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

0,5

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).

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

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 *)

Prog

(Magma) R<x>:=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
(SageMath)
def A147611_list(prec):
    P.<x> = 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

Crossrefs

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

Sequence in context: A055503 A077802 A095151 * A007991 A037294 A371035

Adjacent sequences: A147608 A147609 A147610 * A147612 A147613 A147614

Keyword

easy,nonn

Author

R. J. Mathar, Nov 08 2008

Status

approved