A368011 -id:A368011 - cp's OEIS frontend

A369114 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3) ).

1, 3, 15, 92, 630, 4620, 35494, 282015, 2298417, 19108265, 161418543, 1381606044, 11955789440, 104427062460, 919430773992, 8151530382264, 72711166411422, 652075100808960, 5875868463764446, 53175058170610530, 483082193418731280, 4404057834071995110

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Index entries for reversions of series

Cf. A368011, A369102.

Cf. A097188.

A369114 := proc(n)
    add(binomial(n+k,k) * binomial(4*n+2,n-3*k),k=0..floor(n/3)) ;
    %/(n+1) ;
end proc;
seq(A369114(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^3))/x)

a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(4*n+2, n-3*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(4*n+2,n-3*k).

D-finite with recurrence 81*n*(n-1)*(n+1)*a(n) -945*n^2*(n-1)*a(n-1) +441*(n-1)*(3*n^2+9*n-20)*a(n-2) +3*(1039*n^3 -12393*n^2 +37406*n-33232)*a(n-3) -448*(2*n-5) *(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Jan 25 2024

A369125 Expansion of (1/x) * Series_Reversion( x * ((1-x)^5+x^5) ).

Original entry on oeis.org

1, 5, 40, 385, 4095, 46375, 548300, 6689550, 83593250, 1064463125, 13762667750, 180189122750, 2384130651875, 31829162793750, 428227113655000, 5800188020157500, 79026653220693750, 1082367047392625000, 14893567523068062500, 205796463286063912500

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Index entries for reversions of series

Cf. A097188, A369123, A369124.

Cf. A368011.

A369125 := proc(n)
    add((-1)^k * binomial(n+k,k) * binomial(6*n+4,n-5*k),k=0..floor(n/5)) ;
    %/(n+1) ;
end proc;
seq(A369125(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^5+x^5))/x)

a(n) = sum(k=0, n\5, (-1)^k*binomial(n+k, k)*binomial(6*n+4, n-5*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+k,k) * binomial(6*n+4,n-5*k).

D-finite with recurrence 2*n*(n-1)*(n-2)*(24115*n-65551)*(n+1)*a(n) -5*n*(n-1) *(n-2)*(392783*n^2 -1296338*n +636787)*a(n-1) +100*(n-1)*(n-2) *(86231*n^3 -471376*n^2 +844569*n -522390)*a(n-2) +50*(n-2)*(2114435*n^4 -14778692*n^3 +35712085*n^2 -33505588*n +8727216)*a(n-3) +50*(13474985*n^5 -137009240*n^4 +513119690*n^3 -832716700*n^2 +478740305*n +26151216)*a(n-4) -125*(5*n-21) *(6943*n-12944) *(5*n-19)*(5*n-18)*(5*n-17)*a(n-5)=0. - R. J. Mathar, Jan 25 2024

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A369114 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3) ).

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A369125 Expansion of (1/x) * Series_Reversion( x * ((1-x)^5+x^5) ).

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