A336023 a(0) = a(1) = a(2) = 1; a(n) = (1/2) * Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).
1, 1, 1, 1, 2, 4, 9, 24, 73, 241, 881, 3526, 15252, 70989, 354454, 1887349, 10674773, 63936181, 404240736, 2690174976, 18798077227, 137612318374, 1053181689299, 8410678868749, 69966341141373, 605313810207096, 5438284743262816, 50668461879077851
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Programs
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Mathematica
a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 2, k] a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}] nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2/4 + 1/2 Integrate[Integrate[A[x]^2, x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]! Table[SeriesCoefficient[1 + (72 (WeierstrassP[x, {-1/12, 1/432}] - WeierstrassPPrime[x, {-1/12, 1/432}]))/(1 - 12 WeierstrassP[x, {-1/12, 1/432}])^2, {x, 0, k}] k!, {k, 0, 30}] (* Jan Mangaldan, Nov 27 2020 *) a[ n_] := Module[{b, b0, b1, b2}, b[0]=b0; b[m_] := b[n] = Expand[Dt[b[m-1]] /. {Dt[b0]->b1, Dt[b1]->b2, Dt[b2]->b0*b1}]; b[n] /. {b0->1, b1->1, b2->1}]; (* Michael Somos, Jan 21 2021 *) -
PARI
{a(n) = my(t, b0='b0, b1='b1, b2='b2); t = b0; for(k=1, n, t = deriv(t, b0) * b1 + deriv(t, b1) * b2 + deriv(t, b2) * b0*b1); substvec(t, [b0, b1, b2], [1, 1, 1])}; /* Michael Somos, Jan 21 2021 */
Formula
E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/4 + (1/2) * Integral( Integral A(x)^2 dx) dx.
a(n) ~ n! * c * d^n * n, where d = 0.3327497845052484431663758951399704951054382840130942017786917698589406... is the root of the equation WeierstrassP(1/d, -1/12, 1/432) = 1/12 and c = 1.3286690290594713229820603959597228439699874045711069972066245497... - Vaclav Kotesovec, Jul 05 2020, updated Nov 28 2020
E.g.f.: 12^(1/3) * WeierstrassP((x + c)/12^(1/3), {-12^(1/3), 1/3}), where c = 6.010522299732561104963296256980218197216667207108031458739075310344... is the root of the equation WeierstrassP(c/12^(1/3), {-12^(1/3), 1/3}) = 12^(-1/3). - Vaclav Kotesovec, Jul 05 2020