A147611 The 3rd Witt transform of A000027.

0, 0, 0, 0, 2, 7, 18, 42, 84, 153, 264, 429, 666, 1001, 1456, 2061, 2856, 3876, 5166, 6783, 8778, 11214, 14168, 17710, 21924, 26910, 32760, 39582, 47502, 56637, 67122, 79112, 92752, 108207, 125664, 145299, 167310, 191919, 219336, 249795, 283556

Offset: 0

Author

R. J. Mathar, Nov 08 2008

Keywords

• easy
• nonn

Comments

a(n) is the number of binary Lyndon words of length n+3 having 3 blocks of 0's, see Math.SE. - Andrey Zabolotskiy, Nov 16 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
• Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO].
• Felix Pahl, Find the number of n-length Lyndon words on alphabet {0,1} with k blocks of 0's. (answer), Mathematics StackExchange, 2020.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,6,-9,12,-9,6,-6,4,-1).

Crossrefs

Cf. A006584 (2nd Witt transform of A000027), A049347, A099254, A147618.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); [0,0,0,0] cat Coefficients(R!( x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2) )); // G. C. Greubel, Oct 24 2022

Mathematica

CoefficientList[Series[x^4(2 -x+ 2*x^2)/((1-x)^6*(1 +x +x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 13 2012 *)

SageMath

def A147611_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2) ).list()
A147611_list(50) # G. C. Greubel, Oct 24 2022

Formula

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

a(n) = (1/27)*((3*A049347(n) + A049347(n-1)) - 3*(-1)^n*(A099254(n) - A099254(n- 1)) + n*(3*n^4 - 15*n^2 - 28)/40). - G. C. Greubel, Oct 24 2022