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!)
 A268545 From the diagonal of 1/(1 - (y + z + x w + x z w + x y w)). 71
 1, 10, 246, 7540, 255430, 9163980, 341237820, 13042646760, 508236930630, 20101587623260, 804500381097556, 32508382071448920, 1324112273705453596, 54296281503438398200, 2239266766596344681400, 92809720054802928741840, 3863305447624183692730950, 161427619265399264526790140 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Gheorghe Coserea, Jul 03 2016: (Start) Also diagonal of R(x,y,z) =  1/(1 - x - y - z - x*y - y*z). Annihilating differential operator: x*(4*x+3)*(16*x^2+44*x-1)*Dx^2 + (128*x^3+320*x^2+264*x-3)*Dx + 16*x^2+12*x+30. (End) LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..200 A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015. Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015. FORMULA D-finite with recurrence: n^2*(10*n-13)*a(n) +2*(-220*n^3+506*n^2-334*n+63)*a(n-1) -4*(10*n-3)*(-3+2*n)^2*a(n-2)=0. - R. J. Mathar, Apr 15 2016 a(n) ~ (1+sqrt(5))^(5*n+2) / (5^(1/4) * Pi * n * 2^(3*n+3)). - Vaclav Kotesovec, Jul 01 2016 G.f.: hypergeom([1/12, 5/12],[1],6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3)/(1-40*x+16*x^2)^(1/4). - Gheorghe Coserea, Jul 01 2016 0 = x*(4*x+3)*(16*x^2+44*x-1)*y'' + (128*x^3+320*x^2+264*x-3)*y' + (16*x^2+12*x+30)*y, where y is the g.f. - Gheorghe Coserea, Jul 03 2016 a(n) = Sum_{j = 0..n} C(n,j)*Sum_{k = 0..j} C(j,k)*C(n+k,j)*C(2*n+k,n) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0], [0,1,1]}. - Peter Bala, Jan 26 2018 MAPLE A268545 := proc(n)     1/(1-y-z-x*w-x*z*w-x*y*w) ;     coeftayl(%, x=0, n) ;     coeftayl(%, y=0, n) ;     coeftayl(%, z=0, n) ;     coeftayl(%, w=0, n) ; end proc: seq(A268545(n), n=0..40) ; # R. J. Mathar, Apr 15 2016 #alternative program with(combinat): seq(add(binomial(n, j)*add(binomial(j, k)*binomial(n+k, j)*binomial(2*n+k, n), k = 0..j), j = 0..n), n = 0..20); # Peter Bala, Jan 26 2018 MATHEMATICA a[n_] := a[n] = 1/(1 - y - z - x*w - x*z*w - x*y*w) // SeriesCoefficient[#, {x, 0, n}]& // SeriesCoefficient[#, {y, 0, n}]& // SeriesCoefficient[#, {z, 0, n}]& // SeriesCoefficient[#, {w, 0, n}]&; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 40}] (* or: *) HypergeometricPFQ[{1/12, 5/12}, {1}, (6912*x^3*(-16*x^2 - 44*x + 1))/ (16*x^2 - 40*x + 1)^3]/(16*x^2 - 40*x + 1)^(1/4) + O[x]^41 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 12 2017, after Gheorghe Coserea *) PROG (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 21; x = 'x + O('x^N); Vec(hypergeom([1/12, 5/12], [1], 6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3, N)/(1-40*x+16*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 03 2016 (PARI) diag(expr, N=22, var=variables(expr)) = {   my(a = vector(N));   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));   for (n = 1, N, a[n] = expr;     for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));   return(a); }; diag(1/(1 - x - y - z - x*y - y*z), 18) \\ test: diag(1/(1-x-y-z-x*y-y*z)) == diag(1/(1-(y+z+x*w + x*z*w + x*y*w))) \\ Gheorghe Coserea, Jun 16 2018 CROSSREFS Cf. A268545 - A268555. Sequence in context: A249564 A034222 A197437 * A211093 A108792 A265854 Adjacent sequences:  A268542 A268543 A268544 * A268546 A268547 A268548 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Feb 29 2016 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 2 01:23 EST 2020. Contains 338864 sequences. (Running on oeis4.)