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A267219
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Expansion of exp( Sum_{n >= 1} A002895(n)*x^n/n ).
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3
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1, 4, 22, 152, 1241, 11444, 115390, 1243672, 14104480, 166460800, 2028202288, 25363355200, 324098616925, 4217387014948, 55737166570870, 746544123583928, 10116388473816503, 138496854665195996, 1913322982776458234, 26646647187379206440, 373800949052597088329
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OFFSET
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0,2
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LINKS
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FORMULA
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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.
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MAPLE
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# 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:
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MATHEMATICA
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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 *)
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PROG
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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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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