A077860 -id:A077860 - cp's OEIS frontend

A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

1, 1, 6, 1, 9, 42, 1, 12, 74, 308, 1, 15, 115, 595, 2310, 1, 18, 165, 1020, 4746, 17556, 1, 21, 224, 1610, 8722, 37730, 134596, 1, 24, 292, 2392, 14778, 73080, 299508, 1038312, 1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918

Offset: 0

Paul D. Hanna, Aug 02 2004

Row sums form A097180. Diagonal is A004982. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A048779, where the g.f.s satisfy: A077860(x*A048779(x)) = A048779(x).

			Row polynomials evaluated at y=1/2 equals powers of 4:
4^1 = 1 + 6/2;
4^2 = 1 + 9/2 + 42/2^2;
4^3 = 1 + 12/2 + 74/2^2 + 308/2^3;
4^4 = 1 + 15/2 + 115/2^2 + 595/2^3 + 2310/2^4;
where A077860(y)^(n+1) has the same initial terms as the n-th row:
A077860(y) = 1 +3*y +5*y^2 +5*y^3 +1*y^4 -7*y^5 -15*y^6 -15*y^7 +...
A077860(y)^2 = 1 + 6*y +...
A077860(y)^3 = 1 + 9*y + 42*y^2 +...
A077860(y)^4 = 1 + 12*y + 74*y^2 + 308*y^3 +...
A077860(y)^5 = 1 + 15*y + 115*y^2 + 595*y^3 + 2310*y^4 +...
Rows begin with n=0:
  1;
  1,  6;
  1,  9,  42;
  1, 12,  74,  308;
  1, 15, 115,  595,  2310;
  1, 18, 165, 1020,  4746, 17556;
  1, 21, 224, 1610,  8722, 37730,  134596;
  1, 24, 292, 2392, 14778, 73080,  299508, 1038312;
  1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918; ...

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

Cf. A004982, A048779, A077860, A097180, A097181.

Table[SeriesCoefficient[2*y/((1-8*x*y) +(2*y-1)*(1-8*x*y)^(3/4)), {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/((1-8*x*y) + (2*y-1)*(1-8*x*y)^(3/4)).

G.f.: A(x, y) = A004982(x*y)/(1 - x*A048779(x*y)).

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.

Original entry on oeis.org

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

A084099 Expansion of (1+x)^2/(1+x^2).

Original entry on oeis.org

1, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0

Offset: 0

Paul Barry, May 15 2003

Inverse binomial transform of A077860. Partial sums of A084100.
Transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x)->((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 12 2004
Euler transform of length 4 sequence [2, -3, 0, 1]. - Michael Somos, Aug 04 2009

			G.f. = 1 + 2*x - 2*x^3 + 2*x^5 - 2*x^7 + 2*x^9 - 2*x^11 + 2*x^13 - 2*x^15 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A063886, A077860, A084100, A101455, A180969.

[1] cat [Integers()!((1-(-1)^n)*(-1)^(n*(n-1)/2)): n in [1..100]]; // Wesley Ivan Hurt, Oct 27 2015

A084099:=n->(1-(-1)^n)*(-1)^((2*n-1+(-1)^n)/4): 1,seq(A084099(n), n=1..100); # Wesley Ivan Hurt, Oct 27 2015

CoefficientList[Series[(1+x)^2/(1+x^2),{x,0,110}],x] (* or *) Join[ {1}, PadRight[{},120,{2,0,-2,0}]] (* Harvey P. Dale, Nov 23 2011 *)

{a(n) = if( n<1, n==0, 2 * if( n%2, (-1)^(n\2)) )}; /* Michael Somos, Aug 04 2009 */

a(n) = if(n==0, 1, I*((-I)^n-I^n)) \\ Colin Barker, Oct 27 2015

Vec((1+x)^2/(1+x^2) + O(x^100)) \\ Colin Barker, Oct 27 2015

G.f.: (1+x)^2/(1+x^2).
a(n) = 2 * A101455(n) for n>0. - N. J. A. Sloane, Jun 01 2010
a(n+2) = (-1)^A180969(1,n)*((-1)^n - 1). - Adriano Caroli, Nov 18 2010
G.f.: 4*x + 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013
From Wesley Ivan Hurt, Oct 27 2015: (Start)
a(n) = (1-sign(n)*(-1)^n)*(-1)^floor(n/2).
a(n) = 2*(n mod 2)*(-1)^floor(n/2) for n>0, a(0)=1.
a(n) = (1-(-1)^n)*(-1)^(n*(n-1)/2) for n>0, a(0)=1. (End)
From Colin Barker, Oct 27 2015: (Start)
a(n) = -a(n-2).
a(n) = i*((-i)^n-i^n) for n>0, where i = sqrt(-1).
(End)

A279230 Expansion of 1/((1-x)^2(1-2x+2*x^2)).

Original entry on oeis.org

1, 4, 9, 14, 15, 8, -7, -22, -21, 12, 77, 142, 143, 16, -239, -494, -493, 20, 1045, 2070, 2071, 24, -4071, -8166, -8165, 28, 16413, 32798, 32799, 32, -65503, -131038, -131037, 36, 262181, 524326, 524327, 40, -1048535, -2097110, -2097109, 44, 4194349, 8388654, 8388655

Offset: 0

Philippe Deléham, Dec 08 2016

Partial sums of A077860.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-2).

Cf. A001477, A045618, A077859, A077860, A279231.

Vec(1 / ((1 - x)^2*(1 - 2*x + 2*x^2)) + O(x^50)) \\ Colin Barker, Aug 04 2017

{a(n) = sum(k=0, n\2, (-1)^k*binomial(n+3, 2*k+3))} \\ Seiichi Manyama, Apr 07 2019

a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 2*a(n-4) for n>3.
a(n) = 2*a(n-1) - 2*a(n-2) + n + 1, with a(-1) = a(-2) = 0.
a(n) = (3 - (1-i)^(1+n) - (1+i)^(1+n) + n) where i=sqrt(-1). - Colin Barker, Aug 04 2017
From Seiichi Manyama, Apr 07 2019: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n+3,2*k+3).
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+1,j+1) * binomial(n-i+1,j+1). (End)

cp's OEIS Frontend

A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

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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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A084099 Expansion of (1+x)^2/(1+x^2).

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Magma

Maple

Mathematica

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A279230 Expansion of 1/((1-x)^2*(1-2*x+2*x^2)).

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PARI

Formula

A279230 Expansion of 1/((1-x)^2(1-2x+2*x^2)).