|
|
A240876
|
|
Expansion of (1 + x)^11 / (1 - x)^12.
|
|
11
|
|
|
1, 23, 265, 2047, 11969, 56695, 227305, 795455, 2485825, 7059735, 18474633, 45046719, 103274625, 224298231, 464387817, 921406335, 1759885185, 3248227095, 5812626185, 10113604735, 17152640321, 28418229623, 46082942185, 73265596607, 114375683009
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Also 11-dimensional centered hyperoctahedron numbers (see Deza in References) or Crystal ball sequence for 11-dimensional cubic lattice.
This is row/column 11 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Peter Munn, Jan 05 2023
|
|
REFERENCES
|
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 230 (paragraph 3.6.6).
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
|
|
FORMULA
|
G.f.: (1 + x)^11 / (1 - x)^12.
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12), with initial values as shown.
a(n) = (2*n + 1)*(2*n*(n + 1)*(n^2 + n + 5)*(2*n^2 + 2*n + 51)*(n^4 + 2*n^3 + 68*n^2 + 67*n + 537)/155925 + 1).
Sum_{n >= 0} 1/a(n) = 1.047847848425287358769594801715758965260...
a(n) = Sum_{k = 0..min(11,n)} 2^k*binomial(11,k)*binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014
|
|
MATHEMATICA
|
CoefficientList[Series[(1 + x)^11/(1 - x)^12, {x, 0, 30}], x]
LinearRecurrence[{12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1}, {1, 23, 265, 2047, 11969, 56695, 227305, 795455, 2485825, 7059735, 18474633, 45046719}, 30] (* Harvey P. Dale, Apr 15 2018 *)
|
|
PROG
|
(PARI) Vec((1+x)^11/(1-x)^12+O(x^30))
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)^11/(1-x)^12));
(Maxima) makelist(coeff(taylor((1+x)^11/(1-x)^12, x, 0, n), x, n), n, 0, 30);
(Sage)
m = 30; L.<x> = PowerSeriesRing(ZZ, m)
f = (1+x)^11/(1-x)^12
print(f.coefficients())
|
|
CROSSREFS
|
Subsequence of the odd numbers, A005408.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|