A198953 -id:A198953 - cp's OEIS frontend

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

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)

A200725 G.f. A(x) satisfies A(x) = (1+x^2)(1 + xA(x)^3).

Original entry on oeis.org

1, 1, 4, 16, 76, 399, 2206, 12664, 74790, 451420, 2772313, 17267652, 108821293, 692609446, 4445642625, 28744599748, 187047449289, 1224027357216, 8050074481917, 53179900898596, 352726704965748, 2348036826102013, 15682048658695168, 105052549830928908, 705678173069959645

Offset: 0

Paul D. Hanna, Nov 21 2011

nonn

More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2, q=-3.

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 76*x^4 + 399*x^5 + 2206*x^6 +...
Related expansion:
A(x)^3 = 1 + 3*x + 15*x^2 + 73*x^3 + 384*x^4 + 2133*x^5 + 12280*x^6 +...
where a(3) = 1 + 15; a(4) = 3 + 73; a(5) = 15 + 384; a(6) = 73 + 2133; ...
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x/A^3)*x*A^2 + (1 + 2^2*x/A^3 + x^2/A^6)*x^2*A^4/2 +
(1 + 3^2*x/A^3 + 3^2*x^2/A^6 + x^3/A^9)*x^3*A^6/3 +
(1 + 4^2*x/A^3 + 6^2*x^2/A^6 + 4^2*x^3/A^9 + x^4/A^12)*x^4*A^8/4 +
(1 + 5^2*x/A^3 + 10^2*x^2/A^6 + 10^2*x^3/A^9 + 5^2*x^4/A^12 + x^5/A^15)*x^5*A^10/5 + ...
which involves the squares of the binomial coefficients C(n,k).

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

Cf. A200716, A200717, A200718, A200719, A200074, A200075, A199874, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

nmax=20;aa=ConstantArray[0,nmax]; aa[[1]]=1;Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1+x^2)*(1+x*AGF^3)-AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}];Flatten[{1,aa}] (* Vaclav Kotesovec, Aug 19 2013 *)

{a(n)=local(p=2,q=-3,A=1+x);for(i=1,n,A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n));polcoeff(A,n)}

{a(n)=local(p=2,q=-3,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0,m,binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

{a(n)=local(p=2,q=-3,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * (Sum_{k=0..n} C(n,k)^2 * x^k / A(x)^(3*k)) ).
(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * (1 - x/A(x)^3)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k / A(x)^(3*k) ).
Recurrence: 2*(n-4)*(n-2)*n*(2*n+1)*a(n) = 3*(n-4)*(n-2)*(3*n-2)*(3*n-1)*a(n-1) - 2*(n-4)*(n-2)*n*(2*n-1)*a(n-2) + 6*(n-4)*(3*n-8)*(6*n^2 - 17*n + 2)*a(n-3) + 6*(3*n-14)*(9*n^3 - 66*n^2 + 114*n - 4)*a(n-5) + 6*n*(3*n-20)*(6*n^2 - 47*n + 78)*a(n-7) + 3*(n-2)*n*(3*n-26)*(3*n-19)*a(n-9). - Vaclav Kotesovec, Aug 19 2013
a(n) ~ c*d^n/n^(3/2), where d = 7.1535029565... is the root of the equation -27 - 81*d^2 - 81*d^4 - 27*d^6 + 4*d^7 = 0 and c = 0.26300783791885411389369671... - Vaclav Kotesovec, Aug 19 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-6*k+1,k) * binomial(3*n-6*k+1,n-2*k)/(3*n-6*k+1). - Seiichi Manyama, Dec 17 2024

A216359 G.f. satisfies: A(x) = (1 + xA(x)^2) (1 + x/A(x)).

Original entry on oeis.org

1, 2, 3, 13, 32, 147, 445, 2067, 7019, 32590, 119209, 551551, 2125429, 9795863, 39221165, 180177403, 742575760, 3403131833, 14342166121, 65626369612, 281459129188, 1286834885967, 5596229192396, 25580269950635, 112492633046446, 514323765191879, 2282371511598955

Offset: 0

Paul D. Hanna, Sep 04 2012

The radius of convergence of g.f. A(x) is r = 0.209619875959405379599013693... with A(r) = 2.36951367232829409921688546894691317519410... where y=A(r) satisfies y^7 - 2*y^6 - 4*y^4 + 4*y^3 + 4*y - 2 = 0.

