A235347 -id:A235347 - cp's OEIS frontend

A235352 Series reversion of x(1+3x^2)/(1+x^2) in odd-power terms.

1, -2, 14, -130, 1382, -15906, 192894, -2427522, 31405430, -415086658, 5580629870, -76080887042, 1049295082630, -14613980359010, 205246677882078, -2903566870820610, 41337029956899222, -591796707042765954, 8514525059135909070, -123048063153362454402

Offset: 0

Fung Lam, Jan 16 2014

Conjugate turbulence sequence to A235347.

Georg Fischer, Table of n, a(n) for n = 0..1000 (first 939 terms from _Fung Lam_)

Cf. A235347.

D-finite with recurrence 12*n*(2*n+1)*a(n) +(382*n^2-391*n+90)*a(n-1) +3*(34*n^2-132*n+125)*a(n-2) +(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Mar 24 2023

A235348 Series reversion of x(1-2x-5*x^2)/(1-x^2).

Original entry on oeis.org

1, 2, 12, 82, 636, 5266, 45684, 409706, 3768132, 35346082, 336854844, 3252391170, 31746462732, 312755404818, 3105750620772, 31054695744570, 312404601250644, 3159598296022978, 32108181705850860, 327682918265502002, 3357089384702757276

Offset: 1

Fung Lam, Jan 13 2014

Sum of turbulence series A107841 and A235347.

Fung Lam, Table of n, a(n) for n = 1..1000

Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-5*x^2)/(1-x^2), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Jan 29 2014 *)

Vec( serreverse(x*(1-2*x-5*x^2)/(1-x^2) +O(x^66) ) ) \\ Joerg Arndt, Jan 14 2014

# R. J. Mathar, 2023-03-28
class A235348() :
    def _init_(self) :
        self.a = [1, 2, 12, 82, 636, 5266]
    def at(self, n):
        if n <= len(self.a):
            return self.a[n-1]
        else:
            rhs = -3*(n-1)*(160*n-237)*self.at(n-1) \
            +3*(-422*n**2+1721*n-1713)*self.at(n-2) \
            +2*(-67*n**2+388*n-552)*self.at(n-3) \
            +(137*n**2-1352*n+3279)*self.at(n-4) \
            +(7*n-37)*(n-6)*self.at(n-5) -(n-6)*(n-7)*self.at(n-6)
            rhs //= (-54*n*(n-1))
            self.a.append(rhs)
            return self.a[-1]
a235348 = A235348()
for n in range(1,12):
    print(a235348.at(n))
# a235348.

D-finite with recurrence 54*n*(n-1)*a(n) -3*(n-1)*(160*n-237)*a(n-1) +3*(-422*n^2+1721*n-1713)*a(n-2) +2*(-67*n^2+388*n-552)*a(n-3) +(137*n^2-1352*n+3279)*a(n-4) +(7*n-37)*(n-6)*a(n-5) -(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Mar 24 2023

A235349 Series reversion of x(1-x-2x^2)/(1-x).

Original entry on oeis.org

0, 1, 0, 2, 2, 14, 30, 146, 434, 1862, 6470, 26586, 99946, 406366, 1593774, 6492450, 26100578, 106979894, 436906902, 1803472874, 7446478746, 30945624910, 128821054846, 538584390834, 2256485249682, 9483898177574

Offset: 0

Fung Lam, Jan 16 2014

Derived turbulence series from A235347.

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A235347, A235348, A235350, A235351, A235352.

CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x],x] (* Vaclav Kotesovec, Jan 22 2014 *)

Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ Joerg Arndt, Jan 17 2014

a = [0, 1]
for n in range(20):
    m = len(a)
    d = 0
    for i in range (1, m):
        for j in range (1, m):
            if (i+j)%m == 0 and (i+j) <= m:
                d += a[i]*a[j]
    g = 0
    for i in range (1, m-1):
        for j in range (1, m-1):
            for k in range (1, m-1):
                if (i+j+k)%m == 0 and (i+j+k) <= m:
                    g += a[i]*a[j]*a[k]
    y = 2*g + d - a[m-1]
    a.append(y)
print(a)

G.f.: ( exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/6 )/x, where i=sqrt(-1),
u = 1/6*(-10-63*x+3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3), and
v = 1/6*(-10-63*x-3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3).
a(n) ~ sqrt((1-s)^3 / (2*s*(3 - 3*s + s^2))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where s = 0.31472177038151893868... is the root of the equation 1-2*s-5*s^2+4*s^3 = 0, and r = s*(1-s-2*s^2)/(1-s) = 0.22374229727550306625... - Vaclav Kotesovec, Jan 23 2014
D-finite with recurrence 117*n*(n-1)*a(n) -7*(n-1)*(35*n-66)*a(n-1) +21*(-69*n^2+269*n-254)*a(n-2) +(937*n^2-6403*n+10920)*a(n-3) -28*(n-4)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 24 2023

Prepended a(0)=0 to adapt to offset 0, Joerg Arndt, Jan 23 2014

b-file shifted for offset 0, Vaclav Kotesovec, Jan 23 2014

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 18, 222, 3166, 49098, 804138, 13686198, 239671590, 4290463698, 78160665666, 1444298971662, 27005948771886, 510024567278234, 9714561608833242, 186403770207998310, 3599812021110287862, 69914211761486437026, 1364692279095996581490

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..757

Cf. A025192, A107841, A235347, A364826, A364827.

