A181545 G.f.: A(x) = Sum_{n>=0} (3n)!/(n!)^3 * x^(3n)/(1-x-x^2)^(3n+1).
1, 1, 2, 9, 29, 92, 343, 1281, 4720, 17899, 68933, 266364, 1037423, 4072439, 16065148, 63658521, 253356763, 1012049086, 4055596343, 16299779331, 65683233938, 265310551667, 1073968967929, 4355988107100, 17699727361051
Offset: 0
Keywords
Examples
G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 29*x^4 + 92*x^5 + 343*x^6 + ... which equals the series: A(x) = 1/(1-x-x^2) + 3!/1!^3*x^3/(1-x-x^2)^4 + 6!/2!^3*x^6/(1-x-x^2)^7 + 9!/3!^3*x^9/(1-x-x^2)^10 + 12!/4!^3*x^12/(1-x-x^2)^13 + ... The g.f. also equals the series: A(x) = 1 + x*(1 + x) + x^2*(1 + 2^3*x + x^2) + x^3*(1 + 3^3*x + 3^3*x^2 + x^3) + x^4*(1 + 4^3*x + 6^3*x^2 + 4^3*x^3 + x^4) + x^5*(1 + 5^3*x + 10^3*x^2 + 10^3*x^3 + 5^3*x^4 + x^5) + ... The terms begin: a(0) = a(1) = 1^3; a(2) = 1^3 + 1^3 = 2; a(3) = 1^3 + 2^3 = 9; a(4) = 1^3 + 3^3 + 1^3 = 29; a(5) = 1^3 + 4^3 + 3^3 = 92; a(6) = 1^3 + 5^3 + 6^3 + 1^3 = 343; a(7) = 1^3 + 6^3 + 10^3 + 4^3 = 1281; ...
Links
- Iain Fox, Table of n, a(n) for n = 0..1600
- C. Banderier, P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Proposition 3.1.
- Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
- Edyta Hetmaniok, Barbara Smoleń, Roman Wituła, The Stirling triangles, Proceedings of the Symposium for Young Scientists in Technology, Engineering and Mathematics (SYSTEM 2017), Kaunas, Lithuania, April 28, 2017, p. 35-41.
Programs
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Mathematica
Table[Sum[Binomial[n-k,k]^3,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Jul 31 2014 *) -
PARI
{a(n)=sum(k=0,n\2,binomial(n-k,k)^3)} -
PARI
{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^3*x^k + x*O(x^n))), n)} -
PARI
{a(n)=polcoeff(sum(m=0,n,x^(3*m)/(1-x-x^2+x*O(x^n))^(3*m+1)*(3*m)!/(m!)^3),n)}
Formula
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^3.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^k.
G.f.: A(x) = G( x^3/(1-x-x^2)^3 )/(1-x-x^2) where G(x) satisfies:
* G(x^3) = G( x*(1+3*x+9*x^2)/(1+6*x)^3 )/(1+6*x)
and G(x) is the g.f. of A006480.
Recurrence: (n-3)*n^2*a(n) = (n-3)*(3*n^2 - 3*n + 1)*a(n-1) - (n-1)*a(n-2) + 2*(n-2)*(11*n^2 - 44*n + 34)*a(n-3) + (n-3)*a(n-4) + (n-1)*(3*n^2 - 21*n + 37)*a(n-5) + (n-4)^2*(n-1)*a(n-6). - Vaclav Kotesovec, Jul 31 2014
a(n) ~ sqrt((9+4*sqrt(5))/12) * (2+sqrt(5))^n / (Pi*n). - Vaclav Kotesovec, Jul 31 2014
Equivalently, a(n) ~ phi^(3*n + 3) / (2*sqrt(3)*Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
Comments