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.

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A246056 G.f.: Sum_{n>=0} x^n / (1-2*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k].

Original entry on oeis.org

1, 3, 16, 99, 681, 4950, 37303, 288399, 2272318, 18167553, 146950227, 1199921310, 9875193549, 81811617237, 681621711306, 5706874227051, 47985527200311, 405002888376840, 3429714479025247, 29130993220171449, 248095567594494634, 2118053534177686959, 18122259456592141785
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2014

Keywords

Comments

Conjecture: a(n) == 1 (mod 3) when n = 2*A005836(k) for k >= 0, and a(n) == 0 (mod 3) otherwise, where A005836 gives numbers whose base-3 representation contains no 2.

Examples

			G.f.: A(x) = 1 + 3*x + 16*x^2 + 99*x^3 + 681*x^4 + 4950*x^5 + 37303*x^6 + ...
where the g.f. is given by the binomial series identity:
A(x) = 1/(1-2*x) + x/(1-2*x)^3 * (1 + 2*x) * (1 + 3*x)
+ x^2/(1-2*x)^5 * (1 + 2^2*2*x + 4*x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-2*x)^7 * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3) * (1 + 3^2*3*x + 3^2*9*x^2 + 27*x^3)
+ x^4/(1-2*x)^9 * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4) * (1 + 4^2*3*x + 6^2*9*x^2 + 4^2*27*x^3 + 81*x^4)
+ x^5/(1-2*x)^11 * (1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5) * (1 + 5^2*3*x + 10^2*9*x^2 + 10^2*27*x^3 + 5^2*81*x^4 + 243*x^5) + ...
equals the series
A(x) = 1/(1-x) + x/(1-x)^3 * (1 + x) * (2 + 3*x)
+ x^2/(1-x)^5 * (1 + 2^2*x + x^2) * (4 + 2^2*2*3*x + 9*x^2)
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (8 + 3^2*4*3*x + 3^2*2*9*x^2 + 27*x^3)
+ x^4/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (16 + 4^2*8*3*x + 6^2*4*9*x^2 + 4^2*2*27*x^3 + 81*x^4)
+ x^5/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (32 + 5^2*16*3*x + 10^2*8*9*x^2 + 10^2*4*27*x^3 + 5^2*2*81*x^4 + 243*x^5) + ...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(2 + (1+3*x)) + x^2*(4 + 2^2*2*(1+3*x) + (1+2^2*3*x+9*x^2))
+ x^3*(8 + 3^2*4*(1+3*x) + 3^2*2*(1+2^2*3*x+9*x^2) + (1+3^2*3*x+3^2*9*x^2+27*x^3))
+ x^4*(16 + 4^2*8*(1+3*x) + 6^2*4*(1+2^2*3*x+9*x^2) + 4^2*2*(1+3^2*3*x+3^2*9*x^2+27*x^3) + (1+4^2*3*x+6^2*9*x^2+4^2*27*x^3+81*x^4))
+ x^5*(32 + 5^2*16*(1+3*x) + 10^2*8*(1+2^2*3*x+9*x^2) + 10^2*4*(1+3^2*3*x+3^2*9*x^2+27*x^3) + 5^2*2*(1+4^2*3*x+6^2*9*x^2+4^2*27*x^3+81*x^4) + (1+5^2*3*x+10^2*9*x^2+10^2*27*x^3+5^2*81*x^4+243*x^5)) + ...
equals the series
A(x) = 1 + x*(1 + (2+3*x)) + x^2*(1 + 2^2*(2+3*x) + (4+2^2*2*3*x+9*x^2))
+ x^3*(1 + 3^2*(2+3*x) + 3^2*(4+2^2*2*3*x+9*x^2) + (8+3^2*4*3*x+3^2*2*9*x^2+27*x^3))
+ x^4*(1 + 4^2*(2+3*x) + 6^2*(4+2^2*2*3*x+9*x^2) + 4^2*(8+3^2*4*3*x+3^2*2*9*x^2+27*x^3) + (16+4^2*8*3*x+6^2*4*9*x^2+4^2*2*27*x^3+81*x^4))
+ x^5*(1 + 5^2*(2+3*x) + 10^2*(4+2^2*2*3*x+9*x^2) + 10^2*(8+3^2*4*3*x+3^2*2*9*x^2+27*x^3) + 5^2*(16+4^2*8*3*x+6^2*4*9*x^2+4^2*2*27*x^3+81*x^4) + (32+5^2*16*3*x+10^2*8*9*x^2+10^2*4*27*x^3+5^2*2*81*x^4+243*x^5)) + ...

Links

Crossrefs

Cf. A246423 (dual), A243948, A245929, A227845, A245925, A005836.

