A364374 -id:A364374 - cp's OEIS frontend

A057078 Periodic sequence 1,0,-1,...; expansion of (1+x)/(1+x+x^2).

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

Offset: 0

Wolfdieter Lang, Aug 04 2000

Partial sums of signed sequence is shifted unsigned one: |a(n+2)| = A011655(n+1).
With interpolated zeros, a(n) = sin(5*Pi*n/6 + Pi/3)/sqrt(3) + cos(Pi*n/6 + Pi/6)/sqrt(3); this gives the diagonal sums of the Riordan array (1-x^2, x(1-x^2)). - Paul Barry, Feb 02 2005
From Tom Copeland, Nov 02 2014: (Start)
With a shift and a sign change the o.g.f. of this array becomes the compositional inverse of the shifted Motzkin or Riordan numbers A005043,
(x - x^2) / (1 - x + x^2) = x*(1-x) / (1 - x*(1-x)) = x*(1-x) + [x*(1-x)]^2 + ... . Expanding each term of this series and arranging like powers of x in columns gives skewed rows of the Pascal triangle and reading along the columns gives (mod-signs and indexing) A011973, A169803, and A115139 (see also A091867, A092865, A098925, and A102426 for these term-by-term expansions and A030528). (End)

			G.f. = 1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 - x^11 + x^12 - x^14 + x^15 + ...

Winston de Greef, Table of n, a(n) for n = 0..10000
Ralph E. Griswold, Shaft Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A049310, A010892, A011655.
A049347(n) = a(-n).
Cf. A005043, A011973, A169803, A115139, A091867, A092865, A098925, A102426, A030528.
Cf. A002264, A010060, A010872, A049347, A061347, A364374.

a057078 = (1 -) . (`mod` 3)  -- Reinhard Zumkeller, Mar 22 2013

A057078:=n->1-(n mod 3); seq(A057078(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

a[n_] := {1, 0, -1}[[Mod[n, 3] + 1]] (* Jean-François Alcover, Jul 05 2013 *)
CoefficientList[Series[(1 + x) / (1 + x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 03 2014 *)
LinearRecurrence[{-1, -1},{1, 0},90] (* Ray Chandler, Sep 15 2015 *)

{a(n) = [1, 0, -1][n%3 + 1]}; /* Michael Somos, Oct 15 2008 */

def A057078():
    x, y = -1, 0
    while True:
        yield -x
        x, y = y, -x -y
a = A057078(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(n) = S(n, -1) + S(n - 1, -1) = S(2*n, 1); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, -1) = A049347(n). S(n, 1) = A010892(n).
From Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003: (Start)
a(n) = (1/2)*((-1)^floor(2*n/3) + (-1)^floor((2*n+1)/3)).
a(n) = -a(n-1) - a(n-2).
a(n) = A061347(n) - A049347(n+2). (End)
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(-1)^(n-k) = Sum_{k=0..floor((n+1)/2)} binomial(n+1-k, k)*(-1)^(n-k). - Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2003
Binomial transform is A010892. a(n) = 2*sqrt(3)*sin(2*Pi*n/3 + Pi/3)/3. - Paul Barry, Sep 13 2003
a(n) = cos(2*Pi*n/3) + sin(2*Pi*n/3)/sqrt(3). - Paul Barry, Oct 27 2004
a(n) = Sum_{k=0..n} (-1)^A010060(2n-2k)*(binomial(2n-k, k) mod 2). - Paul Barry, Dec 11 2004
a(n) = (4/3)*(|sin(Pi*(n-2)/3)| - |sin(Pi*n/3)|)*|sin(Pi*(n-1)/3)|. - Hieronymus Fischer, Jun 27 2007
a(n) = 1 - (n mod 3) = 1 + 3*floor(n/3) - n. - Hieronymus Fischer, Jun 27 2007
a(n) = 1 - A010872(n) = 1 + 3*A002264(n) - n. - Hieronymus Fischer, Jun 27 2007
Euler transform of length 3 sequence [0, -1, 1]. - Michael Somos, Oct 15 2008
a(n) = a(n-1)^2 - a(n-2)^2 with a(0) = 1, a(1) = 0. - Francesco Daddi, Aug 02 2011
a(n) = A049347(n) + A049347(n-1). - R. J. Mathar, Jun 26 2013
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, May 16 2023
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Feb 20 2024
From Peter Bala, Sep 08 2024: (Start)
G.f. A(x) satisfies A(x) = (1 + x)*(1 - x*A(x)).
1/x * series_reversion(x/A(x)) = the g.f of A364374. (End)

A216314 G.f. satisfies A(x) = (1 + xA(x)) (1 + 2xA(x)^2).

