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A304121
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Expansion of A(x) = 1 + x + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3.
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0
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1, 2, 3, 8, 24, 73, 230, 751, 2512, 8549, 29520, 103192, 364441, 1298336, 4660273, 16837743, 61187249, 223489715, 820040293, 3021286338, 11172619317, 41454901924, 154285693649, 575826677260, 2154643706124, 8081453618583, 30377809192246, 114421431917805
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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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a(n) = Sum_{i=0..n} Sum_{k=0..n-i} Sum_{j=0..k} C(i,j)*C(j,k-j)*C(k+1,n-k-i)* C(k+i,i)/(k+1).
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MAPLE
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a:= n-> coeff(series(RootOf((A*x)^3+(A*x)^2+
A*(x-1)+x+1, A), x, n+1), x, n):
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MATHEMATICA
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Array[Sum[Sum[Sum[Binomial[i, j] Binomial[j, k - j], {j, 0, k}] Binomial[k + 1, # - k - i] Binomial[k + i, i]/(k + 1), {k, 0, # - i}], {i, 0, # + 1}] &, 31, 0] (* Michael De Vlieger, May 10 2018 *)
n = 27; A = Sum[a[k] x^k, {k, 0, n}] + x O[x]^n; Table[a[k], {k, 0, n}] /.
Solve[LogicalExpand[1 + x + x A + x^2 A^2 + x^3 A^3 == A]] // Flatten (* Peter Luschny, May 14 2018 *)
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PROG
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(Maxima)
a(n):=sum(sum(((sum(binomial(i, j)*binomial(j, k-j), j, 0, k))*binomial(k+1, n-k-i)* binomial(k+i, i))/(k+1), k, 0, n-i), i, 0, n);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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