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + 445*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 10*x^2 + 38*x^3 + 125*x^4 + 500*x^5 + 1839*x^6 +...
A(x)^3 = 1 + 6*x + 21*x^2 + 83*x^3 + 315*x^4 + 1269*x^5 + 5061*x^6 +...
where A(x) = (1-x^2)*A(x)^2 - x*A(x)^3 - x.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + 1/A(x)^3)*x*A(x) + (1 + 2^2/A(x)^3 + 1/A(x)^6)*x^2*A(x)^2/2 +
(1 + 3^2/A(x)^3 + 3^2/A(x)^6 + 1/A(x)^9)*x^3*A(x)^3/3 +
(1 + 4^2/A(x)^3 + 6^2/A(x)^6 + 4^2/A(x)^9 + 1/A(x)^12)*x^4*A(x)^4/4 +
(1 + 5^2/A(x)^3 + 10^2/A(x)^6 + 10^2/A(x)^9 + 5^2/A(x)^12 + 1/A(x)^15)*x^5*A(x)^5/5 +...

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A073157, A198953, A215623, A215624, A364331, A364336.

S:= series(RootOf(x+y+x^2*y^2-y^2+x*y^3, y, 1), x, 41):
seq(coeff(S,x,j),j=0..40); # Robert Israel, Jul 10 2017

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1-x^2)*AGF^2 - x*AGF^3 - x - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 18 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A^2)*(1 + x/(A+x*O(x^n)))); polcoeff(A, 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/A^(3*j))*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

G.f. satisfies:
A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 / A(x)^(3*k) ).
The formal inverse of the g.f. A(x) is (sqrt(1 - 2*x^3 + 4*x^4 + x^6) - (1+x^3))/(2*x^2).
Recurrence: n*(n+1)*(1241*n^5 - 21306*n^4 + 135203*n^3 - 381522*n^2 + 435524*n - 104880)*a(n) = 6*n*(1201*n^4 - 19476*n^3 + 114613*n^2 - 287442*n + 255364)*a(n-1) + 2*(12410*n^7 - 237880*n^6 + 1771109*n^5 - 6388366*n^4 + 11032829*n^3 - 6363274*n^2 - 3856020*n + 4157712)*a(n-2) + 6*(2482*n^7 - 51299*n^6 + 419427*n^5 - 1705769*n^4 + 3477465*n^3 - 2797370*n^2 - 637684*n + 1410288)*a(n-3) + 2*(4964*n^7 - 110044*n^6 + 983093*n^5 - 4442260*n^4 + 10160177*n^3 - 8790970*n^2 - 4722180*n + 9233280)*a(n-4) - 6*(2482*n^7 - 58745*n^6 + 553921*n^5 - 2617109*n^4 + 6255337*n^3 - 6022682*n^2 - 1392300*n + 4289616)*a(n-5) + 60*(n-7)*(2*n - 11)*(n^3 - 40*n^2 + 280*n - 552)*a(n-6) + 2*(n-8)*(2*n - 13)*(1241*n^5 - 15101*n^4 + 62389*n^3 - 91339*n^2 - 930*n + 64260)*a(n-7). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 4.77053998540509708... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 1.27852844884923435863262213680985089152... - Vaclav Kotesovec, Sep 18 2013
In closed form, c = (-4 + (1 + sqrt(1+8/d^2))*d^2) * sqrt((d^3*(1 + sqrt(1+8/d^2) + (4*(4 + d^2*(-3-sqrt(1+8/d^2) + d*(4+d))))/d^6)) / (1 + 1/64*(1 + sqrt(1+8/d^2)-4/d^2)^3*d^3)) / (32*d). - Vaclav Kotesovec, Aug 18 2014
From Peter Bala, Sep 10 2024: (Start)
For n not of the form 3*m + 1, we conjecture that a(n) = Sum_{k = 0..n} binomial(-n+3*k+1, k)*binomial(-n+3*k+1, n-k)/(-n+3*k+1).
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A364336, R^2({a(n)}) = A215623 and R^3({a(n)}) = A364331. Cf. A073157. (End)

A379190 G.f. A(x) satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x))^3.

Original entry on oeis.org

1, 4, 30, 304, 3557, 45150, 604222, 8393282, 119872890, 1749183075, 25964512607, 390828464403, 5951561595889, 91523131078999, 1419293428538496, 22169968253466467, 348507676062911520, 5509187208564734328, 87522347516801353980, 1396619714730284551913, 22375420057050167868366

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A379172, A379188, A379191.
Cf. A002293, A274735, A364475.
Cf. A198953.

a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(3*n+6*k+3, n-k)/(n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n+6*k+3,n-k)/(n+2*k+1).

A379328 G.f. A(x) satisfies A(x) = sqrt( (1 + 2xA(x)^3) * (1 + 2xA(x)) ).

Original entry on oeis.org

1, 2, 8, 44, 272, 1808, 12616, 91136, 675712, 5112576, 39316480, 306402304, 2414543328, 19207303168, 154030314752, 1243912552448, 10107398806016, 82573989969920, 677862373390592, 5588755066388480, 46257005500080128, 384210240316375040, 3201482490107076608

Offset: 0

Seiichi Manyama, Dec 21 2024

nonn

Cf. A379326, A379327.

