A001786 Expansion of 1/((1+x)*(1-x)^11).

1, 10, 56, 230, 771, 2232, 5776, 13672, 30086, 62292, 122464, 230252, 416394, 727672, 1233584, 2035176, 3276559, 5159726, 7963384, 12066626, 17978389, 26373776, 38138464, 54422576, 76705564, 106873832, 147313024, 201017112, 271716644, 364028752, 483631776, 637467632, 833975341

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
• Index entries for linear recurrences with constant coefficients, signature (10,-44,110,-165,132,0,-132,165,-110,44,-10,1).

Crossrefs

Cf. A001780, A158454 (signed column k=5).

Eleventh column of A112465.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( 1/((1+x)*(1-x)^11) )); // G. C. Greubel, Apr 20 2025

Mathematica

CoefficientList[Series[1/((1+x)(1-x)^11),{x,0,50}],x] (* Vincenzo Librandi, Feb 24 2012 *)
LinearRecurrence[{10,-44,110,-165,132,0,-132,165,-110,44,-10,1},{1,10,56,230,771,2232, 5776,13672,30086,62292,122464,230252},30] (* Harvey P. Dale, Oct 22 2015 *)

SageMath

def A001786_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-x)^11) ).list()
print(A001786_list(50)) # G. C. Greubel, Apr 20 2025

Formula

Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (11 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A046521 (here for the unsigned column k = 5 with offset 0). - Wolfdieter Lang, Aug 10 2017