A062109 -id:A062109 - cp's OEIS frontend

A169796 Expansion of ((1-x)/(1-2x))^9.

1, 9, 54, 264, 1134, 4446, 16272, 56412, 187137, 598417, 1854882, 5597172, 16498632, 47638512, 135048672, 376592064, 1034663040, 2804590080, 7509232640, 19880294400, 52088352768, 135173578752, 347680161792, 886900948992, 2245014454272, 5641949085696

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 8 parts equal to 0. - Milan Janjic, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512).

Cf. for ((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

CoefficientList[Series[((1 - x)/(1 - 2 x))^9, {x, 0, 25}], x] (* Michael De Vlieger, Oct 15 2018 *)

G.f.: ((1-x)/(1-2*x))^9.

For n > 0, a(n) = 2^(n-16)*(n+8)*(n^7 + 100*n^6 + 3778*n^5 + 68056*n^4 + 606961*n^3 + 2543284*n^2 + 4524300*n + 2575440)/315. - Bruno Berselli, Aug 07 2011

A370695 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/4) / (1-x))^4.

Original entry on oeis.org

1, 4, 22, 128, 777, 4872, 31330, 205560, 1370868, 9266104, 63343006, 437183260, 3042337215, 21323543252, 150395596016, 1066637271424, 7602188660799, 54422262148632, 391146728466980, 2821396586367568, 20417766975784066, 148200184917042112

Offset: 0

Seiichi Manyama, Mar 27 2024

nonn

Cf. A045902, A062109, A371379, A371486, A371517.

Cf. A270386, A307678, A371516.

A370695 := proc(n)
    4*add(binomial(n-1,n-k)*binomial(3*k+4,k)/(3*k+4),k=0..n) ;
end proc:
seq(A370695(n),n=0..80) ; #R. J. Mathar, Oct 24 2024

a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(3*k+4, k)/(3*k+4));

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+4,k)/(3*k+4).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A307678.
a(n) ~ 9 * 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Mar 29 2024
D-finite with recurrence 2*(n+2)*(2*n+3)*a(n) +(-55*n^2-74*n-15)*a(n-1) +6*(37*n^2-46*n-4)*a(n-2) -(295*n-319)*(n-3)*a(n-3) +124*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Oct 24 2024

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A169796 Expansion of ((1-x)/(1-2x))^9.

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