A132303
Sum of cubes of trinomial coefficients: a(n) = Sum_{k=0..2n} trinomial(n,k)^3 where trinomial(n,k) = [x^k] (1 + x + x^2)^n.
Original entry on oeis.org
1, 3, 45, 831, 17181, 375903, 8530929, 198643455, 4714491357, 113550338127, 2767105469745, 68077260387315, 1688160321677025, 42142679453321307, 1058050429855640217, 26695057057648257231, 676431705046728704733, 17205315843416998571415, 439098128408223839364561, 11239967518370464873317291
Offset: 0
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f:= proc(n) local t; add(subs(x=1,t)^3, t = expand((1+x+x^2)^n)) end proc:
map(f, [$0..20]); # Robert Israel, Aug 29 2025
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a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k)^3)
A168593
G.f.: exp( Sum_{n>=1} A132303(n)*x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.
Original entry on oeis.org
1, 3, 27, 349, 5484, 96408, 1824758, 36393090, 754696998, 16130052394, 353134333470, 7884110379006, 178908263232959, 4115917059924057, 95806493175049929, 2252809457441037107, 53443567449376649304
Offset: 0
G.f.: A(x) = 1 + 3*x + 27*x^2 + 349*x^3 + 5484*x^4 + 96408*x^5 +...
log(A(x)) = 3*x + 45*x^2/2 + 831*x^3/3 + 17181*x^4/4 + 375903*x^5/5 +...+ A132303(n)*x^n/n +...
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{A027907(n,k) = polcoeff((1+x+x^2)^n, k)}
{A132303(n) = sum(k=0, 2*n, A027907(n,k)^3)}
{a(n) = local(A); A = exp(sum(m=1, n+1, A132303(m)*x^m/m) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
Showing 1-2 of 2 results.
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