A387085 -id:A387085 - cp's OEIS frontend

A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

1, 0, 2, 4, 16, 52, 188, 672, 2458, 9052, 33648, 125864, 473500, 1789632, 6791528, 25863568, 98796096, 378411332, 1452886052, 5590262688, 21551271916, 83228809640, 321933018272, 1247062996304, 4837152438556, 18785529571200, 73037938668632, 284268423472432

Offset: 0

Seiichi Manyama, Aug 16 2025

Cf. A183161, A208977, A387084.

Cf. A000108, A000984, A387085.

nmax = 30; CoefficientList[Series[Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] - 1)], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))

Sum_{k=0..n} a(k) * a(n-k) = A387085(n).
G.f.: 1/sqrt( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/sqrt(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g/sqrt((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
a(n) ~ 2^(2*n - 1/2) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2/(16*Pi*sqrt(2*n))). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*(n-1)*a(n) -2*(n-1)*(10*n-17)*a(n-1) +4*(4*n^2-24*n+29)*a(n-2) +32*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

A386371 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n+1,k).

Original entry on oeis.org

1, 3, 31, 317, 3399, 37418, 419229, 4756104, 54463335, 628197809, 7287712566, 84942987198, 993941174829, 11668806723876, 137378189197112, 1621322803014672, 19175540677541991, 227217662222902443, 2696878158795639549, 32057403690640189635, 381573145993865438254

Offset: 0

Seiichi Manyama, Aug 17 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A001449, A079589, A079678, A371753, A385632, A386812.
Cf. A226705, A226733, A226751, A387085.
Cf. A002294.

[&+[(-3)^(n-k) * Binomial(5*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[(-3)^(n-k)*Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n+1, k));

a(n) = [x^n] (1+x)^(5*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(4*n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(5*n-k,n-k).
G.f.: 1/(1 - x*g^3*(-10+13*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g^2/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 7*(B(x)-1)/5), where B(x) is the g.f. of A001449.
D-finite with recurrence 648*n*(135551509682187347695*n -244103380745409504343) *(4*n-1)*(2*n-1)*(4*n-3)*a(n) +(-33979500619583537984836075*n^5 +130803893690808003041848009*n^4 -168380151442376797602371231*n^3 +62069291513227826684567999*n^2 +49760069127090078338544954*n -39530305857276050670355320)*a(n-1) +40*(-108999332467309598098777*n^5 -28981701912184019189355*n^4 -1554974299825191814369159*n^3 +13581461461293413639358363*n^2 -28599284433109723900055776*n +18909354537435947334628944)*a(n-2) +211200*(5*n-11) *(5*n-9)*(28440609019752807*n +93502568692163852)*(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

cp's OEIS Frontend

A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

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