Original entry on oeis.org

1, 3, 17, 121, 965, 8247, 73841, 683713, 6493145, 62898859, 619079889, 6173490857, 62239144525, 633323532783, 6496052173665, 67093423506049, 697181754821297, 7283521984427283, 76455801614169809, 806004056649062937, 8529783421905380629, 90584730265930813607

Offset: 0

Paul D. Hanna, Sep 03 2012

nonn

The radius of convergence of g.f. A(x) is r = 0.08774268876242660659654020... with A(r) = 2.04748732367111203761312028274219344812311691... where y=A(r) satisfies 6*y^3 - 14*y^2 + 4*y - 1 = 0.

r = 1/(((40465 + 387*sqrt(129))^(2/3) + 1174 + 34*(40465 + 387*sqrt(129))^(1/3)) / (40465+387*sqrt(129))^(1/3)/9). - Vaclav Kotesovec, Sep 17 2013

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 121*x^3 + 965*x^4 + 8247*x^5 + 73841*x^6 +...
Related expansions.
A(x)^2 = 1 + 6*x + 43*x^2 + 344*x^3 + 2945*x^4 + 26398*x^5 + 244615*x^6 +...
A(x)^3 = 1 + 9*x + 78*x^2 + 696*x^3 + 6399*x^4 + 60321*x^5 + 580316*x^6 +...
where A(x) = 1 + A(x)*(1+2*A(x))*x + 2*A(x)^3*x^2.
The g.f. also satisfies the series:
A(x) = 1 + 3*x*A(x) + 8*x^2*A(x)^2 + 22*x^3*A(x)^3 + 60*x^4*A(x)^4 + 164*x^5*A(x)^5 + 448*x^6*A(x)^6 +...+ A028859(n)*x^n*A(x)^n +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1*2 + 1/A(x))*x*A(x) + (1*2^2 + 2^2*2/A(x) + 1/A(x)^2)*x^2*A(x)^2/2 +
(1*2^3 + 3^2*2^2/A(x) + 3^2*2/A(x)^2 + 1/A(x)^3)*x^3*A(x)^3/3 +
(1*2^4 + 4^2*2^3/A(x) + 6^2*2^2/A(x)^2 + 4^2*2/A(x)^3 + 1/A(x)^4)*x^4*A(x)^4/4 +
(1*2^5 + 5^2*2^4/A(x) + 10^2*2^3/A(x)^2 + 10^2*2^2/A(x)^3 + 5^2*2/A(x)^4 + 1/A(x)^5)*x^5*A(x)^5/5 +...
Explicitly,
log(A(x)) = 3*x + 25*x^2/2 + 237*x^3/3 + 2361*x^4/4 + 24203*x^5/5 + 252757*x^6/6 + 2674185*x^7/7 + 28567105*x^8/8 +...+ L(n)*x^n/n +...
where L(n) = [x^n] (1+x)^n/(1-2*x-2*x^2)^n.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO]

Cf. A007863, A215661, A215654, A028859, A364374.

