A107841 -id:A107841 - cp's OEIS frontend

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0

Offset: 0

Philippe Deléham, Oct 19 2007

Mirror image of triangle A086810; another version of A126216.
Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007
Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

			Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

[[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 05 2018 *)

for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009

A349256 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 3 * x * A(x)^2)).

Original entry on oeis.org

1, 2, 19, 206, 2563, 34415, 486370, 7128488, 107364421, 1651615568, 25840137724, 409898503763, 6577319627506, 106571487893024, 1741193467526782, 28653852176675324, 474521786894159593, 7902112425718228064, 132243695376774536755, 2222925664652778182060

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..797

Cf. A001764, A107841, A217363, A337169, A346627, A349253, A349254, A349255.

nmax = 19; A[] = 0; Do[A[x] = 1/((1 + x) (1 - 3 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 3 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 19}]
a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^4]; Table[a[n], {n, 0, 19}] (* Peter Luschny, Nov 12 2021 *)

a(n) = (-1)^n + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * 3^k * binomial(3*k,k) / (2*k+1).
a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^4). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(585 + 73*sqrt(65)) * (73 + 9*sqrt(65))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021

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

A235351 Series reversion of x(1-3x-2*x^2)/(1-x).

Original entry on oeis.org

0, 1, 2, 12, 84, 660, 5548, 48836, 444412, 4147220, 39471436, 381671204, 3738957148, 37028943860, 370123733932, 3729092573060, 37831802166076, 386135110256852, 3962278590508812, 40852572573083364, 423006921400424988, 4396894566694687924

Offset: 0

Fung Lam, Jan 16 2014

Derived turbulence series: combined series reversion of A107841 and A235349.

Fung Lam, Table of n, a(n) for n = 0..950

Cf. A107841, A235349.

my(x='x+O('x^25)); concat([0],Vec(serreverse(x*(1-3*x-2*x^2)/(1-x)))) \\ Joerg Arndt, Sep 01 2024

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 = d + 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 = g + a[i]*a[j]*a[k]
    y = 2*g + 3*d - a[m-1]
    a.append(y)
print(a)

G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/2)/x, where i=sqrt(-1),
u = 1/6*(-54-81*x+3*sqrt(-51+522*x+549*x^2-24*x^3))^(1/3), and
v = 1/6*(-54-81*x-3*sqrt(-51+522*x+549*x^2-24*x^3))^(1/3).
D-finite with recurrence 17*n*(n+1)*(11*n-17)*a(n) -n*(1914*n^2-3915*n+1513)*a(n-1) +(-2013*n^3+7137*n^2-7924*n+2640)*a(n-2) +4*(2*n-5)*(11*n-6)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016

a(0) = 0 prepended by Andrey Zabolotskiy, Aug 31 2024

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

A371380 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^2 / (1-x) ).

Original entry on oeis.org

1, 5, 46, 521, 6574, 88658, 1250920, 18236849, 272544886, 4153080950, 64284022516, 1007929418570, 15974993572732, 255522850658564, 4119461259700060, 66869059171095809, 1091990982773631910, 17927521032225339854, 295717190725184361364, 4898634803627227516238

Offset: 0

Seiichi Manyama, Mar 20 2024

nonn

Index entries for reversions of series

Cf. A107841, A371385.

Cf. A371362.

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

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

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

A371385 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^3 / (1-x) ).

Original entry on oeis.org

1, 8, 109, 1808, 33283, 653696, 13419460, 284479136, 6179728951, 136842057800, 3077436307141, 70095952722752, 1613743723323028, 37490308916974496, 877802418598193488, 20693109628871653184, 490732756789852308223, 11699199238845493854872

Offset: 0

Seiichi Manyama, Mar 20 2024

nonn

Index entries for reversions of series

Cf. A107841, A371380.

Cf. A371363.

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-3*x)^3/(1-x))/x)

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

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

cp's OEIS Frontend

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

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A349256 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 3 * x * A(x)^2)).

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Mathematica

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

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Mathematica

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A235351 Series reversion of x*(1-3*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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A235348 Series reversion of x(1-2x-5*x^2)/(1-x^2).

A235351 Series reversion of x(1-3x-2*x^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).