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A051163
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Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).
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11
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1, 2, 5, 12, 30, 76, 194, 496, 1269, 3250, 8337, 21428, 55184, 142376, 367916, 952000, 2466014, 6393372, 16586678, 43054344, 111801908, 290412296, 754543052, 1960808160, 5096293794, 13247503540, 34440553562, 89549255592, 232868582328, 605646682144
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = 1 + Sum_{k=1..n} Sum_{j=0..n-k} C(k, j)*C(n-k, j)*a(j). - Paul D. Hanna, Nov 22 2004
G.f. A(x) satisfies: A(x) = A(x^2/(1-x)^2)/(1-x)^2 and A(x^2) = A(x/(1+x))/(1+x)^2. - Paul D. Hanna, Nov 22 2004
a(0) = 1; a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1) * a(k). - Ilya Gutkovskiy, Apr 07 2022
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MAPLE
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a:= proc(n) option remember; add(`if`(k<2, 1,
a(iquo(k, 2)))*binomial(n, k), k=0..n)
end:
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MATHEMATICA
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a[n_] := a[n] = 1 + Sum[Binomial[k, j]*Binomial[n-k, j]*a[j], {k, 1, n}, {j, 0, n-k}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 11 2015 *)
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PROG
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(PARI) a(n)=1+sum(k=1, n, sum(j=0, n-k, binomial(k, j)*binomial(n-k, j)*a(j)))
(PARI) a(n)=local(A, m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=2; A=subst(A, x, (x/(1-x))^2)/(1-x)); polcoeff(A^2, n))
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CROSSREFS
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KEYWORD
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easy,nonn,eigen
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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