A005407 Number of protruded partitions of n with largest part at most 6. (Formerly M2570)

1, 3, 6, 13, 25, 50, 93, 175, 320, 582, 1041, 1851, 3253, 5682, 9848, 16970, 29070, 49559, 84090, 142107, 239239, 401404, 671386, 1119799, 1862861, 3091708, 5120090, 8462535, 13961695, 22996307, 37819865, 62112581, 101879568, 166912537, 273166466, 446623176

Offset

1,2

References

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232.

Formula

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

Maple

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

Mathematica

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 *)

Prog

(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
(SageMath)
def A005407_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( -1 + (1-x)^6/product(1-x-x^j+x^(2*j+1) for j in (1..6)) ).list()
a=A005407_list(50); a[1:] # G. C. Greubel, Nov 19 2022

Crossrefs

Sequence in context: A285461 A324129 A005406 * A005116 A121349 A215984

Adjacent sequences: A005404 A005405 A005406 * A005408 A005409 A005410

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Dec 19 2004

Status

approved