cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

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

Views

Author

Paul D. Hanna, Nov 21 2011

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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)}
  • PARI
    {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)}
  • PARI
    {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)}

Formula

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

A369600 Expansion of (1/x) * Series_Reversion( x * (1/(1+x)^3 - x^3) ).

Original entry on oeis.org

1, 3, 12, 56, 291, 1638, 9780, 60948, 391821, 2577575, 17256918, 117150228, 804343302, 5575177026, 38957753136, 274143594685, 1941037464402, 13818185220783, 98848503602394, 710185896393792, 5122358166219855, 37076879861508830, 269235792063692580
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Links

Crossrefs

Cf. A198888, A369599.
Cf. A274379, A274735.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1/(1+x)^3-x^3))/x)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(3*n+3*k+3, n-3*k))/(n+1);

Formula

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

A211248 G.f. satisfies: A(x) = (1 + x*A(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

Views

Author

Paul D. Hanna, Apr 05 2012

Keywords

Comments

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.

Examples

			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 +...

Crossrefs

Cf. A200716, A200717, A199876, A211249, A200731, A199874, A199877.
Cf. A198951, A198953, A198957, A192415, A198888, A036765.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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)}
  • PARI
    {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)}
  • PARI
    {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)}

Formula

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 + x*A(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

Views

Author

Paul D. Hanna, Apr 05 2012

Keywords

Comments

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.

Examples

			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 +...

Crossrefs

Cf. A200716, A200717, A199876, A211248, A200731, A200075, A199877.
Cf. A199874, A198951, A198953, A198957, A192415, A198888, A036765.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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)}
  • PARI
    {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)}
  • PARI
    {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)}

Formula

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

A369599 Expansion of (1/x) * Series_Reversion( x * (1/(1+x)^2 - x^3) ).

Original entry on oeis.org

1, 2, 5, 15, 54, 223, 993, 4580, 21521, 102563, 495318, 2422302, 11979965, 59824535, 301202673, 1527118720, 7789673832, 39947163395, 205835776301, 1065155017623, 5533253267649, 28844759080896, 150846487065730, 791163319140664, 4160593763997122
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Links

Crossrefs

Cf. A198888, A369600.
Cf. A274378, A274734.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1/(1+x)^2-x^3))/x)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(2*n+2*k+2, n-3*k))/(n+1);

Formula

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

A379082 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^3)^2 ).

Original entry on oeis.org

1, 2, 5, 16, 64, 288, 1354, 6496, 31728, 157818, 798098, 4091712, 21211165, 110969430, 585116287, 3106334810, 16590881379, 89085610328, 480627775528, 2604103448334, 14163573236255, 77302955664902, 423245859576867, 2324046398587426, 12795255089638583, 70617777139027756
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A198888, A379083.
Cf. A379085, A379089.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379085(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^3*A(x)^(7/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A379089.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^3 )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..floor(n/3)} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-3*k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-3*k).

A379083 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^3)^3 ).

Original entry on oeis.org

1, 3, 12, 58, 321, 1941, 12405, 82188, 558567, 3870694, 27245268, 194269872, 1400352702, 10187886330, 74710928103, 551676261727, 4098401671788, 30610414484517, 229717037309281, 1731295701244008, 13098454442320593, 99444838611953627, 757393732018935552, 5785220154325055826
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A198888, A379082.
Cf. A379087, A379090.

Programs

  • PARI
    a(n) = 3*sum(k=0, n\3, binomial(3*n+k+3, k)*binomial(3*n+k+3, n-3*k)/(3*n+k+3));

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379087(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^3*A(x)^(10/3)) )^3.
(3) A(x) = B(x)^3 where B(x) is the g.f. of A379090.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^3 )^(3*(n+1)).
a(n) = 3 * Sum_{k=0..floor(n/3)} binomial(3*n+k+3,k) * binomial(3*n+k+3,n-3*k)/(3*n+k+3) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+k+2,k) * binomial(3*n+k+3,n-3*k).

A199247 G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^2*A(x)^3 + x^3*A(x)^4).

