A103779 -id:A103779 - cp's OEIS frontend

A350383 a(n) = [x^n] 1/(1 + x + x^2)^n.

1, -1, 1, 2, -15, 49, -98, 48, 561, -2860, 8151, -12948, -9282, 149226, -594320, 1428952, -1448655, -5538975, 37450900, -122995950, 239589735, -37528755, -1886983020, 8939152560, -24579514050, 35197176924, 51580335366, -541312482256, 2033695030128, -4624358661240

Offset: 0

Seiichi Manyama, Dec 29 2021

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

Cf. A002426, A027907, A103779, A110556, A165817, A213684, A228960, A350406, A350407, A362087.

a := n -> (-1)^n*hypergeom([-n/3, 1/3 - n/3, 2/3 - n/3, n], [1/3, 2/3, 1], 1): seq(simplify(a(n)), n = 0..30); # Peter Bala, Apr 17 2023

a[n_] := Coefficient[Series[1/(1 + x + x^2)^n, {x, 0, n}], x, n]; Array[a, 30, 0] (* Amiram Eldar, Dec 29 2021 *)

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1+k,k) * binomial(n,3*k).
Recurrence: 3*(n-1)*n*(4*n - 7)*a(n) = -2*(n-1)*(28*n^2 - 63*n + 27)*a(n-1) - 3*(3*n - 5)*(3*n - 4)*(4*n - 3)*a(n-2). - Vaclav Kotesovec, Mar 18 2023
From Peter Bala, Apr 15 2023: (Start)
a(n) = (-1)^n*hypergeom([-n/3, 1/3 - n/3, 2/3 - n/3, n], [1/3, 2/3, 1], 1).
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. Cf. A228960.
More generally, let k be a positive integer, m an integer and let f(x) = g(x)/h(x), where g(x) and h(x) are both finite products of cyclotomic polynomials. Then we conjecture that the same supercongruences hold, except for a finite number of primes p depending on f(x), for the sequence {a_(k,m,f)(n): n >= 0} defined by a_(k,m,f)(n) = [x^(k*n)] f(x)^(m*n). (End)
From Peter Bala, Mar 11 2025: (Start)
G.f.: A(x) = 1 + x*d/dx(log(G(x)/x)), where G(x) = x - x^2 + x^3 - 4*x^5 + 14*x^6 - 30*x^7 + ... is the g.f. of A103779.
The following formulas hold for n >= 1:
a(n) = [x^n] T(2*n, (1 - x)/2), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = Sum_{k = 0..n} (-1)^(n+k) * n/(2*n-k) * binomial(2*n-k, k)*binomial(2*n-2*k, n).
a(n) = (1/2)*(-1)^n*binomial(2*n, n)*hypergeom([-n/2, (-n+1)/2], [-2*n+1], 4). Cf. A213684. (End)

A212260 G.f. A(x) satisfies A(x)+A(x)^2+A(x)^3 = (1-sqrt(1-4*x))/2.

Original entry on oeis.org

0, 1, 0, 1, 3, 5, 22, 64, 198, 710, 2332, 8105, 28665, 100653, 361104, 1301180, 4713267, 17217021, 63140534, 232702261, 861507251, 3200666821, 11933894310, 44636509320, 167427781950, 629691033738, 2373987233286, 8970240131032, 33965443165016, 128857452216256

Offset: 0

Vladimir Kruchinin, May 12 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A103779.

a:= n-> coeff(series(RootOf(A+A^2+A^3=(1-sqrt(1-4*x))/2, A), x, n+1), x, n): seq(a(n), n=0..40); # Alois P. Heinz, May 12 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 0, 1, 3][n+1],
     (2*(n-1)*(n-2)*(n-3)*(21574*n^2-148237*n+252420)*a(n-1)
     +(n-2)*(n-3)*(30485*n^3-173514*n^2+191353*n+116820)*a(n-2)
     -4*(n-3)*(12730*n^4-266121*n^3+1766621*n^2-4771248*n+4563630)*a(n-3)
     -36*(6*n-23)*(67*n-183)*(6*n-25)*(3*n-10)*(3*n-11)*a(n-4))/
     (132*(67*n-250)*(n-1)*(n-2)*(n-3)*n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 27 2013

nn=29;f[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0==Series[f[x]+f[x]^2 +f[x]^3-(1-(1-4x)^(1/2))/2,{x,0,nn}],x][[3]];Table[a[n],{n,0,nn}]/.sol (* Geoffrey Critzer, Sep 27 2013 *)

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

a(n) = (sum(k=2..n, C(2*n-k-1,n-1)*(sum(i=1..k-1, (-1)^i*C(i,k-i-1) * C(k+i-1,k-1)))) +C(2*n-2,n-1))/n if n>0, a(0) = 0.

a(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = 0.12273243737616788383659461976... is the real root of the equation 8*c*(134*c^2 - 1) = 1. - Vaclav Kotesovec, Nov 20 2017

A217361 Series reversion of x+x^2+2*x^3.

