A108623 -id:A108623 - cp's OEIS frontend

A104507 Row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

1, 0, -2, -3, 2, 15, 19, -28, -134, -129, 353, 1254, 791, -4238, -11818, -3123, 49162, 110007, -17783, -554458, -996323, 690932, 6096792, 8624747, -12287153, -65419110, -69285296, 178655307, 684550946, 483569751, -2354830741, -6970706252, -2324044054, 29195280375, 68793790705

Offset: 0

Paul D. Hanna, Mar 11 2005

Cf. A104505, A1086223, A246437, A370616, A386548.

CoefficientList[Series[(x/((1 - x)) + 1/((-Sqrt[5 x^2 - 2 x + 1] + x + 1)) x (1 - (5 x - 1)/(Sqrt[5 x^2 - 2 x + 1]))), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 05 2015 *)

a(n):=sum((-1)^j*binomial(n,j)*binomial(n-j-1,n-2*j),j,0,n/2); /* Vladimir Kruchinin, Oct 04 2015 */

a(n)=sum(k=0,n,polcoeff((1+x-x^2)^n,n+k))

a(n) = sum(k=0, n/2, (-1)^k*binomial(n,k)*binomial(n-k-1,n-2*k));
vector(40, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

G.f.: (x/((1-x))+1/((-sqrt(5*x^2-2*x+1)+x+1))*x*(1-(5*x-1)/(sqrt(5*x^2-2*x+1)))). - Vladimir Kruchinin, Oct 04 2015
a(n) = Sum_{j=0..n/2}((-1)^j*binomial(n,j)*binomial(n-j-1,n-2*j)). - Vladimir Kruchinin, Oct 04 2015
From Peter Bala, Jul 24 2025: (Start)
a(n) = [x^n] (1 - x^2/(1 - x))^n. Cf. A246437.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all primes p and all positive integers n and k.
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 - x^2 - x^3 + x^4 + 4*x^5 + ... is the g.f. of A108623.(End)

A108624 G.f. satisfies x = A(x)*(1+A(x))/(1-A(x)-(A(x))^2).

Original entry on oeis.org

1, 0, -1, 1, 1, -4, 3, 8, -23, 10, 67, -153, 9, 586, -1081, -439, 5249, -7734, -7941, 47501, -53791, -105314, 430119, -343044, -1249799, 3866556, -1730017, -13996097, 34243897, -1947204, -150962373, 296101864, 121857185

Offset: 1

Christian G. Bower, Jun 12 2005

sign

Row sums of triangle A202327. - Peter Luschny, Apr 26 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A039980, A202327.

Except for signs, same as A108623.

function A108624_list(len::Int)
    len <= 0 && return BigInt[]
    T = zeros(BigInt, len, len); T[1,1] = 1
    S = Array(BigInt, len); S[1] = 1
    for n in 2:len
        T[n,n] = 1
        for k in 1:n-1
            T[n,k] = (k > 1 ? T[n-1,k-1] : 0) - T[n-1,k] - T[n-1,k+1]
        end
        S[n] = sum(T[n,k] for k in 1:n)
    end
S end
println(A108624_list(33)) # Peter Luschny, Apr 27 2017

R:=PowerSeriesRing(Rationals(), 41);
Coefficients(R!( (-1+x+Sqrt(1+2*x+5*x^2))/(2*(1+x)) )); // G. C. Greubel, Oct 20 2023

a[n_]:= Sum[k Sum[(-1)^(j-k) Binomial[j, 2j-n-k] Binomial[n, j], {j, 0, n}], {k, 1, n}]/n;
Array[a, 33] (* Jean-François Alcover, Jun 13 2019, after Vladimir Kruchinin *)
Rest@CoefficientList[Series[(-1+x+Sqrt[1+2*x+5*x^2])/(2*(1+x)), {x,0, 41}], x] (* G. C. Greubel, Oct 20 2023 *)

def A108624_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (-1+x+sqrt(1+2*x+5*x^2))/(2*(1+x))).list()
a=A108624_list(41); a[1:] # G. C. Greubel, Oct 20 2023

a(n) = (1/n)*Sum_{k=1..n} ( k * Sum_{j=0..n} (-1)^(k+j)*binomial(j, 2*j-n-k)*binomial(n,j) ). - Vladimir Kruchinin, May 19 2012

G.f.: (-1 + x + sqrt(1+2*x+5*x^2))/(2*(1+x)). - G. C. Greubel, Oct 20 2023

A366081 Expansion of (1/x) * Series_Reversion( x*(1-x)^2/(1-x-x^2) ).

Original entry on oeis.org

1, 1, 1, 0, -5, -22, -68, -165, -285, -96, 1892, 10574, 38436, 107175, 217063, 165232, -1150565, -7780744, -31173680, -94537100, -212903852, -239418048, 788015576, 6734057510, 29396759220, 95418332383, 233697161887, 334222633632, -514863450175, -6299672869750

Offset: 0

Seiichi Manyama, Sep 28 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A007440, A108623, A366082, A366083.

Cf. A109081.

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

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

A366082 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x-x^2) ).

Original entry on oeis.org

1, 2, 6, 21, 79, 308, 1219, 4826, 18857, 71574, 257553, 837114, 2140496, 1379550, -35589730, -370646635, -2719034151, -17429175486, -103771133876, -588804389677, -3225403649859, -17180039158530, -89342552789741, -454604059204324, -2265246385921936

Offset: 0

Seiichi Manyama, Sep 28 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A007440, A108623, A366081, A366083.

Cf. A366049.

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

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

A366083 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x-x^2) ).

Original entry on oeis.org

1, 3, 14, 78, 478, 3109, 21063, 146997, 1049302, 7624330, 56198481, 419155136, 3157356819, 23984387314, 183519131353, 1413099475142, 10941294442694, 85132006090350, 665294548097852, 5219591907202092, 41095469624286421, 324595783790966343

Offset: 0

Seiichi Manyama, Sep 28 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A007440, A108623, A366081, A366082.

Cf. A366050.

a(n) = sum(k=0, n\2, (-1)^k*binomial(n+1, k)*binomial(4*n-k+2, n-2*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+1,k) * binomial(4*n-k+2,n-2*k).

A039978 An example of a d-perfect sequence.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. DOI: 10.1007/978-1-4471-0551-0_23

a(n) = A108623(n) mod 3 - Christian G. Bower, Jun 12 2005

More terms from Christian G. Bower, Jun 12 2005

cp's OEIS Frontend

A104507 Row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

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A108624 G.f. satisfies x = A(x)*(1+A(x))/(1-A(x)-(A(x))^2).

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A366081 Expansion of (1/x) * Series_Reversion( x*(1-x)^2/(1-x-x^2) ).

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A366082 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x-x^2) ).

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A366083 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x-x^2) ).

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A039978 An example of a d-perfect sequence.

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