A300048 -id:A300048 - cp's OEIS frontend

A364736 G.f. satisfies A(x) = 1 + xA(x) / (1 + xA(x)^3).

1, 1, 0, -3, -3, 17, 45, -90, -546, 130, 5832, 7074, -53625, -159214, 374517, 2419131, -728364, -30011530, -37519884, 307731042, 940757526, -2343385995, -15421126275, 5164279686, 203045257272, 255851517115, -2186669342070, -6760669947375, 17391580425180

Offset: 0

Seiichi Manyama, Aug 05 2023

sign

Cf. A364735, A364737, A364738.

Cf. A300048.

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

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

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).

Original entry on oeis.org

1, 1, 3, 14, 76, 450, 2818, 18352, 123028, 843345, 5884227, 41650479, 298352365, 2158751879, 15754446893, 115830820439, 857147952469, 6379136387303, 47715901304501, 358529599468636, 2704884469806606, 20481615947325089, 155605509972859999, 1185779099027494848

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

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

Cf. A090192, A106228, A364759, A378919.
Cf. A001764, A219537, A364865, A365224.
Cf. A300048, A364747, A378889.

a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);

a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 11 2024

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 11 2024: (Start)
G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 + x*A(x))).
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). (End)

A349017 G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)))^3.

Original entry on oeis.org

1, 3, 9, 34, 147, 684, 3341, 16896, 87702, 464566, 2501178, 13646625, 75289022, 419301351, 2354121750, 13309905653, 75715795119, 433063793430, 2488921730886, 14366319150072, 83246947358766, 484082947060300, 2823980738817453, 16522429720210884, 96928401308507100

Offset: 0

Seiichi Manyama, Nov 06 2021

nonn

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

Cf. A262441, A349018.

Cf. A300048.

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

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
a(n) ~ sqrt((1 - r*s)*(1 - r - r*s) / (1 - r*(s-1))) / (2*sqrt(Pi)*n^(3/2)* r^(n+1)), where r = 0.16019884639474132810520949540299923469792581229191347... and s = 2.80076422793129845097661115192234873280320027349745080... are real roots of the system of equations (-1 + r*s)^3/(-1 + r + r*s)^3 = s, (3*r^2*(-1 + r*s)^2)/(-1 + r + r*s)^4 = 1. - Vaclav Kotesovec, Nov 15 2021
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A300048. - Seiichi Manyama, Dec 04 2024

A378685 G.f. A(x) satisfies A(x) = 1 + xA(x)^7/(1 - xA(x)^3).

Original entry on oeis.org

1, 1, 8, 88, 1126, 15716, 232069, 3564835, 56382489, 912031280, 15018257510, 250913307393, 4242722219425, 72470224174650, 1248608968982903, 21673752440979879, 378677335852165297, 6654158090059397480, 117523324766568499072, 2085095374834405245007

Offset: 0

Seiichi Manyama, Dec 04 2024

nonn

Cf. A243659, A300048.
Cf. A108447, A365192.
Cf. A378686.

a(n, r=1, s=1, t=7, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^6/(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(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A363560 Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

Original entry on oeis.org

1, 1, 3, 18, 126, 966, 7863, 66696, 583111, 5217513, 47547405, 439777242, 4117802109, 38956162023, 371795456373, 3575401032544, 34611064585803, 336998629754631, 3298200003722997, 32428037256038775, 320151289224740949, 3172536384239678856, 31544584654878015766

Offset: 0

Paul D. Hanna, Aug 12 2023

nonn

Compare to: G(x)^3 = 1 + x*(G(x) + G(x)^2 + G(x)^3) holds when G(x) = 1/(1-x).

Conjecture: a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 966*x^5 + 7863*x^6 + 66696*x^7 + 583111*x^8 + 5217513*x^9 + 47547405*x^10 + ...
such that
A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
Also,
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  18,  126,   966,   7863,   66696, ...];
n=2: [1, 2,  7,  42,  297,  2292,  18738,  159450, ...];
n=3: [1, 3, 12,  73,  522,  4059,  33354,  284886, ...];
n=4: [1, 4, 18, 112,  811,  6360,  52566,  450888, ...];
n=5: [1, 5, 25, 160, 1175,  9301,  77370,  666780, ...];
n=6: [1, 6, 33, 218, 1626, 13002, 108919,  943524, ...];
n=7: [1, 7, 42, 287, 2177, 17598, 148540, 1293937, ...];
n=8: [1, 8, 52, 368, 2842, 23240, 197752, 1732928, ...];
n=9: [1, 9, 63, 462, 3636, 30096, 258285, 2277756, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) )
where
G(x) = 1 + x + 2*x^2 + 11*x^3 + 61*x^4 + 380*x^5 + 2502*x^6 + 17163*x^7 + 121312*x^8 + 877370*x^9 + 6461765*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^5) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
TERMS MODULO 3.
It appears that a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).
The residues of a(7*k + 1) modulo 3, for k >= 0, begin
a(7*k + 1) (mod 3) = [1, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A300048, A161634.

{a(n) = my(A=1+x); for(i=1, n, A = (1 + x*(A + A^2 + A^9) +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 + A(x)^5 - A(x)^6 + A(x)^8) .
(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^4 - A(x)^6 + A(x)^7 + A(x)^10).
(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 - A(x)^6 + A(x)^8 + A(x)^11).
(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 + A(x)^9 + A(x)^12).
(7) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) ) ) ).

