A027271 a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.

1, 4, 18, 48, 180, 432, 1512, 3456, 11664, 25920, 85536, 186624, 606528, 1306368, 4199040, 8957952, 28553472, 60466176, 191476224, 403107840, 1269789696, 2660511744, 8344332288, 17414258688, 54419558400, 113192681472, 352638738432, 731398864896, 2272560758784

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,12,0,-36).

Formula

From Paul Barry, Mar 03 2004: (Start)

G.f.: (1+4*x+6*x^2)/(1-6*x^2)^2 = (d/dx)((1+3*x)/(1-6*x^2)).

a(n) = 6^(n/2)*((3-sqrt(6))*(-1)^n + (3+sqrt(6)))*(n+1)/6. (End)

a(n) = 4*b(n) + b(n+1) + 6*b(n-1) with b(n)= 0, 1, 0, 12, 0, 108, 0, 864, ... (aerated A053469). - R. J. Mathar, Sep 29 2012

E.g.f.: (1 + 2*x)*cosh(sqrt(6)*x) + sqrt(2/3)*(1 + 3*x)*sinh(sqrt(6)*x). - Stefano Spezia, May 07 2023

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
A027271[n_]:= A027271[n]= Sum[(k+1)*T[n, k], {k, 0, 2*n}];
Table[A027271[n], {n, 0, 40}] (* G. C. Greubel, Apr 12 2022 *)

Prog

(PARI) A027271(n)=my(b(n)=if(!bittest(n, 0), n\2*6^(n\2-1))); 4*b(n+1)+b(n+2)+6*b(n) \\ could be made more efficient and explicit by simplifying the formula for n even and for n odd separately. - M. F. Hasler, Sep 29 2012
(Magma) [Round(6^(n/2)*( 3*((n+1) mod 2) + Sqrt(6)*(n mod 2) )*(n+1)/3): n in [0..40]]; // G. C. Greubel, Apr 12 2022
(SageMath) [6^(n/2)*( 3*((n+1)%2) + sqrt(6)*(n%2) )*(n+1)/3 for n in (0..40)] # G. C. Greubel, Apr 12 2022

Crossrefs

Cf. A026536, A053469, A199299 (bisection).

Sequence in context: A228108 A259451 A181860 * A073991 A052642 A102928

Adjacent sequences: A027268 A027269 A027270 * A027272 A027273 A027274

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved