A015085 -id:A015085 - cp's OEIS frontend

A376113 a(0) = 1; a(n) = (1/3) * Sum_{k=1..n} (4^k-1) * a(k-1) * a(n-k).

1, 1, 6, 137, 11938, 4095882, 5599192492, 30588428274345, 668265444025582410, 58395039572032120897838, 20410643002515607839683651348, 28536181214271796693200339702494058, 159585939576145805663910944364491926768148, 3569877304419418296304606194938539586766279745396

Offset: 0

Ilya Gutkovskiy, Sep 10 2024

nonn

Cf. A015085, A337556, A376111, A376112.

a[0] = 1; a[n_] := a[n] = (1/3) Sum[(4^k - 1) a[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 13}]
nmax = 13; A[] = 0; Do[A[x] = 3/(3 + x A[x] - 4 x A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 3 / (3 + x * A(x) - 4 * x * A(4*x)).

A349287 G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(4*x)^2).

Original entry on oeis.org

1, 1, 9, 321, 42937, 22259313, 45726174057, 374866565186721, 12285883413435994137, 1610409077693221284887505, 844327818646575560326075164105, 1770688839714867344554954935264852993, 14853625190589908388648838739441430566681721

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

Cf. A001764, A015085, A171192, A349286.

nmax = 12; A[] = 0; Do[A[x] = 1/(1 - x A[4 x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[4^(i + j) a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 12}]

a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 4^(i+j) * a(i) * a(j) * a(n-i-j-1).

a(n) ~ c * 2^(n^2), where c = 0.6660597482166910709619924328518595274303795046... - Vaclav Kotesovec, Nov 14 2021

cp's OEIS Frontend

A376113 a(0) = 1; a(n) = (1/3) * Sum_{k=1..n} (4^k-1) * a(k-1) * a(n-k).

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