Original entry on oeis.org

1, 1, 2, 7, 25, 92, 359, 1453, 6018, 25411, 109032, 473942, 2082550, 9235675, 41284297, 185819487, 841433773, 3830604764, 17521832924, 80490034307, 371169646860, 1717567062240, 7973153760616, 37119622029816, 173272771061677, 810810134833720, 3802675087749650
Offset: 0

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Author

Paul D. Hanna, Nov 04 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 25*x^4 + 92*x^5 + 359*x^6 + 1453*x^7 +...
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 135*x^4 + 543*x^5 + 2243*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 233*x^4 + 976*x^5 + 4154*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 370*x^4 + 1611*x^5 + 7065*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^3 + 2*x^3*A(x)^4 + x^4*A(x)^5.

Crossrefs

Cf. A198888.

Programs

  • Mathematica
    Table[Sum[Binomial[n+k, k]*Binomial[n+2*k+1, n-2*k]/(n+1), {k,0,Floor[n/2]}], {n,0,30}] (* Vaclav Kotesovec, Nov 18 2017 *)
  • PARI
    {a(n)=sum(k=0, n\2, binomial(n+k, k)*binomial(n+2*k+1, n-2*k))/(n+1)}
  • PARI
    {a(n)=local(A=1+x); A=1/x*serreverse(x/(1+x+x*O(x^n)) - x^3 - x^4); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^2*A^3 + x^3*A^4)+x*O(x^n)); polcoeff(A, n)}

Formula

a(n) = Sum_{k=0..[n/2]} C(n+k, k)*C(n+2*k+1, n-2*k)/(n+1).
G.f.: A(x) = (1/x)*Series_Reversion( x/(1+x) - x^3 - x^4 ).
Recurrence: 240*(n-2)*(n-1)*n*(n+1)*(543314047*n^6 - 9777816570*n^5 + 72419307730*n^4 - 282185906280*n^3 + 609197162263*n^2 - 689640519870*n + 319111630200)*a(n) = 16*(n-2)*(n-1)*n*(40205239478*n^7 - 743661045919*n^6 + 5720613587045*n^5 - 23561094982780*n^4 + 55538240643272*n^3 - 73680608104486*n^2 + 49378914311580*n - 12008736298800)*a(n-1) - 4*(n-2)*(n-1)*(91820073943*n^8 - 1836091148216*n^7 + 15607730833978*n^6 - 73389282297476*n^5 + 207862429740607*n^4 - 361251960415604*n^3 + 373910114125992*n^2 - 209189908378584*n + 47967495287040)*a(n-2) + 4*(n-2)*(516148344650*n^9 - 11611593292425*n^8 + 113759368018383*n^7 - 635856019816596*n^6 + 2229739867262616*n^5 - 5073403355773305*n^4 + 7464399000631567*n^3 - 6815882210854914*n^2 + 3481564826740104*n - 750085518551040)*a(n-3) - (710111459429*n^10 - 18460497932422*n^9 + 212532989720754*n^8 - 1424989554134304*n^7 + 6151577737700817*n^6 - 17829379077999138*n^5 + 35045804727952456*n^4 - 45976065299760536*n^3 + 38349466932479664*n^2 - 18244541292137280*n + 3720781171814400)*a(n-4) + 8*(n-4)*(2*n - 7)*(4*n - 17)*(4*n - 15)*(543314047*n^6 - 6517932288*n^5 + 31679935585*n^4 - 79420560120*n^3 + 107526834808*n^2 - 73755881832*n + 19667171520)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 2*r*s^2 + 6*r^2*s^3 + 4*r^3*s^4) / (3 + 12*r*s + 10*r^2*s^2)) / (2*sqrt(Pi) * n^(3/2) * r^(n + 1/2)), where r = 0.2013887134255166663337905234200508058745432798749... and s = 1.748883682651423548151134706780057317341305059059... are roots of the system of equations (1 + r*s)*(1 + r^2*s^3 + r^3*s^4) = s, r*(1 + 3*r*s^2 + 8*r^2*s^3 + 5*r^3*s^4) = 1. - Vaclav Kotesovec, Nov 18 2017
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