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A199247
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G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^2*A(x)^3 + x^3*A(x)^4).
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0
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1, 1, 2, 7, 25, 92, 359, 1453, 6018, 25411, 109032, 473942, 2082550, 9235675, 41284297, 185819487, 841433773, 3830604764, 17521832924, 80490034307, 371169646860, 1717567062240, 7973153760616, 37119622029816, 173272771061677, 810810134833720, 3802675087749650
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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) = Sum_{k=0..[n/2]} C(n+k, k)*C(n+2*k+1, n-2*k)/(n+1).
G.f.: A(x) = (1/x)*Series_Reversion( x/(1+x) - x^3 - x^4 ).
Recurrence: 240*(n-2)*(n-1)*n*(n+1)*(543314047*n^6 - 9777816570*n^5 + 72419307730*n^4 - 282185906280*n^3 + 609197162263*n^2 - 689640519870*n + 319111630200)*a(n) = 16*(n-2)*(n-1)*n*(40205239478*n^7 - 743661045919*n^6 + 5720613587045*n^5 - 23561094982780*n^4 + 55538240643272*n^3 - 73680608104486*n^2 + 49378914311580*n - 12008736298800)*a(n-1) - 4*(n-2)*(n-1)*(91820073943*n^8 - 1836091148216*n^7 + 15607730833978*n^6 - 73389282297476*n^5 + 207862429740607*n^4 - 361251960415604*n^3 + 373910114125992*n^2 - 209189908378584*n + 47967495287040)*a(n-2) + 4*(n-2)*(516148344650*n^9 - 11611593292425*n^8 + 113759368018383*n^7 - 635856019816596*n^6 + 2229739867262616*n^5 - 5073403355773305*n^4 + 7464399000631567*n^3 - 6815882210854914*n^2 + 3481564826740104*n - 750085518551040)*a(n-3) - (710111459429*n^10 - 18460497932422*n^9 + 212532989720754*n^8 - 1424989554134304*n^7 + 6151577737700817*n^6 - 17829379077999138*n^5 + 35045804727952456*n^4 - 45976065299760536*n^3 + 38349466932479664*n^2 - 18244541292137280*n + 3720781171814400)*a(n-4) + 8*(n-4)*(2*n - 7)*(4*n - 17)*(4*n - 15)*(543314047*n^6 - 6517932288*n^5 + 31679935585*n^4 - 79420560120*n^3 + 107526834808*n^2 - 73755881832*n + 19667171520)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 2*r*s^2 + 6*r^2*s^3 + 4*r^3*s^4) / (3 + 12*r*s + 10*r^2*s^2)) / (2*sqrt(Pi) * n^(3/2) * r^(n + 1/2)), where r = 0.2013887134255166663337905234200508058745432798749... and s = 1.748883682651423548151134706780057317341305059059... are roots of the system of equations (1 + r*s)*(1 + r^2*s^3 + r^3*s^4) = s, r*(1 + 3*r*s^2 + 8*r^2*s^3 + 5*r^3*s^4) = 1. - Vaclav Kotesovec, Nov 18 2017
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 25*x^4 + 92*x^5 + 359*x^6 + 1453*x^7 +...
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 135*x^4 + 543*x^5 + 2243*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 233*x^4 + 976*x^5 + 4154*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 370*x^4 + 1611*x^5 + 7065*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^3 + 2*x^3*A(x)^4 + x^4*A(x)^5.
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MATHEMATICA
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Table[Sum[Binomial[n+k, k]*Binomial[n+2*k+1, n-2*k]/(n+1), {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Nov 18 2017 *)
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PROG
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(PARI) {a(n)=sum(k=0, n\2, binomial(n+k, k)*binomial(n+2*k+1, n-2*k))/(n+1)}
(PARI) {a(n)=local(A=1+x); A=1/x*serreverse(x/(1+x+x*O(x^n)) - x^3 - x^4); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^2*A^3 + x^3*A^4)+x*O(x^n)); polcoeff(A, 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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