The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A228511 a(n) = sum_{k=0}^n binomial(n,k)^2*4^k*A000108(k). 1
 1, 5, 49, 645, 9921, 167909, 3030705, 57284901, 1120905985, 22531796805, 462793508529, 9674942743365, 205261950829761, 4409503432713765, 95746612458475569, 2098428359692863717, 46366172896708865025, 1031886636204630031493, 23112239140054942651185, 520644236358436868354565, 11789139538117859937032385 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: Let p be any odd prime. (i) Let A(p) be the p X p determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,p-1. Then we have A(p) == (-1)^{(p-1)/2} (mod p). (ii) Let B(p) be the p X p determinant with (i,j)-entry equal to b(i+j) for all i,j = 0,...,p-1, where b(n) denotes sum_{k=0}^n binomial(n,k)^2*binomial(2k,k)*4^k or sum_{k=0}^n binomial(n,k)^2*binomial(2k,k)*(-2)^(n-k). Then B(p) is congruent to the Legendre symbol (p/3) modulo p. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..100 Zhi-Wei Sun, On some determinants with Legendre symbol entries, preprint, arXiv:1308.2900 [math.NT], 2013-2019. FORMULA By Zeilberger's algorithm, we have the following recurrence: 225*(12*n+43)*(n+1)^2*(n+2)^2*a(n)   - (n+2)^2*(3108*n^3+20869*n^2+42172*n+26271)*a(n+1)   + (n+3)*(420*n^4+4037*n^3+13835*n^2+19872*n+9840)*a(n+2) = (n+1)*(n+3)*(12*n+31)*(n+4)^2*a(n+3). a(n) ~ 5^(2*n+5/2)/(32*Pi*n^2). - Vaclav Kotesovec, Aug 25 2013 MATHEMATICA a[n_]:=Sum[Binomial[n, k]^2*4^k*CatalanNumber[k], {k, 0, n}] Table[a[n], {n, 0, 20}] CROSSREFS Cf. A000108, A086618. Sequence in context: A216483 A243945 A297513 * A116873 A324361 A089914 Adjacent sequences:  A228508 A228509 A228510 * A228512 A228513 A228514 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 23 2013 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 November 27 06:19 EST 2020. Contains 338678 sequences. (Running on oeis4.)