A363573 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

Original entry on oeis.org

1, 1, 3, 16, 99, 670, 4804, 35855, 275635, 2167577, 17354844, 140994899, 1159398760, 9631155422, 80703507043, 681333999628, 5789823864323, 49484286592503, 425092050147999, 3668385302806058, 31786451503719132, 276447315011186576, 2412336247105063011, 21114946136742383146

Offset: 0

Paul D. Hanna, Aug 14 2023

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 670*x^5 + 4804*x^6 + 35855*x^7 + 275635*x^8 + 2167577*x^9 + 17354844*x^10 + ...
such that
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  16,   99,   670,   4804,   35855,   275635, ...];
n=2: [1, 2,  7,  38,  239,  1634,  11798,   88506,   683045, ...];
n=3: [1, 3, 12,  67,  429,  2967,  21594,  162945,  1263183, ...];
n=4: [1, 4, 18, 104,  679,  4756,  34922,  265244,  2066591, ...];
n=5: [1, 5, 25, 150, 1000,  7101,  52645,  402725,  3155125, ...];
n=6: [1, 6, 33, 206, 1404, 10116,  75775,  584148,  4603911, ...];
n=7: [1, 7, 42, 273, 1904, 13930, 105490,  819918,  6503553, ...];
n=8: [1, 8, 52, 352, 2514, 18688, 143152, 1122312,  8962615, ...];
n=9: [1, 9, 63, 444, 3249, 24552, 190326, 1505727, 12110400, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) )
where
G(x) = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 219*x^5 + 1202*x^6 + 6867*x^7 + 40378*x^8 + 242782*x^9 + 1485836*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^3) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

Cf. A219537, A271469, A300048, A161634, A363560.

a:= n-> coeff(series(RootOf(1-A+x*(A-A^3+A^5), A), x, n+1), x, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 14 2023

{a(n) = my(A=1+x); for(i=1, n, A = 1 + x*(A - A^3 + A^5) +x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 - A(x)^4 + A(x)^5 + A(x)^6).
(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 - A(x)^4 + A(x)^6 + A(x)^7).
(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^7 + A(x)^8).
(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^8 + A(x)^9).
(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^9 + A(x)^10).
(7) A(x)^7 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10 + A(x)^11).
(8) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) ) ) ).
(9) A(x) = 1 / A(-x*A(x)^5).

A364723 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^4).

Original entry on oeis.org

1, 1, 2, 8, 38, 196, 1073, 6120, 35968, 216304, 1324676, 8232981, 51796538, 329229344, 2111031444, 13638557196, 88695018723, 580153216512, 3814285704000, 25192499164320, 167075960048996, 1112162062296061, 7428213584196010, 49766086788057256

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A000108, A106228, A300048, A364734.

Cf. A364739.

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

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

A364734 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^5).

Original entry on oeis.org

1, 1, 2, 9, 48, 276, 1687, 10750, 70597, 474478, 3247844, 22563904, 158693152, 1127661358, 8083795761, 58390722901, 424562043703, 3104994695198, 22825260066996, 168564068029385, 1249985066423749, 9303815610715531, 69483859839881494, 520527161650519576

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A000108, A106228, A300048, A364723.

Cf. A364740.

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

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

A378425 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^2 * (1 + x)^3) ).

Original entry on oeis.org

1, 1, 2, 7, 24, 82, 297, 1121, 4317, 16900, 67185, 270480, 1100122, 4513809, 18661618, 77666327, 325117967, 1368001765, 5782686120, 24545144206, 104573104040, 447036252525, 1916918691196, 8243075111450, 35538551601880, 153584392913986, 665201585797986, 2887012910233897

Offset: 0

Seiichi Manyama, Nov 25 2024

nonn

Index entries for reversions of series

Cf. A036765, A378426.
Cf. A300048, A378427.
Cf. A378406.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2*(1+x)^3))/x)

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

G.f.: exp( Sum_{k>=1} A378406(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] (1 + x + x^2 * (1 + x)^3)^(n+1).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(n+2*k+1,n-2*k).

A378427 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^3 * (1 + x)^3) ).

Original entry on oeis.org

1, 1, 1, 2, 8, 29, 88, 253, 775, 2575, 8797, 29833, 100635, 342408, 1181727, 4120223, 14435969, 50738813, 179038408, 634696939, 2259677734, 8072923814, 28924907573, 103915759961, 374302237154, 1351541722226, 4891132336481, 17736792240766, 64440831300682

Offset: 0

Seiichi Manyama, Nov 25 2024

nonn

Index entries for reversions of series

Cf. A300048, A378425.

Cf. A198951, A378407.

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

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

G.f.: exp( Sum_{k>=1} A378407(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] (1 + x + x^3 * (1 + x)^3)^(n+1).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+1,k) * binomial(n+2*k+1,n-3*k).

cp's OEIS Frontend

A364736 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x)^3).

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A364758 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)).

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A349017 G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)))^3.

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A378685 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)^3).

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A363560 Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

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A363573 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

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A364723 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x*A(x)^4).

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A364734 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x*A(x)^5).

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A378425 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^2 * (1 + x)^3) ).

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A364736 G.f. satisfies A(x) = 1 + xA(x) / (1 + xA(x)^3).

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).

A378685 G.f. A(x) satisfies A(x) = 1 + xA(x)^7/(1 - xA(x)^3).

A364723 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^4).

A364734 G.f. satisfies A(x) = 1 + xA(x) / (1 - xA(x)^5).