A348859 -id:A348859 - cp's OEIS frontend

A348857 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(2*x))).

1, 2, 7, 44, 481, 9254, 326395, 21927776, 2874607189, 744650622170, 383510575423471, 393869218949592212, 807827718206737362889, 3311287802485779192925838, 27136007596894473408507305443, 444677773080105539125038867872456, 14572535437424416878539776253365375549

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A007317, A015083, A348858, A348859.

nmax = 16; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[2 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[2^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

a(n) = 1 + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).

a(n) ~ c * 2^(n*(n-1)/2), where c = 10.96416094535958612421479005398505892527943513193882801485045169159164... - Vaclav Kotesovec, Nov 02 2021

A348858 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(3*x))).

Original entry on oeis.org

1, 2, 9, 103, 3101, 261192, 64285189, 47059492688, 103060910397021, 676492249628112382, 13317427360663454672669, 786420726604930579016189223, 139314431838014895142151741877241, 74037818920801629179455290512454633872, 118040419689979917511971388549088825283510249

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A007317, A015084, A348857, A348859.

nmax = 14; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[3 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

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

a(n) ~ c * 3^(n*(n-1)/2), where c = 4.508135635010167805309616576501854361005320931661829410476785686203732753... - Vaclav Kotesovec, Nov 02 2021

A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 1, -1, 1, 8, 1, -28, 1, 134, 1, -649, 1, 3320, 1, -17497, 1, 94526, 1, -520507, 1, 2910896, 1, -16487794, 1, 94393106, 1, -545337199, 1, 3175320608, 1, -18615098836, 1, 109783526822, 1, -650884962907, 1, 3877184797784, 1, -23193307022860, 1, 139271612505362

Offset: 0

Seiichi Manyama, Jul 17 2025

sign

Cf. A007317, A348857, A348858, A348859.

Cf. A064641, A386300.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, (-1)^j*v[j+1]*v[i-j])); v;

G.f. A(x) satisfies:
(1) A(x) = 1/( (1-x) * (1-x*A(-x)) ).
(2) A(x)*A(-x) = B(-x^2), where B(x) is the g.f. of A064641.
(3) A(x) = 1/(1-x) + 2*x/(1+x^2 + sqrt(1+6*x^2-3*x^4)).
a(2*n) = 1 and a(2*n+1) = 1 + (-1)^n * A064641(n) for n >= 0.

cp's OEIS Frontend

A348857 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(2*x))).

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A348858 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(3*x))).

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Mathematica

Formula

A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

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