Original entry on oeis.org

1, -1, 0, 5, -16, 14, 96, -495, 880, 2002, -17888, 48178, 19040, -665380, 2501312, -1983543, -23639952, 124654250, -216770400, -722621130, 5941209120, -15657865020, -12958545600, 267306817050, -972419359392, 534946077108, 11045425672512

Offset: 1

R. J. Mathar, Oct 01 2012

			If y=x+x^2+2*x^3, then x= y -y^2 +5*y^4 -16*y^5 +14*y^6 +96*y^7 -...

R. J. Mathar, Table of n, a(n) for n = 1..103
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A103779 (x+x^2+x^3), A006013 (x-2*x^2+x^3).

Rest[CoefficientList[InverseSeries[Series[x+x^2+2*x^3,{x,0,20}],x],x]] (* Vaclav Kotesovec, Sep 10 2013 *)

D-finite with recurrence 7*n*(n-1)*a(n) +16*(n-1)*(2*n-3)*a(n-1) +12*(3*n-5)*(3*n-7)*a(n-2)=0.

Lim sup n->infinity |a(n)|^(1/n) = 6*sqrt(3/7) = 3.927922... - Vaclav Kotesovec, Sep 10 2013

A217365 Series reversion of x + x^2 + x^3 + x^4 + x^5.

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -6, 27, -83, 209, -455, 845, -1169, 272, 5916, -29070, 98040, -274075, 660859, -1351756, 2110020, -1186110, -8227260, 47128770, -170898624, 505121130, -1281947030, 2772309230, -4708067030, 3936320480, 13030540120, -90168747031, 348836671587, -1077316101393

Offset: 1

R. J. Mathar, Oct 01 2012

sign

Appears to obey an 8-term hypergeometric recurrence with 4th-order polynomial coefficients.

			If y=x+x^2+x^3+x^4+x^5, then x=y -y^2 +y^3 -y^4 +y^5 -6*y^7 +27*y^8 -83*y^9 +...

R. J. Mathar and Robert Israel, Table of n, a(n) for n = 1..2066 (1..105 from R. J. Mathar)
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 (2012)

Cf. A063019 (x-x^2+x^3-x^4), A103779 (x+x^2+x^3).

rec := 5*n*(5*n-1)*(5*n+1)*(5*n+2)*(5*n+3)*a(n)-(n+1)*(27906*n^4+198109*n^3+447051*n^2+405674*n+128400)*a(n+1)+(25*(n+2))*(1875*n^4+28312*n^3+141513*n^2+287228*n+204072)*a(n+2)+(250*(n+2))*(n+3)*(250*n^3+3031*n^2+11433*n+13668)*a(n+3)+(625*(n+2))*(n+3)*(n+4)*(75*n^2+662*n+1403)*a(n+4)+(3125*(n+2))*(n+3)*(n+4)*(n+5)*(6*n+29)*a(n+5)+(3125*(n+2))*(n+3)*(n+4)*(n+5)*(n+6)*a(n+6):
f:= gfun:-rectoproc({rec,a(1) = 1, a(2) = -1, a(3) = 1, a(4) = -1, a(5) = 1, a(6) = 0},a(n),remember):
map(f, [$1..50]); # Robert Israel, Jul 10 2015

InverseSeries[x + x^2 + x^3 + x^4 + x^5 + O[x]^50][[3]] (* Vladimir Reshetnikov, Jul 09 2015 *)
RecurrenceTable[{5 (-8+3 n) (-21+5 n) (-19+5 n) (-18+5 n) (-17+5 n) (-17+6 n) (-11+6 n) a[-4+n]+4 (-3+n) (-11+6 n) (-4792620+7647427 n-4855116 n^2+1533029 n^3-240768 n^4+15048 n^5) a[-3+n]+30 (-3+n) (-2+n) (-1267420+2386123 n-1760222 n^2+636639 n^3-113004 n^4+7884 n^5) a[-2+n]+100 (-3+n) (-2+n) (-1+n) (-23+6 n) (-2310+2921 n-1152 n^2+144 n^3) a[-1+n]+125 (-3+n) (-2+n) (-1+n) n (-11+3 n) (-23+6 n) (-17+6 n) a[n]==0, a[1]==1, a[2]==-1, a[3]==1, a[4]==-1}, a, {n, 1, 40}] (* Vaclav Kotesovec, Aug 18 2015 *)

