A240876 Expansion of (1 + x)^11 / (1 - x)^12.

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

Offset: 0

Author

Bruno Berselli, Apr 16 2014

Keywords

• nonn
• easy

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

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17.
• OEIS Wiki, Centered orthoplex numbers, see Table of formulas and values (row 11).
• Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Crossrefs

Row/column 11 of A008288.

Cf. similar sequences with g.f. (1+x)^m/(1-x)^(m+1): A005408 (m=1), A001844 .. A001849 (m=2..7), A008417 (m=8), A008419 (m=9), A008421 (m=10), this sequence (m=11), A053805 (m=12).

Subsequence of the odd numbers, A005408.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(),m); Coefficients(R!((1+x)^11/(1-x)^12));

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 *)

Maxima

makelist(coeff(taylor((1+x)^11/(1-x)^12, x, 0, n), x, n), n, 0, 30);

PARI

Vec((1+x)^11/(1-x)^12+O(x^30))

Sage

m = 30; L. = PowerSeriesRing(ZZ, m)
f = (1+x)^11/(1-x)^12
print(f.coefficients())

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).

a(n) = A008421(n) + 2*Sum_{i=0..n-1} A008421(i) for n > 0, a(0) = 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

Extensions

Edited by M. F. Hasler, May 07 2018