A267219 Expansion of exp( Sum_{n >= 1} A002895(n)*x^n/n ).

1, 4, 22, 152, 1241, 11444, 115390, 1243672, 14104480, 166460800, 2028202288, 25363355200, 324098616925, 4217387014948, 55737166570870, 746544123583928, 10116388473816503, 138496854665195996, 1913322982776458234, 26646647187379206440, 373800949052597088329

Offset: 0

Author

Peter Bala, Jan 12 2016

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A002895, A267220.

Programs

Maple

# define the Domb numbers
A002895 := n -> add(binomial(n,k)^2*binomial(2*n-2*k,n-k)*binomial(2*k,k), k = 0..n):
A267219 := proc (n) option remember; if n = 0 then 1 else 1/n*add( A002895(n-k)*A267219(k), k = 0..n-1) end if; end proc:
seq(A267219(n), n = 0..20);

Mathematica

m = 21;
domb[n_] := Sum[Binomial[n, k]^2 Binomial[2n - 2k, n - k] Binomial[2k, k], {k, 0, n}];
Exp[Sum[domb[n] x^n/n, {n, 1, m}]] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 04 2021 *)

PARI

\\ here b(n) is A002895(n).
b(n)={sum(k=0, n, binomial(n, k)^2 * binomial(2*n-2*k, n-k) * binomial(2*k, k) )}
seq(n)={Vec(exp(sum(k=1, n, b(k)*x^k/k, O(x*x^n))))} \\ Andrew Howroyd, Dec 23 2019

Formula

n*a(n) = Sum_{k = 0..n-1} A002895(n-k)*a(k).

O.g.f. A(x) = exp( Sum_{n >= 1} A002895(n)*x^n/n ) = 1 + 4*x + 22*x^2 + 152*x^3 + 1241*x^4 + ....

The o.g.f. A(x) satisfies 1 + x*d/dx(log(A(x))) = Sum_{n >= 0} A002895(n)*x^n.

A(x)^(1/4) = 1 + x + 4*x*2 + 25*x^3 + 199*x^4 + 1837*x^5 + ... appears to have integer coefficients.

For integer m, define a sequence {u_m(n) : n >= 0} by u_m(n) = [x^n] A(x)^(m*n/4). Conjecture: the supercongruences u_m(n*p^r) == u_m(n*p^(r-1)) (mod p^(2*r)) hold for all primes p >= 5 and positive integers n and r. - Peter Bala, Oct 17 2024

a(n) ~ c * 16^n / n^(5/2), where c = 0.51464454254223965069014111993955211551145811088623409257449917225276305467... - Vaclav Kotesovec, Oct 31 2024

Extensions

Terms a(17) and beyond from Andrew Howroyd, Dec 23 2019