A364400 -id:A364400 - cp's OEIS frontend

A364407 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^3).

1, 2, -6, 42, -350, 3234, -31878, 328426, -3494142, 38093442, -423344966, 4778162922, -54621614814, 631114404258, -7358619459654, 86472788963370, -1023093071862526, 12177054520248834, -145700860758056838, 1751559565664348842, -21145576694586256734

Offset: 0

Seiichi Manyama, Jul 23 2023

sign

Cf. A364393, A364408, A364409.

Cf. A349310, A364400.

A364407 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(n+3*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364407(n),n=0..70); # R. J. Mathar, Jul 25 2023

nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x*(1 + 1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

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

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349310.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+3*k-2,n-1) for n > 0.
D-finite with recurrence 3*n*(52*n-187)*(3*n-1) *(3*n-2)*a(n) +(14392*n^4 -70190*n^3 +56951*n^2 +50237*n -49500)*a(n-1) +3*(-17252*n^4 +205959*n^3 -851664*n^2 +1432459*n -815652)*a(n-2) +18*(-472*n^4 +1294*n^3 +36359*n^2 -226731*n +361171)*a(n-3) -27*(n-5)*(404*n^3 -2235*n^2 -4058*n +26406)*a(n-4) -81*(n-5)*(n-6) *(8*n^2+358*n-1785)*a(n-5) +243*(n-5)*(n-6) *(n-7)*(4*n-31)*a(n-6)=0. - R. J. Mathar, Jul 25 2023

A364398 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)).

Original entry on oeis.org

1, 2, -14, 162, -2270, 35234, -582958, 10076354, -179802046, 3287029698, -61246957902, 1158889656930, -22207636788894, 430106644358242, -8405699952109166, 165557885912786818, -3282954949273886590, 65487784219460233602, -1313225110482709157518

Offset: 0

Seiichi Manyama, Jul 22 2023

sign

Cf. A112478, A364394, A364396.
Cf. A364399, A364400.
Cf. A260332.

A364398 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(4*n+k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364398(n),n=0..70); # R. J. Mathar, Jul 25 2023

nmax = 18; A[] = 1; Do[A[x] = 1+x/A[x]^3*(1+1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *)

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

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k-2,n-1) for n > 0.
D-finite with recurrence 2*n*(462919*n -714364)*(4*n-3) *(2*n-1)*(4*n-1)*a(n) +(625365036*n^5 -2723245780*n^4 +4202103460*n^3 -2471353250*n^2 +81675089*n +289227120)*a(n-1) +(-484851248*n^5 +5501638270*n^4 -25122933600*n^3 +57439557800*n^2 -65490996232*n +29691239955)*a(n-2) +(2*n-5)*(652184*n -1103659)*(4*n-13) *(n-3)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Jul 25 2023
a(n) ~ c*(-1)^(n-1)*256^n*27^(-n)*2F1([1-n, 4*n], [3*n], -1)*n^(-3/2), with c = sqrt(3/(32*Pi)). - Stefano Spezia, Oct 21 2023

A364399 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -16, 212, -3400, 60384, -1142960, 22598832, -461250208, 9644611008, -205537131008, 4447969973888, -97482797466624, 2159242220999936, -48260706692535552, 1087076798266594048, -24652590023639251456, 562396337623786449920

Offset: 0

Seiichi Manyama, Jul 22 2023

sign

Cf. A364393, A364395, A364397.
Cf. A364398, A364400.
Cf. A363380.

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

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A363380.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n-1)*256^n*27^(-n)*3F2([-n, 2*n, 2*n-1/2], [3*n/2, (3*n+1)/2], -1)*n^(-3/2), with c = (1/8)*sqrt(3/(2*Pi)). - Stefano Spezia, Oct 21 2023

A371562 G.f. A(x) satisfies A(x) = 1 + x/A(x)^3 * (1 + A(x)^5).

Original entry on oeis.org

1, 2, -2, 30, -166, 1514, -12474, 114006, -1050830, 10005138, -96772786, 951500686, -9469982966, 95267209850, -966979784554, 9891522355270, -101866781649310, 1055294818173474, -10989809960251490, 114983445265899774, -1208092406024272710

Offset: 0

Seiichi Manyama, Apr 13 2024

sign

Cf. A364395, A364396, A364398, A364399, A364400, A364407.

a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n-5*k-2, n-1))/n);

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n-5*k-2,n-1) for n > 0.

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A364407 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^3).

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A364398 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)).

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A364399 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)^2).

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A371562 G.f. A(x) satisfies A(x) = 1 + x/A(x)^3 * (1 + A(x)^5).

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