CoefficientList[1/x * InverseSeries[Series[x*(1 - 2*x - 2*x^2)/(1+x),{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 2*x*(A+x*O(x^n))^2)); polcoeff(A, n)}

{a(n)=polcoeff( (1/x)*serreverse( x*(1-2*x-2*x^2)/(1+x +x*O(x^n))), n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*2^(m-j)/A^j)*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * 2^(n-k)/A(x)^k ).
(2) A(x) = (1/x) * Series_Reversion( x*(1 - 2*x - 2*x^2)/(1+x) ).
(3) A(x) = Sum_{n>=0} A028859(n) * x^n * A(x)^n, where g.f. of A028859 = (1+x)/(1-2*x-2*x^2).
The formal inverse of the g.f. A(x) is (sqrt(1-4*x+12*x^2) - (1+2*x))/(4*x^2).
a(n) = [x^n] ( (1+x)/(1-2*x-2*x^2) )^(n+1) / (n+1).
Recurrence: 3*n*(n+1)*(43*n-76)*a(n) = n*(1462*n^2 - 3315*n + 1274)*a(n-1) + (86*n^3 - 324*n^2 + 523*n - 330)*a(n-2) + (n-2)*(2*n-5)*(43*n-33)*a(n-3)
a(n) ~ 1/516*sqrt(86)*sqrt((1448486261 + 1803807*sqrt(129))^(1/3)*((1448486261 + 1803807*sqrt(129))^(2/3) + 1280110 + 1118*(1448486261 + 1803807*sqrt(129))^(1/3)))/(1448486261 + 1803807*sqrt(129))^(1/3) * (((40465 + 387*sqrt(129))^(2/3) + 1174 + 34*(40465 + 387*sqrt(129) )^(1/3)) / (40465+387*sqrt(129))^(1/3)/9)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
a(n) = Sum_{k=0..n} 2^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024

A364375 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 2, 0, -11, 28, 1, -206, 564, 38, -4711, 13329, 1273, -119762, 344707, 41884, -3251250, 9445976, 1381154, -92305098, 269504686, 45848871, -2707126108, 7921304973, 1532928960, -81375728566, 238196143730, 51591751698, -2493907008116, 7293147604136

Offset: 0

Seiichi Manyama, Jul 21 2023

Cf. A364371, A364372, A364374, A364376.

Cf. A198953.

A364375 := proc(n)
    add( (-1)^k*binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k)/(n+2*k+1),k=0..n) ;
end proc:
seq(A364375(n),n=0..80); # R. J. Mathar, Jul 25 2023

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1).
D-finite with recurrence +2*n*(191553133*n -462036810)*(2*n+1) *(n+1)*a(n) +2*n*(6735679202*n^3 -31340869996*n^2 +39568451245*n -13340358389)*a(n-1) +6*(13937077342*n^4 -106287464449*n^3 +278022830194*n^2 -296712736455*n +108876423952)*a(n-2) +6*(42118990776*n^4 -422141236704*n^3 +1546534911485*n^2 -2448212978721*n +1409411956166)*a(n-3) +6*(72631772298*n^4 -948761263665*n^3 +4512788370945*n^2 -9254886913710*n +6888712179986)*a(n-4) +3*(10147840245*n^4 -513806508936*n^3 +5519825354705*n^2 -22028093493130*n +30003008863784)*a(n-5) +6*(-9503341830*n^4 +235269814455*n^3 -2064338754902*n^2 +7709425316943*n -10409244067330)*a(n-6) +18*(3*n-20)*(n-6) *(156488131*n-746235854) *(3*n-13)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
G.f. A(x) satisfies (1/x) * series_reversion(x*A(x)) = 1/G(x), where G(x) is the g.f. of A364371.
P-recursive: fifth-order recurrence:      (2*n+1)*(2*n+2)*(3045*n^5-26680*n^4+84901*n^3-123566*n^2+86300*n-25368)*n*a(n) + 6*(18270*n^7-160080*n^6+500851*n^5-666969*n^4+307749*n^3+70849*n^2-76222*n+8288)*n*a(n-1) +  6*(54810*n^8-562455*n^7+2191158*n^6-3956204*n^5+2960986*n^4+88959*n^3-1045774*n^2+187688*n+69888)*a(n-2) + 6*(109620*n^8-1289340*n^7+5897421*n^6-13016841*n^5+13725877*n^4-5967199*n^3+2484230*n^2-3359528*n+1002624)*a(n-3) - 6*(54810*n^8-726885*n^7+3719313*n^6-9080919*n^5+10367473*n^4-4378276*n^3+1152956*n^2-2297912*n+768096)*a(n-4) + (3*n-7)*(3*n-12)*(3*n-14)*(3045*n^5-11455*n^4+8631*n^3+1507*n^2+2376*n-1368)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 2 and a(4) = 0. (End)

A364376 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^4).

