OFFSET
0,3
FORMULA
G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
a(n) = Sum_{k=0..n-1} binomial(6*k-2+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n-1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k-1,k).
Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025
From Vaclav Kotesovec, Nov 09 2025: (Start)
Recurrence (of order 2): 15625*(n-1)*(5*n - 4)*(5*n - 3)*(5*n - 2)*(5*n - 1)*(22500*n^3 - 103050*n^2 + 156145*n - 78261)*a(n) = 360*(18225000000*n^8 - 151814250000*n^7 + 538281450000*n^6 - 1057888147500*n^5 + 1255277812500*n^4 - 915721202475*n^3 + 398019419015*n^2 - 93306702862*n + 8915190072)*a(n-1) - 3359232*(2*n - 3)*(3*n - 5)*(3*n - 4)*(6*n - 11)*(6*n - 7)*(22500*n^3 - 35550*n^2 + 17545*n - 2666)*a(n-2).
a(n) ~ 2^(6*n-2) * 3^(6*n-1) / 5^(5*n) * (1 - 4/(3*sqrt(15*Pi*n))). (End)
From Seiichi Manyama, May 06 2026: (Start)
a(n+1) = (1/n) * Sum_{k=0..n-1} (k+1) * (6*k+16) * 5^k * binomial(6*n+5,n-1-k) for n > 0.
a(n+1) = (1/n) * Sum_{k=0..n-1} (k+1) * (5*k+16) * 6^k * binomial(6*n+3-k,n-1-k) for n > 0. (End)
EXAMPLE
(1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...
MAPLE
A386368 := proc(n::integer)
add(binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1), k=0..n-1) ;
end proc:
seq(A386368(n), n=0..80) ; # R. J. Mathar, Jul 30 2025
MATHEMATICA
Table[Sum[5^(n-k-1) * Binomial[6*n-1, k], {k, 0, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 09 2025 *)
PROG
(PARI) a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));
(PARI) my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 19 2025
STATUS
approved