Cf. A198953, A379280.

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

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

A364340 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 15, 179, 2502, 38262, 619991, 10459410, 181771289, 3231782239, 58505593456, 1074766446526, 19984671314164, 375414901633692, 7113886504446443, 135820770971898805, 2610186429457347486, 50452256583633573513, 980187901557594671335, 19130197594133100828170, 374894511736219913097375

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Cf. A239109, A364333, A364339.

Cf. A007863, A198953, A215623, A215624.

a(n) = sum(k=0, n, binomial(n+5*k+1, k)*binomial(n+5*k+1, n-k)/(n+5*k+1));

a(n) = Sum_{k=0..n} binomial(n+5*k+1,k) * binomial(n+5*k+1,n-k) / (n+5*k+1).

A211248 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^4).

Original entry on oeis.org

1, 1, 4, 20, 114, 703, 4565, 30752, 212921, 1505916, 10833164, 79018804, 583062388, 4344431508, 32641910199, 247033970128, 1881402836376, 14408753414558, 110897147057354, 857307054338476, 6653979156676983, 51831065993122915, 405060413133136902, 3175019470333290488

Offset: 0

Paul D. Hanna, Apr 05 2012

nonn

More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2 and q=1.

			G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 114*x^4 + 703*x^5 + 4565*x^6 +...
where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 522*x^4 + 3381*x^5 + 22735*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 132*x^3 + 841*x^4 + 5588*x^5 + 38288*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 343*x^3 + 2429*x^4 + 17430*x^5 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^4 + x^3*A(x)^7.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x))*x*A(x)^2 + (1 + 2^2*x*A(x) + x^2*A(x)^2)*x^2*A(x)^4/2 +
(1 + 3^2*x*A(x) + 3^2*x^2*A(x)^2 + x^3*A(x)^3)*x^3*A(x)^6/3 +
(1 + 4^2*x*A(x) + 6^2*x^2*A(x)^2 + 4^2*x^3*A(x)^3 + x^4*A(x)^4)*x^4*A(x)^8/4 +
(1 + 5^2*x*A(x) + 10^2*x^2*A(x)^2 + 10^2*x^3*A(x)^3 + 5^2*x^4*A(x)^4 + x^5*A(x)^5)*x^5*A(x)^10/5 +
(1 + 6^2*x*A(x) + 15^2*x^2*A(x)^2 + 20^2*x^3*A(x)^3 + 15^2*x^4*A(x)^4 + 6^2*x^5*A(x)^5 + x^6*A(x)^6)*x^6*A(x)^12/6 +...
more explicitly,
log(A(x)) = x + 7*x^2/2 + 49*x^3/3 + 359*x^4/4 + 2706*x^5/5 + 20767*x^6/6 +...

Cf. A200716, A200717, A199876, A211249, A200731, A199874, A199877.

Cf. A198951, A198953, A198957, A192415, A198888, A036765.

CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1 - x - x^3)^2/(1 + x^2)^2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Nov 22 2017 *)

{a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-x-x^3)^2/(1+x^2+x*O(x^n))^2 ) ), n)}
for(n=0,30,print1(a(n),", "))

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

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

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

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x-x^3)^2/(1+x^2)^2 ) ).
(2) A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
(3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(2*n+2) / (n+1).
(4) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^k) * x^n*A(x)^(2*n)/n ).
(5) A(x) = exp( Sum_{n>=1} (1-x*A(x))^(2*n+1) * (Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^k) * x^n*A(x)^(2*n)/n ).
(6) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) = (1 + x*G(x)^2)*(1 + x^2*G(x)^2) is the g.f. of A199874.
a(n) ~ s * sqrt((1 + 2*r*s + 3*r^2*s^4) / (3*Pi*(1 + 2*r*s + 7*r^2*s^4))) / (2*n^(3/2)*r^n), where r = 0.1194948955213353102456218138370139612914667337222... and s = 1.428770161302757679335810379290625953730830139744... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^4) = s, r*s^2*(3 + 4*r*s + 7*r^2*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017

A211249 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^5).

Original entry on oeis.org

1, 1, 4, 21, 126, 819, 5611, 39900, 291719, 2179181, 16560175, 127617168, 994951887, 7833555324, 62196300997, 497425570173, 4003607595960, 32404662671330, 263586896132154, 2153631763231319, 17666722629907960, 145449082369322208, 1201414340736684702

Offset: 0

Paul D. Hanna, Apr 05 2012

nonn

More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2 and q=2.

