A097182 -id:A097182 - cp's OEIS frontend

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).

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

Paul D. Hanna, Aug 03 2004

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).

			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; ...

G. C. Greubel, Rows n = 0..50 of triangle, flattened

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

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 *)

{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))))}

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)).

A097184 G.f. A(x) satisfies A097182(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097181.

Original entry on oeis.org

1, 7, 70, 805, 9982, 129766, 1742572, 23960365, 335445110, 4763320562, 68418604436, 992069764322, 14499481170860, 213349508656940, 3157572728122712, 46968894330825341, 701770538825272742, 10526558082379091130, 158452400608443161220

Offset: 0

Paul D. Hanna, Aug 03 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A097181, A097182, A097183.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-16*x)^(1/8))/(2*x) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-(1-16*x)^(1/8))/(2*x), x, n+2), x, n), n = 0..20); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[(1-(1-16*x)^(1/8))/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
Table[FullSimplify[16^n*Gamma[n+7/8]/(Gamma[7/8]*Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)

a(n)=polcoeff((1-(1-16*x+x^2*O(x^n))^(1/8))/(2*x),n,x)

def A097184_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1-(1-16*x)^(1/8))/(2*x)).list()
A097184_list(20) # G. C. Greubel, Sep 17 2019

G.f.: A(x) = (1-(1-16*x)^(1/8))/(2*x).
G.f.: A(x) = (1/x)*(series reversion of x/A097182(x)).
a(n) = A097183(n)/(n+1).
D-finite with recurrence: (n+1)*a(n) +2*(-8*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012
a(n) = 16^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014
a(n) ~ 16^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Feb 09 2014

More terms from Vincenzo Librandi, Feb 10 2014

A097183 Main diagonal of triangle A097181, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097182(y)^(n+1), where R_n(1/2) = 8^n for all n>=0.

Original entry on oeis.org

1, 14, 210, 3220, 49910, 778596, 12198004, 191682920, 3019005990, 47633205620, 752604648796, 11904837171864, 188493255221180, 2986893121197160, 47363590921840680, 751502309293205456, 11930099160029636614

Offset: 0

Paul D. Hanna, Aug 03 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A097181, A097182, A097184.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-16*x)^(-7/8) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-16*x)^(-7/8), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

Table[FullSimplify[(n+1)*16^n*Gamma[n+7/8]/(Gamma[7/8]*Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-16*x)^(-7/8), {x,0,20}], x] (* G. C. Greubel, Sep 17 2019 *)

a(n)=polcoeff(1/(1-16*x+x*O(x^n))^(7/8),n,x)

def A097183_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1-16*x)^(-7/8)).list()
A097183_list(20) # G. C. Greubel, Sep 17 2019

G.f.: A(x) = 1/(1-16*x)^(7/8).
a(n) = (n+1)*A097184(n).
a(n) = (n+1) * 16^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014

A097185 Row sums of triangle A097181, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097182(y)^(n+1), where R_n(1/2) = 8^n for all n>=0.

Original entry on oeis.org

1, 15, 232, 3627, 57016, 899298, 14216560, 225110307, 3568890328, 56635884470, 899474459280, 14294357356110, 227286593929136, 3615608476770340, 57538659207907552, 915981394162628387, 14586262906867731096

Offset: 0

Paul D. Hanna, Aug 03 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A097181, A097182.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/((1-16*x) + (1-16*x)^(7/8)) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series(2/((1-16*x) + (1-16*x)^(7/8)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[2/((1-16*x) +(1-16*x)^(7/8)), {x,0,30}], x] (* G. C. Greubel, Sep 17 2019 *)

a(n)=polcoeff(2/((1-16*x)+(1-16*x+x*O(x^n))^(7/8)),n,x)

def A097185_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(2/((1-16*x) + (1-16*x)^(7/8))).list()
A097185_list(30) # G. C. Greubel, Sep 17 2019

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

cp's OEIS Frontend

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).

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A097184 G.f. A(x) satisfies A097182(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097181.

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A097183 Main diagonal of triangle A097181, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097182(y)^(n+1), where R_n(1/2) = 8^n for all n>=0.

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A097185 Row sums of triangle A097181, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097182(y)^(n+1), where R_n(1/2) = 8^n for all n>=0.

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Magma

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