A381828 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^2 ) )^(1/2).
1, 2, 10, 65, 480, 3824, 32039, 278256, 2482578, 22617830, 209540672, 1968031520, 18696064179, 179332892186, 1734451272240, 16895744042472, 165621305486976, 1632518433458400, 16170959983623314, 160888256475481560, 1607061512154585046, 16110030923830784248
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^2)/x)^(1/2))
Formula
G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*n-2*k,n-k)/(2*n+k+1).
D-finite with recurrence +432*n*(n-1)*(n-2)*(2*n+1)*(2*n-1)*(2*n-3)*(262261060139434887136491*n -880264534325728808928710)*a(n) +24*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(9441398165019655936913676*n^3 -1563359509176097527827297363*n^2 +8122005300033248841454135898*n -10005843136737488906545668303)*a(n-1) -8*(n-2)*(2*n-3)*(26904862014415612504704360259*n^5 -439294650192331167438487778367*n^4 +2462557164881954865201862193560*n^3 -6116391863054255517662202621591*n^2 +6730597164009721987374566778403*n -2508886036978141982914230533400)*a(n-2) +2*(3280856375160701992555505608813*n^7 -60505233834440544774094319915261*n^6 +458650706405377012453301766859297*n^5 -1843996542698657351167896639498197*n^4 +4199211312282774397146042070543498*n^3 -5283107978583820687249123910721062*n^2 +3195330463869279708956264243293272*n -571272270914692694572799416918200)*a(n-3) +3*(-10499174187769013704183946812135*n^7 +189831332911960443054698384732480*n^6 -1395267797131742288585801071743534*n^5 +5221938509132769354051685228032464*n^4 -9839826026184653630837080778918103*n^3 +6229383740555425356174546560814416*n^2 +6216439623275682391743799709941612*n -8390747283534155728971424365124320)*a(n-4) -112*(7*n-31)*(7*n-32) *(2094251874056865218841652*n -5622141652266976856940223)*(7*n-29)*(7*n-26) *(7*n-30)*(7*n-27)*a(n-5)=0. - R. J. Mathar, Mar 10 2025