cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A097188 G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.

Original entry on oeis.org

1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Links

Crossrefs

Cf. A004988, A025748, A034164, A057083, A097186.
Essentially identical to A025748.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 - (1-9*x)^(1/3))/(3*x) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series((1-(1-9*x)^(1/3))/(3*x), x, n+2), x, n), n = 0..25); # G. C. Greubel, Sep 17 2019
  • Mathematica
    Table[FullSimplify[9^n * Gamma[n+2/3] / ((n+1) * Gamma[2/3] * Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-(1 - 9 x)^(1/3))/(3 x), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2014 *)
  • PARI
    a(n)=polcoeff((1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x),n,x)
  • Sage
    def A097188_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 - (1-9*x)^(1/3))/(3*x)).list()
A097188_list(25) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = (1 - (1-9*x)^(1/3))/(3*x).
G.f.: A(x) = (1/x)*(series reversion of x/A057083(x)).
a(n) = A004988(n)/(n+1).
a(n) = A025748(n+1).
a(n) = 3*A034164(n-1) for n>=1.
x*A(x) is the compositional inverse of x-3*x^2+3*x^3. - Ira M. Gessel, Feb 18 2012
a(n) = 1/(n+1) * Sum_{k=1..n} binomial(k,n-k) * 3^(k)*(-1)^(n-k) * binomial(n+k,n), if n>0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012
a(n) = 9^n * Gamma(n+2/3) / ((n+1) * Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014
Sum_{n>=0} 1/a(n) = 21/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022

A097189 Row sums of triangle A097186, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A057083(y)^(n+1), where R_n(1/3) = 3^n for all n >= 0.

Original entry on oeis.org

1, 7, 55, 451, 3781, 32131, 275563, 2378971, 20640907, 179791327, 1571002291, 13762897435, 120832716655, 1062818450155, 9363143224315, 82600459304203, 729572125425661, 6450872644562491, 57092964352312951, 505729048454449651
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Links

Crossrefs

Cf. A004988, A057083, A097186, A097188.

Programs

  • GAP
    List([0..30], n-> 1 + Sum([0..n-1], k-> Sum([0..n-k], j-> (-1)^(n-k-j)*3^j*Binomial(j, n-k-j)*Binomial(n+j, n) )) ); # G. C. Greubel, Sep 17 2019
  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 3/((1-9*x) + 2*(1-9*x)^(2/3)) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series(3/((1-9*x) + 2*(1-9*x)^(2/3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 17 2019
  • Mathematica
    CoefficientList[Series[3/((1-9*x) + 2*(1-9*x)^(2/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)
  • Maxima
    a(n):=sum(sum(binomial(k,n-m-k)*3^k*(-1)^(n-m-k)*binomial(n+k,n),k,0,n-m),m,0,n-1)+1; /* Vladimir Kruchinin, Sep 09 2019 */
  • PARI
    a(n)=polcoeff(3/((1-9*x)+2*(1-9*x+x*O(x^n))^(2/3)),n,x)
  • Sage
    def A097189_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 3/((1-9*x) + 2*(1-9*x)^(2/3)) ).list()
A097189_list(30) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 3/((1-9*x) + 2*(1-9*x)^(2/3)).
G.f.: A(x) = A004988(x)/(1 - x*A097188(x)).
a(n) = 1 + Sum_{m=0..n-1} Sum_{k=0..n-m} C(k,n-m-k)*3^k*(-1)^(n-m-k)*C(n+k,n). - Vladimir Kruchinin, Sep 17 2019
Conjecture: n*(n-1)*a(n) - (19*n-18)*(n-1)*a(n-1) + 9*(11*n^2-31*n+22)*a(n-2) - 9*(3*n-4)*(3*n-5)*a(n-3) = 0. - R. J. Mathar, Nov 16 2012
a(n) ~ 3^(2*n+1)/(2*Gamma(2/3) * n^(1/3))*(1 - sqrt(3)*Gamma(2/3)^2 / (4*Pi*n^(1/3))). - Vaclav Kotesovec, Feb 04 2014

A097187 Antidiagonal sums of triangle A097186, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A057083(y)^(n+1), where R_n(1/3) = 3^n for all n>=0.

