 2197653240, 11472991008, 60023749566, 314621200260, 1651883008050, 8685998428800, 45734484854520, 241098942106440, 1272406536645660 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,0). - Joerg Arndt, Jun 30 2011 Diagonal of rational function 1/(1 - (x + y + y^2)). - Gheorghe Coserea, Aug 06 2018 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3 example 7. FORMULA a(n) = Sum_{k=0..n} C(n+k,k)*C(k,n-k). - Paul Barry, May 13 2006 a(n) = Sum_{j=0..n} binomial(2*j, n)*binomial(n+j, 2*j). - Zerinvary Lajos, Aug 22 2006 a(n) = [x^n] (1/(1-x-x^2))^(n+1). - Paul Barry, Mar 23 2011 a(n) = (n+1)*A001002(n+1). G.f.: Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n!. - Paul D. Hanna, Aug 04 2012 Recurrence: 5*(n-1)*n*a(n) = 11*(n-1)*(2*n-1)*a(n-1) + 3*(3*n-4)*(3*n-2)*a(n-2). - Vaclav Kotesovec, Oct 08 2012 a(n) ~ 3^(3*n+3/2)/(2^(3/2)*5^(n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012 G.f.: A(x) where (x+1)*(27*x-5)*A(x)^3 + 4*A(x) + 1 = 0. - Mark van Hoeij, May 01 2013 EXAMPLE G.f.: A(x) = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 + ... MAPLE a:=n->sum(binomial(2*j, n)*(binomial(n+j, 2*j)), j=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, Aug 22 2006 series(RootOf((x+1)*(27*x-5)*A^3+4*A+1, A), x=0, 30); # Mark van Hoeij, May 01 2013 MATHEMATICA Table[Sum[Binomial[n+k, k]Binomial[k, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Sep 30 2012 *) Table[Binomial[2 n, n] Hypergeometric2F1[1/2 - n/2, -n/2, -2 n, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 19 2016 *) PROG (PARI) {a(n) = if( n<0, 0, sum(k=0, n\2, (2*n-k)!/ (k! * (n-2*k)!)) / n!)}; /* Michael Somos, Sep 29 2003 */ (PARI) {a(n) = if( n<0, 0, n++; n * polcoeff(serreverse( x - x^2 - x^3 + x * O(x^n)), n))}; /* Michael Somos, Sep 29 2003 */ (PARI) /* same as in A092566 but use */ steps=[[0, 1], [1, 0], [2, 0]]; /* Joerg Arndt, Jun 30 2011 */ (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F {a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^m/m!)); polcoeff(A, n)} \\ Paul D. Hanna, Aug 04 2012 (GAP) List([0..25], n->Sum([0..n], k->Binomial(n+k, k)*Binomial(k, n-k))); # Muniru A Asiru, Aug 06 2018 CROSSREFS Sequence in context: A107979 A021001 A231134 * A268039 A235596 A052512 Adjacent sequences:  A038109 A038110 A038111 * A038113 A038114 A038115 KEYWORD nonn,easy AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)