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A267219 Expansion of exp( Sum_{n >= 1} A002895(n)*x^n/n ). 3
1, 4, 22, 152, 1241, 11444, 115390, 1243672, 14104480, 166460800, 2028202288, 25363355200, 324098616925, 4217387014948, 55737166570870, 746544123583928, 10116388473816503, 138496854665195996, 1913322982776458234, 26646647187379206440, 373800949052597088329 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

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.

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-Fran├žois Alcover, Jan 04 2021 *)

PROG

(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

CROSSREFS

Cf. A002895, A267220.

Sequence in context: A307439 A189845 A039304 * A152404 A062817 A196275

Adjacent sequences:  A267216 A267217 A267218 * A267220 A267221 A267222

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jan 12 2016

EXTENSIONS

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

STATUS

approved

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Last modified February 28 12:51 EST 2021. Contains 341707 sequences. (Running on oeis4.)