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A027271
a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.
3
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
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
KEYWORD
nonn,easy
STATUS
approved