Original entry on oeis.org

1, 0, -1, 3, -4, -9, 73, -212, 111, 1956, -10078, 21466, 29823, -418183, 1561911, -1722963, -13205004, 86962328, -232448945, -109578204, 3849218852, -17135183489, 27800381006, 113891855632, -966644138742, 3075070731677, -833503324311, -41673632701038

Offset: 0

Seiichi Manyama, Jul 21 2023

Cf. A364372, A364374, A364375.

Cf. A215623.

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

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

G.f.: x/series_reversion(x*G(x)), where G(x) = 1 - x^2 + 3*x^3 - 6*x^4 + 6*x^5 + 15*x^6 - ... is the g.f. of A364372. - Peter Bala, Aug 27 2024

A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + xA(x)) (1 + 3xA(x)^2).

Original entry on oeis.org

1, 4, 31, 301, 3274, 38158, 465919, 5883040, 76189177, 1006440238, 13508178448, 183689450959, 2525336086630, 35041483528522, 490125130328455, 6902993856515389, 97814486474787898, 1393470813699724726, 19946461692566594413, 286742046721454817358, 4138001844031453456120

Offset: 0

Paul D. Hanna, Sep 07 2024

nonn

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

			G.f. A(x) = 1 + 4*x + 31*x^2 + 301*x^3 + 3274*x^4 + 38158*x^5 + 465919*x^6 + 5883040*x^7 + 76189177*x^8 + 1006440238*x^9 + 13508178448*x^10 + ...
where A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 4*x + 15*x^2 + 57*x^3 + 216*x^4 + 819*x^5 + 3105*x^6 + 11772*x^7 + ... + A125145(n)*x^n + ...
where B(x) = (1 + x)/(1 - 3*x - 3*x^2).

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A125145, A215661, A375435, A375436, A375437.

Cf. A007863, A216314, A364374.

{a(n) = my(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = polcoef( (1/x)*serreverse( x*(1-3*x-3*x^2)/(1+x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * 3^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies:
(1) A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 3*x^2)/(1 + x) ).
(4) A(x) = Sum_{n>=0} A125145(n) * x^n * A(x)^n, where g.f. of A125145 = (1 + x)/(1 - 3*x - 3*x^2).
(5) x = (sqrt(21*A(x)^2 - 6*A(x) + 1) - (1 + 3*A(x)))/(6*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024
a(n) ~ ((36 + (48266 - 714*sqrt(17))^(1/3) + (48266 + 714*sqrt(17))^(1/3))/7)^n / (sqrt(6*Pi*((20517 - 4861*sqrt(17))^(1/3) + (20517 + 4861*sqrt(17))^(1/3) - 42)) * n^(3/2)). - Vaclav Kotesovec, Sep 14 2024

A386548 a(n) = [x^n] ((1 - x)/(1 - x + x^2))^n.

Original entry on oeis.org

1, 0, -2, -3, 6, 25, 1, -147, -218, 591, 2223, -484, -14871, -18759, 68353, 222697, -116058, -1629671, -1656989, 8275203, 23266031, -20154144, -184550412, -141418628, 1019061001, 2468408775, -3122976521, -21213927840, -10837119735, 126256071125, 262294667301, -456407675223

Offset: 0

Peter Bala, Jul 25 2025

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.

Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and all positive integers n and k.

