A097179 -id:A097179 - cp's OEIS frontend

A097180 Row sums of triangle A097179, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A077860(y)^(n+1), where R_n(1/2) = 4^n for all n>=0.

1, 7, 52, 395, 3036, 23506, 182904, 1428387, 11185900, 87789702, 690212744, 5434455182, 42841215704, 338081920260, 2670388231152, 21109070463267, 166980248599884, 1321686452484286, 10467203182893800, 82936871755938970

Offset: 0

Paul D. Hanna, Aug 02 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A077860, A097179.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 2/((1-8*x) + (1-8*x)^(3/4)) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series(2/((1-8*x) + (1-8*x)^(3/4)), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[2/((1-8*x) + (1-8*x)^(3/4)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

a(n)=polcoeff(2/((1-8*x)+(1-8*x+x*O(x^n))^(3/4)),n,x)

def A097180_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(2/((1-8*x) + (1-8*x)^(3/4))).list()
A097180_list(20) # G. C. Greubel, Sep 17 2019

G.f.: A(x) = 2/((1-8*x) + (1-8*x)^(3/4)).
Conjecture: n*(n-1)*(n+1)*a(n) -12*n*(2*n-1)*(n-1)*a(n-1) +12*(n-1) * (16*n^2-32*n+17)*a(n-2) -16*(4*n-5)*(4*n-7)*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Nov 16 2012
a(n) ~ 2^(3*n+1) / (Gamma(3/4)*n^(1/4)) * (1 - Gamma(3/4) / (n^(1/4) * sqrt(Pi))). - Vaclav Kotesovec, Feb 04 2014

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

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)).

cp's OEIS Frontend

A097180 Row sums of triangle A097179, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A077860(y)^(n+1), where R_n(1/2) = 4^n for all n>=0.

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