|
| |
|
|
A226751
|
|
G.f.: 1 / (1 + 6*x*G(x) - 7*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
|
|
1
|
|
|
|
1, 1, 9, 48, 289, 1761, 10932, 68664, 435201, 2777763, 17829489, 114968052, 744178716, 4832624044, 31469746632, 205422018288, 1343734578561, 8806130111847, 57805893969531, 380013533789928, 2501507255441049, 16486378106441697, 108773240389894056
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} C(2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(-2*k, k).
G.f.: 1/(1 - x*G(x) - 7*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
Conjecture: 18*n*(2*n-1)*(55*n-76)*a(n) +(-11605*n^3+28521*n^2-20870*n+4536)*a(n-1) -24*(55*n-21)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + 9*x^2 + 48*x^3 + 289*x^4 + 1761*x^5 + 10932*x^6 +...
A related series is G(x) = 1 + x*G(x)^3, where
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
such that A(x) = 1/(1 + 6*x*G(x) - 7*x*G(x)^2).
|
|
|
MATHEMATICA
|
Table[Sum[Binomial[n+2*k, n-k]*Binomial[2*n-2*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 17 2013 *)
|
|
|
PROG
|
(PARI) {a(n)=sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(3*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^3+x*O(x^n)); polcoeff(1/(1+6*x*G-7*x*G^2), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^3+x*O(x^n)); polcoeff(1/(1-x*G-7*x^2*G^4), n)}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
