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A249015
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A binomial convolution.
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1
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1, 1, 5, 17, 69, 339, 1677, 9321, 55137, 343659, 2285289, 15910857, 116120781, 886308147, 7033465989, 58008074409, 495792941337, 4381170220251, 39980186877537, 376025841184329, 3640077999981189, 36224841818288547, 370112212444620861, 3878334404076375657
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = 0^0 + Sum_{k=0..n-1} binomial(n,k)*b(k)*c(n-k-1),
where the numbers b(n) = A049425(n+1) have e.g.f. (1+t)^2*exp(t+t^2+t^3/3)
and the numbers c(n) have e.g.f. exp(-(t+t^2+t^3/3)).
D-finite with recurrence: a(n+4) + n*a(n+3) - 3*(n+3)*a(n+2) - 3*(n+3)*(n+2)*a(n+1) - (n+3)*(n+2)*(n+1)*a(n) = 0.
E.g.f.: A(t) = 1+(1+t)^2*exp(t+t^2+t^3/3)*Integral_{u=0..t} exp(-(u+u^2+u^3/3)) du.
Differential equation for the e.g.f.: (1+t)*A''(t) - (2+3*t+3*t^2+t^3)*A'(t) - 3*(1+t)^2*A(t) = 0.
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MATHEMATICA
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b[n_] := Sum[(n!/k!)Sum[Binomial[k, i]Binomial[k-i+2, n-2i-k]/3^i, {i, 0, k}], {k, 0, n}]
c[n_] := Sum[(n!/k!)(-1)^k Sum[Binomial[k, i]Binomial[k-i, n-2i-k]/3^i, {i, 0, k}], {k, 0, n}]
Table[If[n==0, 1, 0]+Sum[Binomial[n, k]b[k]c[n-k-1], {k, 0, n-1}], {n, 0, 40}]
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PROG
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(Maxima) b(n) := sum((n!/k!)*sum(binomial(k, i)*binomial(k-i+2, n-2*i-k)/3^i, i, 0, k), k, 0, n);
c(n) := sum((n!/k!)*(-1)^k*sum(binomial(k, i)*binomial(k-i, n-2*i-k)/3^i, i, 0, k), k, 0, n);
makelist((if n=0 then 1 else 0)+sum(binomial(n, k)*b(k)*c(n-k-1), k, 0, n-1), n, 0, 20);
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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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