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

%I A267219 #25 Mar 06 2025 04:44:19
%S A267219 1,4,22,152,1241,11444,115390,1243672,14104480,166460800,2028202288,
%T A267219 25363355200,324098616925,4217387014948,55737166570870,
%U A267219 746544123583928,10116388473816503,138496854665195996,1913322982776458234,26646647187379206440,373800949052597088329
%N A267219 Expansion of exp( Sum_{n >= 1} A002895(n)*x^n/n ).
%H A267219 Andrew Howroyd, <a href="/A267219/b267219.txt">Table of n, a(n) for n = 0..200</a>
%F A267219 n*a(n) = Sum_{k = 0..n-1} A002895(n-k)*a(k).
%F A267219 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 + ....
%F A267219 The o.g.f. A(x) satisfies 1 + x*d/dx(log(A(x))) = Sum_{n >= 0} A002895(n)*x^n.
%F A267219 A(x)^(1/4) = 1 + x + 4*x*2 + 25*x^3 + 199*x^4 + 1837*x^5 + ... appears to have integer coefficients.
%F A267219 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
%F A267219 a(n) ~ c * 16^n / n^(5/2), where c = 0.51464454254223965069014111993955211551145811088623409257449917225276305467... - _Vaclav Kotesovec_, Oct 31 2024
%p A267219 # define the Domb numbers
%p A267219 A002895 := n -> add(binomial(n,k)^2*binomial(2*n-2*k,n-k)*binomial(2*k,k), k = 0..n):
%p A267219 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:
%p A267219 seq(A267219(n), n = 0..20);
%t A267219 m = 21;
%t A267219 domb[n_] := Sum[Binomial[n, k]^2 Binomial[2n - 2k, n - k] Binomial[2k, k], {k, 0, n}];
%t A267219 Exp[Sum[domb[n] x^n/n, {n, 1, m}]] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Jan 04 2021 *)
%o A267219 (PARI) \\ here b(n) is A002895(n).
%o A267219 b(n)={sum(k=0, n, binomial(n, k)^2 * binomial(2*n-2*k, n-k) * binomial(2*k, k) )}
%o A267219 seq(n)={Vec(exp(sum(k=1, n, b(k)*x^k/k, O(x*x^n))))} \\ _Andrew Howroyd_, Dec 23 2019
%Y A267219 Cf. A002895, A267220.
%K A267219 nonn,easy
%O A267219 0,2
%A A267219 _Peter Bala_, Jan 12 2016
%E A267219 Terms a(17) and beyond from _Andrew Howroyd_, Dec 23 2019