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A177111
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Sequence defined by the recurrence formula a(n+1) = sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=1, k=-2 and l=0.
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2
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1, 1, -2, -9, -30, -84, -204, -389, -326, 1780, 13156, 57452, 197552, 551846, 1138832, 752911, -8109806, -57353648, -255573404, -898715548, -2539157248, -5106161134, -1629827488, 50275158584, 330772150256, 1453122571658
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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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G.f.: f(z) = (1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=-2, l=0).
Conjecture: +(n+1)*a(n) +(-7*n+2)*a(n-1) +3*(5*n-7)*a(n-2) +3*(n-8)*a(n-3) +2*(-10*n+41)*a(n-4) +8*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2016
Conjectured recurrence follows from the differential equation (8*z^6 - 20*z^5 + 3*z^4 + 15*z^3 - 7*z^2 + z) * f'(z) + (2*z^4 - 15*z^3 + 9*z^2 - 5*z + 1) * f(z) + 4*z^4 - 11*z^3 + 9*z^2 + 3*z - 1 = 0 satisfied by the g.f. - Robert Israel, Jan 03 2024
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EXAMPLE
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a(2)=2*1*1-4=-2. a(3)=2*1*(-2)-4+1^2-2=-9. a(4)=2*1*(-9)-4+2*1*(-2)-4=-30.
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MAPLE
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l:=0: : k := -2 : m:=1:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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