A005407 - OEIS

%I M2570 #24 Nov 19 2022 04:31:56

%S 1,3,6,13,25,50,93,175,320,582,1041,1851,3253,5682,9848,16970,29070,

%T 49559,84090,142107,239239,401404,671386,1119799,1862861,3091708,

%U 5120090,8462535,13961695,22996307,37819865,62112581,101879568,166912537,273166466,446623176

%N Number of protruded partitions of n with largest part at most 6.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. P. Stanley, Ordered structures and partitions, Memoirs of the Amer. Math. Soc., no. 119 (1972).

%H G. C. Greubel, <a href="/A005407/b005407.txt">Table of n, a(n) for n = 1..1000</a>

%H R. P. Stanley, <a href="http://www.fq.math.ca/Scanned/13-3/stanley.pdf">A Fibonacci lattice</a>, Fib. Quart., 13 (1975), 215-232.

%F G.f.: (1-x)^6/Product_{i=1..6} (1-x-x^i+x^(1+2*i)) - 1. - _Emeric Deutsch_, Dec 19 2004

%p G:=(1-x)^6/Product(1-x-x^i+x^(1+2*i),i=1..6)-1: Gser:=series(G,x=0,39): seq(coeff(Gser,x^n),n=1..37); # _Emeric Deutsch_, Dec 19 2004

%t CoefficientList[Series[(1-x)^6/Product[1-x-x^i+x^(1+2i),{i,6}]-1,{x,0,40}],x] (* _Harvey P. Dale_, Jan 23 2015 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( -1 + (1-x)^6/(&*[1-x-x^j+x^(2*j+1): j in [1..6]]) )); // _G. C. Greubel_, Nov 19 2022

%o (SageMath)

%o def A005407_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( -1 + (1-x)^6/product(1-x-x^j+x^(2*j+1) for j in (1..6)) ).list()

%o a=A005407_list(50); a[1:] # _G. C. Greubel_, Nov 19 2022

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Dec 19 2004