Original entry on oeis.org

1, 1, 7, 10, 58, 94, 499, 868, 4360, 7951, 38407, 72508, 339997, 659380, 3019639, 5984968, 26880052, 54249628, 239683171, 491235070, 2139947788, 4444675456, 19125212575, 40190140696, 171064560433, 363227946394, 1531088393647
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Links

Crossrefs

Cf. A004988, A057083, A097186, A097188.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 3*x/((1-9*x^2) + (3*x-1)*(1-9*x^2)^(2/3)) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series(3*x/((1-9*x^2) +(3*x-1)*(1-9*x^2)^(2/3)), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019
  • Mathematica
    CoefficientList[Series[3*x/((1-9*x^2) +(3*x-1)*(1-9*x^2)^(2/3)), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    a(n)=polcoeff(3*x/((1-9*x^2)+(3*x-1)*(1-9*x^2+x^2*O(x^n))^(2/3)), n,x)
  • Sage
    def A097187_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(3*x/((1-9*x^2) + (3*x-1)*(1-9*x^2)^(2/3))).list()
A097187_list(30) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 3*x/((1-9*x^2) + (3*x-1)*(1-9*x^2)^(2/3)).
G.f.: A(x) = A004988(x^2)/(1 - x*A097188(x^2)).

A097181 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1).

Original entry on oeis.org

1, 1, 14, 1, 21, 210, 1, 28, 378, 3220, 1, 35, 595, 6475, 49910, 1, 42, 861, 11396, 108402, 778596, 1, 49, 1176, 18326, 207074, 1791930, 12198004, 1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920, 1, 63, 1953, 39585, 587727, 6783147
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Comments

Row sums form A097185. Diagonal is A097183.
Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097184, where the g.f.s satisfy: A097182(x*A097184(x)) = A097184(x).

Examples

			Row polynomials evaluated at y=1/2 equals powers of 8:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4;
where A097182(y)^(n+1) has the same initial terms as the n-th row:
A097182(y) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 -+...
A097182(y)^2 = 1 + 14y +...
A097182(y)^3 = 1 + 21y + 210y^2 +...
A097182(y)^4 = 1 + 28y + 378y^2 + 3220y^3 +...
A097182(y)^5 = 1 + 35y + 595y^2 + 6475y^3 + 49910y^4 +...
Rows begin with n=0:
  1;
  1, 14;
  1, 21,  210;
  1, 28,  378,  3220;
  1, 35,  595,  6475,  49910;
  1, 42,  861, 11396, 108402,  778596;
  1, 49, 1176, 18326, 207074, 1791930, 12198004;
  1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920;
  1, 63, 1953, 39585, 587727, 6783147, 62974371, 479497491, 3019005990; ...

Links

Crossrefs

Cf. A097179, A097182, A097183, A097184, A097185, A097186.

Programs

  • Mathematica
    Table[SeriesCoefficient[2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)), {x, 0,n}, {y,0,k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    {T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 2^n*(4^n -sum(j=0,n-1, T(n,j)/2^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}

Formula

G.f.: A(x, y) = 2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)).
G.f.: A(x, y) = A097183(x*y)/(1 - x*A097184(x*y)).

A097190 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 9^n, where R_n(y) forms the initial (n+1) terms of g.f. A097191(y)^(n+1).

Original entry on oeis.org

1, 1, 24, 1, 36, 612, 1, 48, 1104, 15912, 1, 60, 1740, 32130, 417690, 1, 72, 2520, 56700, 912492, 11027016, 1, 84, 3444, 91350, 1750014, 25562628, 292215924, 1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552, 1, 108, 5724, 197802, 4992354
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Examples

			Row polynomials evaluated at y=1/3 equals powers of 9:
9^1 = 1 + 24/3;
9^2 = 1 + 36/3 + 612/3^2;
9^3 = 1 + 48/3 + 1104/3^2 + 15912/3^3;
9^4 = 1 + 60/3 + 1740/3^2 + 32130/3^3 + 417690/3^4;
where A097191(y)^(n+1) has the same initial terms as the n-th row:
A097191(y) = 1 + 12y + 60y^2 + 90y^3 - 558y^4 - 2916y^5 + 2160y^6 +...
A097191(y)^2 = 1 + 24y +...
A097191(y)^3 = 1 + 36y + 612y^2 +...
A097191(y)^4 = 1 + 48y + 1104y^2 + 15912y^3 +...
A097191(y)^5 = 1 + 60y + 1740y^2 + 32130y^3 + 417690y^4 +...
Rows begin with n=0:
  1;
  1, 24;
  1, 36,  612;
  1, 48, 1104,  15912;
  1, 60, 1740,  32130,  417690;
  1, 72, 2520,  56700,  912492, 11027016;
  1, 84, 3444,  91350, 1750014, 25562628, 292215924;
  1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552; ...

Links

Crossrefs

Cf. A097186, A097191, A097192, A097193, A097194, A097195.

Programs

  • Mathematica
    Table[SeriesCoefficient[3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)), {x, 0,n}, {y,0,k}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    {T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 3^n*(9^n-sum(j=0,n-1, T(n,j)/3^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}

Formula

G.f.: A(x, y) = 3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)).
G.f.: A(x, y) = A097192(x*y)/(1 - x*A097193(x*y)).
Showing 1-5 of 5 results.

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