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A289694
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The number of partitions of [n] with exactly 4 blocks without peaks.
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2
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0, 0, 0, 1, 4, 16, 64, 236, 818, 2736, 8934, 28622, 90324, 281792, 871556, 2677750, 8184383, 24913238, 75593383, 228793147, 691094857, 2084237036, 6277871658, 18890568921, 56798001639, 170665733660, 512554832309, 1538718547049
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OFFSET
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1,5
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (10,-45,130,-280,471,-643,734,-701,575,-400,237,-121,49,-18,4,-1).
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FORMULA
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G.f. x^4*(x^2-x+1)*(x^4-x^3+3*x^2-2*x+1)*(x^6-x^5+5*x^4-4*x^3+6*x^2-3*x+1) / ( (x-1)*(x^5-x^4+4*x^3-3*x^2+3*x-1)*(x^7-x^6+6*x^5-5*x^4+10*x^3-6*x^2+4*x-1)*(x^3-x^2+2*x-1) ). - R. J. Mathar, Mar 11 2021
a(n)= 10*a(n-1) -45*a(n-2) +130*a(n-3) -280*a(n-4) +471*a(n-5) -643*a(n-6) +734*a(n-7) -701*a(n-8) +575*a(n-9) -400*a(n-10) +237*a(n-11) -121*a(n-12) +49*a(n-13) -18*a(n-14) +4*a(n-15) -a(n-16). - R. J. Mathar, Mar 11 2021
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MAPLE
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with(orthopoly) :
nmax := 15:
tpr := 1+x^2/2 :
k := 4:
g := x^k ;
for j from 1 to k do
if j> 1 then
g := g*( U(j-1, tpr)-(1+x)*U(j-2, tpr)) / ((1-x)*U(j-1, tpr)-U(j-2, tpr)) ;
else
# note that U(-1, .)=0, U(0, .)=1
g := g* U(j-1, tpr) / ((1-x)*U(j-1, tpr)) ;
end if;
end do:
simplify(%) ;
taylor(g, x=0, nmax+1) ;
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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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STATUS
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approved
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