A376490 -id:A376490 - cp's OEIS frontend

A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

1, 0, 0, 1, 1, 1, 3, 5, 7, 14, 26, 43, 79, 148, 264, 483, 903, 1664, 3080, 5771, 10795, 20209, 38059, 71799, 135569, 256762, 487310, 925981, 1762841, 3361897, 6419595, 12275301, 23505143, 45061424, 86485016, 166176499, 319630115, 615387675, 1185940209, 2287527119, 4416083429

Offset: 0

Ilya Gutkovskiy, Jul 21 2021

nonn

Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.

nmax = 40; A[] = 0; Do[A[x] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 40}]

a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 30 2021
From Seiichi Manyama, Sep 26 2024: (Start)
G.f.: 2/(1 + sqrt(1 - 4*x^3/(1 - x))).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)

A376489 G.f. satisfies A(x) = 1 / (1 - x^2*A(x)^2 / (1 - x)).

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 22, 49, 143, 359, 1025, 2742, 7812, 21666, 62044, 175927, 507484, 1460297, 4243802, 12340559, 36108354, 105839241, 311551092, 919000678, 2719362502, 8063263402, 23967845874, 71379427920, 213010634136, 636757780808, 1906765570820

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

Cf. A002212, A376490, A376491.

A376489 := proc(n)
    add(binomial(3*k,k)*binomial(n-k-1,n-2*k)/(2*k+1),k=0..floor(n/2)) ;
end proc:
seq(A376489(n),n=0..70) ; # R. J. Mathar, Sep 26 2024

a(n) = sum(k=0, n\2, binomial(3*k, k)*binomial(n-k-1, n-2*k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*k,k) * binomial(n-k-1,n-2*k) / (2*k+1).

D-finite with recurrence 8*n*(n+1)*a(n) -28*n*(n-1)*a(n-1) +2*(-9*n^2-n+14)*a(n-2) +(115*n^2-463*n+426)*a(n-3) +4*(-26*n^2+168*n-265)*a(n-4) +3*(3*n-13)*(3*n-14)*a(n-5)=0. - R. J. Mathar, Sep 26 2024

A376491 G.f. satisfies A(x) = 1 / (1 - x^4*A(x)^4 / (1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 6, 11, 16, 21, 61, 136, 246, 391, 856, 1926, 3886, 7021, 14146, 30606, 64276, 125561, 251147, 527752, 1115877, 2273557, 4611992, 9583058, 20198698, 41982193, 86481758, 179676908, 377608039, 791559669, 1649078139, 3441054929

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

Cf. A002212, A376489, A376490.

a(n) = sum(k=0, n\4, binomial(5*k, k)*binomial(n-3*k-1, n-4*k)/(4*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(5*k,k) * binomial(n-3*k-1,n-4*k) / (4*k+1).

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A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

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A376489 G.f. satisfies A(x) = 1 / (1 - x^2*A(x)^2 / (1 - x)).

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A376491 G.f. satisfies A(x) = 1 / (1 - x^4*A(x)^4 / (1 - x)).

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