Vec(serreverse(x + x^2 + x^3 + x^4 + x^5 + O(x^50))) \\ Michel Marcus, Aug 03 2015

D-finite with recurrence: 5 n (5 n - 1) (5 n + 1) (5 n + 2) (5 n + 3) a(n) - (n + 1) (27906 n^4 + 198109 n^3 + 447051 n^2 + 405674 n + 128400) a(n + 1) + 25 (n + 2) (1875 n^4 + 28312 n^3 + 141513 n^2 + 287228 n + 204072) a(n + 2) + 250 (n + 2) (n + 3) (250 n^3 + 3031 n^2 + 11433 n + 13668) a(n + 3) + 625 (n + 2) (n + 3) (n + 4) (75 n^2 + 662 n + 1403) a(n + 4) + 3125 (n + 2) (n + 3) (n + 4) (n + 5) (6 n + 29) a(n + 5) + 3125(n + 2) (n + 3) (n + 4) (n + 5) (n + 6) a(n + 6) = 0. - Vladimir Reshetnikov, Jul 09 2015
From Robert Israel, Jul 10 2015: (Start)
G.f. G(x) satisfies G + G^2 + G^3 + G^4 + G^5 = x
and the differential equation
-2184*x^3+32760*x^2-163800*x+273000+(-10920*x^3+163800*x^2-819000*x+1365000)*G(x)+(2457000*x^4-24555336*x^3+33314040*x^2-22930200*x-10608000)*G'(x)+(11602500*x^5-88671024*x^4+64015500*x^3-18674400*x^2-27414000*x-9780000)*G''(x)+(7962500*x^6-48147528*x^5+12768480*x^4-2457200*x^3-7171500*x^2-6885000*x-1975000)*G'''(x)+(1137500*x^7-5485909*x^6-750720*x^5-1121525*x^4-654500*x^3-744375*x^2-475000*x-109375)*G''''(x) = 0
from which we can obtain the 8-term recurrence mentioned in the Comments:
1820*(5*n+1)*(5*n+2)*(5*n+3)*(5*n-1)*a(n)-13*(421993*n^4+2859670*n^3+6398855*n^2+5850050*n+1876272)*a(n+1)-60*(12512*n^4-187784*n^3-1717861*n^2-4206649*n-3230668)*a(n+2)-25*(44861*n^4+367454*n^3+1830175*n^2+6002422*n+7768608)*a(n+3)-500*(n+4)*(1309*n^3+22197*n^2+140942*n+279612)*a(n+4)-1875*(n+5)*(n+4)*(397*n^2+5657*n+18614)*a(n+5)-25000*(19*n+136)*(n+6)*(n+5)*(n+4)*a(n+6)-109375*(n+5)*(n+4)*(n+7)*(n+6)*a(n+7) = 0.
From the Lagrange inversion theorem,
  a(n) = 1/n! * (d/dx)^(n-1) (p^n)(0) where p(x) = 1/(1+x+x^2+x^3+x^4).
(End)
Recurrence: 125*(n-3)*(n-2)*(n-1)*n*(3*n - 11)*(6*n - 23)*(6*n - 17)*a(n) = -100*(n-3)*(n-2)*(n-1)*(6*n - 23)*(144*n^3 - 1152*n^2 + 2921*n - 2310)*a(n-1) - 30*(n-3)*(n-2)*(7884*n^5 - 113004*n^4 + 636639*n^3 - 1760222*n^2 + 2386123*n - 1267420)*a(n-2) - 4*(n-3)*(6*n - 11)*(15048*n^5 - 240768*n^4 + 1533029*n^3 - 4855116*n^2 + 7647427*n - 4792620)*a(n-3) - 5*(3*n - 8)*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(6*n - 17)*(6*n - 11)*a(n-4). - Vaclav Kotesovec, Aug 18 2015

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A350383 a(n) = [x^n] 1/(1 + x + x^2)^n.

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A212260 G.f. A(x) satisfies A(x)+A(x)^2+A(x)^3 = (1-sqrt(1-4*x))/2.

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A217361 Series reversion of x+x^2+2*x^3.

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A217365 Series reversion of x + x^2 + x^3 + x^4 + x^5.

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A381840 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 - x^2*A(x)^7.

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A381840 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 - x^2A(x)^7.