A363982 -id:A363982 - cp's OEIS frontend

A291534 Expansion of the series reversion of x/((1 + x)*(1 - x^2)).

1, 1, 0, -3, -7, -4, 24, 85, 99, -215, -1196, -2100, 1420, 17512, 42160, 9477, -252073, -815965, -736456, 3365813, 15248793, 22861712, -37036000, -273657748, -575046252, 180950476, 4658415696, 13042693000, 6717278152, -73400374512, -275797704864, -321427878811, 1012425395135

Offset: 1

Ilya Gutkovskiy, Aug 25 2017

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Reversion of g.f. for the canonical enumeration of integers (A001057).

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Series Reversion
Index entries for reversions of series

Cf. A001057, A001764, A036765.

Cf. A363982, A364051, A364764.

Rest[CoefficientList[InverseSeries[Series[x/((1 + x) (1 - x^2)), {x, 0, 33}], x], x]]
Table[HypergeometricPFQ[{(1 - n)/2, 1 - n/2, -n}, {1, 3/2}, 1], {n, 1, 33}] (* Vladimir Reshetnikov, Oct 15 2018 *)

a(n) = sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(2*n, n-1-k))/n; \\ Seiichi Manyama, Aug 05 2023

G.f. A(x) satisfies: A(x)/((1 + A(x))*(1 - A(x)^2)) = x.
a(n) = hypergeom([(1 - n)/2, 1 - n/2, -n], [1, 3/2], 1). - Vladimir Reshetnikov, Oct 15 2018
From Vladimir Reshetnikov, Oct 18 2018: (Start)
G.f.: 2^(1/3)*(6 - 8*x - 2^(1/3)*t^2)/(6*sqrt(x)*t), where t = (3*sqrt(12 - 39*x + 96*x^2) - (9 + 16*x)*sqrt(x))^(1/3).
D-finite with recurrence: 64*n*(n + 1)*(2*n + 1)*a(n) - 4*(n + 1)*(37*n^2 + 134*n + 120)*a(n + 1)  + (n + 2)*(55*n^2 + 235*n + 240)*a(n + 2) - 2*(6*n + 21)*(n + 2)*(n + 3)*a(n + 3) = 0. (End)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(2*n,n-1-k). - Seiichi Manyama, Aug 05 2023
From Seiichi Manyama, Aug 11 2023: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(3*n-k,n-1-k). (End)

A364864 G.f. A(x) satisfies A(x) = 1 + xA(x)^3 / (1 + xA(x)^3).

Original entry on oeis.org

1, 1, 2, 4, 6, -1, -58, -304, -1090, -2876, -4216, 9244, 106746, 529962, 1874628, 4669760, 4309742, -35179252, -277928680, -1269921008, -4214431912, -9197175241, 30113526, 128659598896, 822227670866, 3453484223084, 10519017940952, 18490932535144

Offset: 0

Seiichi Manyama, Aug 11 2023

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A291534, A364865, A364866, A365218.
Cf. A001764, A271469, A363982, A364736, A378892.
Cf. A002293, A336538.

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

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

A378892 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)^3).

Original entry on oeis.org

1, 1, 5, 37, 322, 3067, 30951, 325171, 3519038, 38959997, 439177850, 5023590609, 58163050071, 680308820750, 8026782091957, 95419476630100, 1141762194395927, 13740910664096101, 166216043531507231, 2019807368837970964, 24644779751103948475, 301818330734940817283

Offset: 0

Seiichi Manyama, Dec 10 2024

nonn

Cf. A001764, A271469, A363982, A364736, A364864.

Cf. A378891.

a(n, r=1, s=-1, t=6, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

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

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).

A364051 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 + xA(x)^5).

Original entry on oeis.org

1, 1, 1, -3, -21, -41, 166, 1460, 3445, -13503, -136721, -364412, 1285021, 14694643, 43144726, -132548857, -1709480698, -5456400119, 14285376285, 209281385564, 720201663662, -1572818128366, -26541960203077, -97918748134874, 173825501585400, 3453517916428141

Offset: 0

Seiichi Manyama, Aug 05 2023

sign

Cf. A291534, A363982, A364764.

Cf. A364740.

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

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

A364764 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 + xA(x)^4).

Original entry on oeis.org

1, 1, 1, -2, -14, -27, 70, 625, 1457, -3541, -37403, -98547, 207098, 2564079, 7448923, -12940485, -190014459, -600991549, 827159379, 14802832468, 50584687754, -52159768068, -1193457862093, -4384199208207, 3090291576246, 98618925147291, 388126462227091

Offset: 0

Seiichi Manyama, Aug 05 2023

sign

Cf. A291534, A363982, A364051.

Cf. A364739.

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

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

cp's OEIS Frontend

A291534 Expansion of the series reversion of x/((1 + x)*(1 - x^2)).

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A364864 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 / (1 + x*A(x)^3).

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A378892 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 + x*A(x)^3).

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A364051 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 + x*A(x)^5).

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A364764 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 + x*A(x)^4).

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A364864 G.f. A(x) satisfies A(x) = 1 + xA(x)^3 / (1 + xA(x)^3).

A378892 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)^3).

A364051 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 + xA(x)^5).

A364764 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 + xA(x)^4).