Cf. A104507, A246437, A364374, A370616.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 0 elif n = 2 then -2 else
( 2*(n-1)*(2*n-3)*(19*n^2-60*n+36)*a(n-1) - 2*(190*n^4-1170*n^3+2519*n^2-2229*n+666)*a(n-2) - 2*(n-3)*(2*n-3)*(19*n^2-41*n+18)*a(n-3) )/(3*n*(n-1)*(19*n^2-79*n+78)) fi; end:
seq(a(n), n = 0..30);

a[n_]:=SeriesCoefficient[((1 - x)/(1 - x + x^2))^n,{x,0,n}]; Array[a,32,0] (* Stefano Spezia, Jul 29 2025 *)

a(n) = my(x='x+O('x^(n+1))); polcoef(((1 - x)/(1 - x + x^2))^n, n); \\ Michel Marcus, Aug 03 2025

a(n) = Sum_{k = 0..floor(n/2)} binomial(-n, k)*binomial(n-k-1, n-2*k) = Sum_{k = 0
..floor(n/2)} (-1)^k*binomial(n+k-1, k)*binomial(n-k-1, n-2*k). Cf. A246437.
a(n) = -n*hypergeom([n+1, 1 - (1/2)*n, 3/2 - (1/2)*n], [2, 2 - n], 4) for n >= 3.
P-recursive: 3*n*(n - 1)*(19*n^2 - 79*n + 78)*a(n) = 2*(n - 1)*(2*n - 3)*(19*n^2 - 60*n + 36)*a(n-1) - 2*(190*n^4 - 1170*n^3 + 2519*n^2 - 2229*n + 666)*a(n-2) - 2*(n - 3)*(2*n - 3)*(19*n^2 - 41*n + 18)*a(n-3) with a(0) = 1, a(1) = 0 and a(2) = -2.
exp( Sum_{n >= 1} a(n)*(-x)^n/n ) = 1 - x^2 + x^3 + 2*x^4 - 6*x^5 - x^6 +  ... is the g.f. of A364374.

A366115 Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^5 ).

Original entry on oeis.org

1, 4, 21, 125, 801, 5390, 37558, 268656, 1961355, 14555266, 109472688, 832625469, 6393072182, 49488174700, 385795571040, 3026190911853, 23867383581009, 189156323865632, 1505649098866535, 12031665674394905, 96486323017581420, 776255276240140980

Offset: 0

Seiichi Manyama, Sep 29 2023

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

Cf. A127632, A364374, A366114.

seq(simplify(1/(n+1)*binomial(4*n+4, n)*hypergeom([n+1, -(1/2)*n, (1/2)*(1-n)], [3*n+5, -4*(n+1)], 4)), n = 0..20); # Peter Bala, Aug 22 2024

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(4*n-k+4,n-2*k).
From Peter Bala, Aug 22 2024: (Start)
P-recursive: 6*n*(2*n+3)*(n^2-1)*(2021*n^3-8037*n^2+9946*n-3672)*a(n) = 4*n*(n-1)*(94987*n^5-282752*n^4+140624*n^3+146936*n^2-56373*n-18522)*a(n-1) - 6*(n-1)*(367822*n^6-1646645*n^5+2610582*n^4-1674935*n^3+259948*n^2+125940*n-35712)*a(n-2) + 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(2021*n^3-1974*n^2-65*n+258)*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 21.
G.f. A(x) satisfies 1 + x*A(x) = (1/x) * series_reversion( x/c(x*c(x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A366114 Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^3 ).

Original entry on oeis.org

1, 2, 4, 7, 9, 2, -34, -130, -284, -284, 730, 4864, 14860, 27134, 6462, -170865, -771303, -2005828, -2751028, 3491747, 36288137, 130265102, 283131062, 210905402, -1317613954, -7461822262, -22297519418, -38398674146, 10151248222, 355843715494, 1495838414326

Offset: 0

Seiichi Manyama, Sep 29 2023

sign

Cf. A127632, A364374, A366115.

Cf. A279565.

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

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

cp's OEIS Frontend

A057078 Periodic sequence 1,0,-1,...; expansion of (1+x)/(1+x+x^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Sage

Formula

A216314 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + 2*x*A(x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A364375 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A364376 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^4).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A386548 a(n) = [x^n] ((1 - x)/(1 - x + x^2))^n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366115 Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^5 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

A216314 G.f. satisfies A(x) = (1 + xA(x)) (1 + 2xA(x)^2).

A364375 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^3).

A364376 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^4).

A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + xA(x)) (1 + 3xA(x)^2).