Cf. A144097, A243659.

A364825 := proc(n)
    (-1)^n*add( (-3)^k*binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364825(n),n=0..80); # R. J. Mathar, Aug 10 2023

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

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +2079*n*(3*n-1)*(3*n+1)*a(n) +(-347173*n^3 +395007*n^2 -41030*n -43092)*a(n-1) +18*(-59207*n^3 +325826*n^2 -590255*n +352406)*a(n-2) +3*(-3299*n^3 +35998*n^2 -125399*n +141144)*a(n-3) +9*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 10 2023

A364826 G.f. satisfies A(x) = 1 - xA(x)^4 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 22, 338, 6038, 117570, 2420758, 51833106, 1142472150, 25749801986, 590737764118, 13748997055826, 323842714201622, 7704914865207362, 184899022770465558, 4470200057557410834, 108776308617293352534, 2662072268791363675650

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..704

Cf. A025192, A107841, A235347, A364825, A364827.

Cf. A243667, A260332.

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

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(4*n+k+1,n) / (4*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0.

A364827 G.f. satisfies A(x) = 1 - xA(x)^5 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 26, 478, 10254, 240122, 5950530, 153417542, 4072868742, 110585691634, 3056671795946, 85722961493742, 2433127206219582, 69763483031049066, 2017643094336224914, 58789801741123032918, 1724199860717303739062, 50858327392484088101346

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..666

Cf. A025192, A107841, A235347, A364825, A364826.

Cf. A243668, A363006.

a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(5*n+k+1, n)/(5*n+k+1));

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(5*n+k+1,n) / (5*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(5*n,k-1) for n > 0.

A235350 Series reversion of x(1-2x-x^2)/(1-x^2).

Original entry on oeis.org

1, 2, 8, 42, 248, 1570, 10416, 71474, 503088, 3612226, 26353720, 194806458, 1455874792, 10982013250, 83504148192, 639360351074, 4925190101600, 38144591091970, 296837838901992, 2319880586624714, 18200693844341720, 143294043656426082, 1131747417739664528

Offset: 1

Fung Lam, Jan 16 2014

Derived series from A107841. The reversion has a quadratic power in x in the denominator. The general form reads x*(1-p*x-q*x^2)/(1-q*x^2).

Fung Lam, Table of n, a(n) for n = 1..1000

Cf. A107841, A235347, A235348, A235349, A235351, A235352.

Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1-x^2), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Jan 29 2014 *)

Vec(serreverse(x*(1-2*x-x^2)/(1-x^2)+O(x^66))) \\ Joerg Arndt, Jan 17 2014

a = [0, 1]
for n in range(20):
    m = len(a)
    d = 0
    for i in range (1, m):
        for j in range (1, m):
            if (i+j)%(m-1) == 0 and (i+j) < m:
                d += a[i]*a[j]
    f = 0
    for i in range (1, m):
        for j in range (1, m):
            if (i+j)%m == 0 and (i+j) <= m:
                f += a[i]*a[j]
    g = 0
    for i in range (1, m):
        for j in range (1, m):
            for k in range (1, m):
                if (i+j+k)%m == 0 and (i+j+k) <= m:
                    g += a[i]*a[j]*a[k]
    y = g + 2*f - d
    a.append(y)
print(a[1:]) # Edited by Andrey Zabolotskiy, Sep 04 2024

G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 2/3)/x, where i=sqrt(-1),
u = 1/3*(-17+3*x-6*x^2+x^3+3*sqrt(-6+54*x-30*x^2+18*x^3-3*x^4))^(1/3), and
v = 1/3*(-17+3*x-6*x^2+x^3-3*sqrt(-6+54*x-30*x^2+18*x^3-3*x^4))^(1/3).
D-finite with recurrence 6*n*(n-1)*a(n) -(n-1)*(52*n-75)*a(n-1) +(2*n+3)*(5*n-11)*a(n-2) +2*(5*n^2-62*n+150)*a(n-3) +(-13*n^2+130*n-321)*a(n-4) +(7*n-37)*(n-6)*a(n-5) -(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Mar 24 2023

cp's OEIS Frontend

A235352 Series reversion of x*(1+3*x^2)/(1+x^2) in odd-power terms.

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A235348 Series reversion of x*(1-2*x-5*x^2)/(1-x^2).

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A235349 Series reversion of x*(1-x-2*x^2)/(1-x).

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A364825 G.f. satisfies A(x) = 1 - x*A(x)^3 * (1 - 3*A(x)).

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A364826 G.f. satisfies A(x) = 1 - x*A(x)^4 * (1 - 3*A(x)).

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A364827 G.f. satisfies A(x) = 1 - x*A(x)^5 * (1 - 3*A(x)).

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A235350 Series reversion of x*(1-2*x-x^2)/(1-x^2).

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A235352 Series reversion of x(1+3x^2)/(1+x^2) in odd-power terms.

A235348 Series reversion of x(1-2x-5*x^2)/(1-x^2).

A235349 Series reversion of x(1-x-2x^2)/(1-x).

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

A364826 G.f. satisfies A(x) = 1 - xA(x)^4 (1 - 3*A(x)).

A364827 G.f. satisfies A(x) = 1 - xA(x)^5 (1 - 3*A(x)).

A235350 Series reversion of x(1-2x-x^2)/(1-x^2).