			G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 819*x^5 + 5611*x^6 +...
where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 88*x^3 + 564*x^4 + 3828*x^5 + 27040*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 195*x^3 + 1335*x^4 + 9486*x^5 + 69305*x^6 +...
A(x)^8 = 1 + 8*x + 60*x^2 + 448*x^3 + 3374*x^4 + 25704*x^5 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^5 + x^3*A(x)^8.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x)^2)*x*A(x)^2 +
(1 + 2^2*x*A(x)^2 + x^2*A(x)^4)*x^2*A(x)^4/2 +
(1 + 3^2*x*A(x)^2 + 3^2*x^2*A(x)^4 + x^3*A(x)^6)*x^3*A(x)^6/3 +
(1 + 4^2*x*A(x)^2 + 6^2*x^2*A(x)^4 + 4^2*x^3*A(x)^6 + x^4*A(x)^8)*x^4*A(x)^8/4 +
(1 + 5^2*x*A(x)^2 + 10^2*x^2*A(x)^4 + 10^2*x^3*A(x)^6 + 5^2*x^4*A(x)^8 + x^5*A(x)^10)*x^5*A(x)^10/5 +
(1 + 6^2*x*A(x)^2 + 15^2*x^2*A(x)^4 + 20^2*x^3*A(x)^6 + 15^2*x^4*A(x)^8 + 6^2*x^5*A(x)^10 + x^6*A(x)^12)*x^6*A(x)^12/6 +...
more explicitly,
log(A(x)) = x + 7*x^2/2 + 52*x^3/3 + 403*x^4/4 + 3211*x^5/5 + 26050*x^6/6 +...

Cf. A200716, A200717, A199876, A211248, A200731, A200075, A199877.

Cf. A199874, A198951, A198953, A198957, A192415, A198888, A036765.

CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1-2*x-x^2+x^4 + (1-x-x^2) * Sqrt[(1+x+x^2)*(1-3*x+x^2)])/2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Nov 22 2017 *)

{a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-2*x-x^2+x^4 + (1-x-x^2)*sqrt( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n)))/2 ) ), n)}
for(n=0,30,print1(a(n),", "))

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

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

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

G.f.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-x^2+x^4 + (1-x-x^2)*sqrt( (1+x+x^2)*(1-3*x+x^2) ))/2 ) ).
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) is the g.f. of A200075.
(2) A(x) = sqrt( (1/x)*Series_Reversion( x/G(x)^2 ) ) where A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) and 1+x*G(x) is the g.f. of A004148.
(3) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(2*k)) * x^n*A(x)^(2*n)/n ).
(4) A(x) = exp( Sum_{n>=1} (1-x*A(x)^2)^(2*n+1) * (Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^(2*k)) * x^n*A(x)^(2*n)/n ).
a(n) ~ s * sqrt((1 + 2*r*s^2 + 3*r^2*s^5) / (Pi*(3 + 10*r*s^2 + 28*r^2*s^5))) / (2*n^(3/2)*r^n), where r = 0.1130413665724951344267888513870607581680912144315... and s = 1.385648922830296011590145919380626723251960276539... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^5) = s, r*s^2*(3 + 5*r*s^2 + 8*r^2*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017

A379249 G.f. A(x) satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x))^2.

Original entry on oeis.org

1, 3, 18, 148, 1403, 14417, 156161, 1755664, 20293341, 239654554, 2879027132, 35072400492, 432238230583, 5379422216020, 67513288465855, 853481985400772, 10858099927189575, 138912471444569435, 1786014309638224994, 23065160118446902506, 299062458173041384523

Offset: 0

Seiichi Manyama, Dec 18 2024

nonn

Cf. A198953, A379190.

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

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

cp's OEIS Frontend

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

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A200725 G.f. A(x) satisfies A(x) = (1+x^2)*(1 + x*A(x)^3).

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A216359 G.f. satisfies: A(x) = (1 + x*A(x)^2) * (1 + x/A(x)).

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A379190 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * (1 + x*A(x))^3.

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A379328 G.f. A(x) satisfies A(x) = sqrt( (1 + 2*x*A(x)^3) * (1 + 2*x*A(x)) ).

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A364340 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^6).

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A211248 G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^4).

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A211249 G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^5).

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

A200725 G.f. A(x) satisfies A(x) = (1+x^2)(1 + xA(x)^3).

A216359 G.f. satisfies: A(x) = (1 + xA(x)^2) (1 + x/A(x)).

A379190 G.f. A(x) satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x))^3.

A379328 G.f. A(x) satisfies A(x) = sqrt( (1 + 2xA(x)^3) * (1 + 2xA(x)) ).

A364340 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^6).

A211248 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^4).

A211249 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^5).

A379249 G.f. A(x) satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x))^2.