OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..1032
FORMULA
a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*F(x)^2*A(x)^3 where F(x) is the g.f. of A001764.
G.f. satisfies: A(x/G(x)) = F(x*G(x)) where G(x) = F(x/G(x)) is the g.f. of A000108 and F(x) is the g.f. of A001764.
D-finite with recurrence: 4734532422978435*(9*n + 2)*(9*n + 4)*(9*n + 8)*(9*n + 1)*(9*n + 5)*(9*n + 7)*a(n) - 58458510*(226857785761746*n^6 + 2185740541961001*n^5 + 8767692038928429*n^4 + 18650016970223661*n^3 + 22118870293993413*n^2 + 13843014390537390*n + 3568227591118400)*a(n + 1) - 3542940*(973890386456076*n^6 + 23574804066437256*n^5 + 198923512574757123*n^4 + 821039635494392634*n^3 + 1810592157537134471*n^2 + 2057482005581249960*n + 950260348692776660)*a(n + 2) - 5832*(622341820494029016*n^6 + 16653021195510684324*n^5 + 180718374529495525230*n^4 + 1020146193840964736745*n^3 + 3168923106282387369624*n^2 + 5151338599669666177671*n + 3432538495922736797590)*a(n + 3) + 3456*(580538219634667926*n^6 + 15579961381983939138*n^5 + 172086817893491644245*n^4 + 1001964747516620440275*n^3 + 3244433560875259547094*n^2 + 5539805730294705724662*n + 3895792350327985718200)*a(n + 4) - 52992*(2*n + 11)*(n + 5)*(2485664904850902*n^4 + 42655722680155068*n^3 + 263769250266150291*n^2 + 687746852730720747*n + 621065456917750012)*a(n + 5) + 37377024*(2*n + 11)*(n + 6)*(n + 5)*(2*n + 13)*(46555410069*n^2 + 81213811407*n - 663147243142)*a(n + 6) + 44746481407768576*(2*n + 11)*(n + 7)*(n + 6)*(n + 5)*(2*n + 15)*(2*n + 13)*a(n + 7) = 0. - Robert Israel, Mar 18 2026
EXAMPLE
G.f.: A(x) = F(x*F(x)^2) = 1 + x + 5*x^2 + 31*x^3 + 211*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
MAPLE
f:= gfun:-rectoproc({ 4734532422978435*(9*n + 2)*(9*n + 4)*(9*n + 8)*(9*n + 1)*(9*n + 5)*(9*n + 7)*a(n) - 58458510*(226857785761746*n^6 + 2185740541961001*n^5 + 8767692038928429*n^4 + 18650016970223661*n^3 + 22118870293993413*n^2 + 13843014390537390*n + 3568227591118400)*a(n + 1) - 3542940*(973890386456076*n^6 + 23574804066437256*n^5 + 198923512574757123*n^4 + 821039635494392634*n^3 + 1810592157537134471*n^2 + 2057482005581249960*n + 950260348692776660)*a(n + 2) - 5832*(622341820494029016*n^6 + 16653021195510684324*n^5 + 180718374529495525230*n^4 + 1020146193840964736745*n^3 + 3168923106282387369624*n^2 + 5151338599669666177671*n + 3432538495922736797590)*a(n + 3) + 3456*(580538219634667926*n^6 + 15579961381983939138*n^5 + 172086817893491644245*n^4 + 1001964747516620440275*n^3 + 3244433560875259547094*n^2 + 5539805730294705724662*n + 3895792350327985718200)*a(n + 4) - 52992*(2*n + 11)*(n + 5)*(2485664904850902*n^4 + 42655722680155068*n^3 + 263769250266150291*n^2 + 687746852730720747*n + 621065456917750012)*a(n + 5) + 37377024*(2*n + 11)*(n + 6)*(n + 5)*(2*n + 13)*(46555410069*n^2 + 81213811407*n - 663147243142)*a(n + 6) + 44746481407768576*(2*n + 11)*(n + 7)*(n + 6)*(n + 5)*(2*n + 15)*(2*n + 13)*a(n + 7), a(0) = 1, a(1) = 1, a(2) = 5, a(3) = 31, a(4) = 211, a(5) = 1516, a(6) = 11295}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 18 2026
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*binomial(3*(n-k)+2*k, n-k)*2*k/(3*(n-k)+2*k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 14 2009
STATUS
approved