Programs

  • Mathematica
    Table[Sum[3^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 2^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Aug 24 2014 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-2*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 2^k * x^k) * sum(k=0, m, binomial(m, k)^2 * 3^k *x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*2^(m-k)*3^k*x^k) * sum(k=0, m, binomial(m, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 2^(m-k)* sum(j=0, k, binomial(k, j)^2 * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * 2^(k-j) * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 3^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 2^j))}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 2^(n-k) * 3^k * x^k].
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 2^(k-j) * 3^j * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 2^(n-k) * Sum_{j=0..k} C(k,j)^2 * 3^j * x^j.
a(n) = Sum_{k=0..[n/2]} 3^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 2^j.
Recurrence: (n-5)*(n-4)*(n-2)*n^2*a(n) = 3*(n-5)*(n-4)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - (n-5)*(n-4)*(n-1)*(26*n^2 - 78*n + 61)*a(n-2) - 3*(n-5)*(n-2)*(8*n^3 - 56*n^2 + 118*n - 79)*a(n-3) + (n-3)*(125*n^4 - 1500*n^3 + 6197*n^2 - 10182*n + 5414)*a(n-4) - 9*(n-4)*(n-1)*(8*n^3 - 88*n^2 + 310*n - 341)*a(n-5) - 9*(n-5)*(n-2)*(n-1)*(26*n^2 - 234*n + 529)*a(n-6) + 81*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - 81*(n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Aug 24 2014
a(n) ~ c * d^n / n, where d = 8.9576182866823126497141284131... is the root of the equation 81 - 324*d + 234*d^2 + 72*d^3 - 125*d^4 + 24*d^5 + 26*d^6 - 12*d^7 + d^8 = 0, and c = 0.455454371861834589008839056170849399984539880764403809033969331822... . - Vaclav Kotesovec, Aug 24 2014

A245929 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * (-2*x)^j.

Original entry on oeis.org

1, -26, 1926, -179780, 18601030, -2040558156, 232474675356, -27194647204296, 3243950895157830, -392816395353590780, 48137032861960009396, -5956254538539775751736, 742934976698338610043676, -93295798612937748051169400, 11783597764983598765508801400
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2014

Keywords

Examples

			G.f.: A(x) = 1 - 26*x^4 + 1926*x^8 - 179780*x^12 + 18601030*x^16 -+...
where the g.f. is given by the binomial series:
A(x) = 1 + x*(1 - (1-2*x)) + x^2*(1 - 2^2*(1-2*x) + (1-2^2*2*x+4*x^2))
+ x^3*(1 - 3^2*(1-2*x) + 3^2*(1-2^2*2*x+4*x^2) - (1-3^2*2*x+3^2*4*x^2-8*x^3))
+ x^4*(1 - 4^2*(1-2*x) + 6^2*(1-2^2*2*x+4*x^2) - 4^2*(1-3^2*2*x+3^2*4*x^2-8*x^3) + (1-4^2*2*x+6^2*4*x^2-4^2*8*x^3+16*x^4))
+ x^5*(1 - 5^2*(1-2*x) + 10^2*(1-2^2*2*x+4*x^2) - 10^2*(1-3^2*2*x+3^2*4*x^2-8*x^3) + 5^2*(1-4^2*2*x+6^2*4*x^2-4^2*8*x^3+16*x^4) - (1-5^2*2*x+10^2*4*x^2-10^2*8*x^3+5^2*16*x^4-32*x^5))
+ x^6*(1 - 6^2*(1-2*x) + 15^2*(1-2^2*2*x+4*x^2) - 20^2*(1-3^2*2*x+3^2*4*x^2-8*x^3) + 15^2*(1-4^2*2*x+6^2*4*x^2-4^2*8*x^3+16*x^4) - 6^2*(1-5^2*2*x+10^2*4*x^2-10^2*8*x^3+5^2*16*x^4-32*x^5) + (1-6^2*2*x+15^2*4*x^2-20^2*8*x^3+15^2*16*x^4-6^2*32*x^5+64*x^6)) +...
We can also express the g.f. by the binomial series identity:
A(x) = 1/(1+x) + x/(1+x)^3*(1-x)*(1+2*x)
+ x^2/(1+x)^5*(1 - 2^2*x + x^2)*(1 + 2^2*2*x + 4*x^2)
+ x^3/(1+x)^7*(1 - 3^2*x + 3^2*x^2 - x^3)*(1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3)
+ x^4/(1+x)^9*(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)*(1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4)
+ x^5/(1+x)^11*(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)*(1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5)
+ x^6/(1+x)^13*(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)*(1 + 6^2*2*x + 15^2*4*x^2 + 20^2*8*x^3 + 15^2*16*x^4 + 6^2*32*x^5 + 64*x^6) +...
Note that all coefficients of x^k in A(x) vanish except for k = 4*n, n>=0.

Links

Crossrefs

Cf. A227845, A245925.

Programs

  • Mathematica
    Table[Sum[Sum[Binomial[4*n-k, k+j]^2 * Binomial[k+j, k]^2 * (-1)^j * 2^k,{j,0,4*n-2*k}],{k,0,2*n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2014 *)
  • PARI
    /* By definition: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, (-1)^k*binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*(-2)^j*x^j)+x*O(x^n))), n)}
for(n=0, 20, print1(a(4*n), ", "))
  • PARI
    /* From binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1+x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*(2*x)^k) * sum(k=0, m, binomial(m, k)^2*(-x)^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 20, print1(a(4*n), ", "))
  • PARI
    /* Formula for a(n), skipping zero-valued terms: */
{a(n)=sum(k=0,n\2,sum(j=k,n-k,binomial(n-k,j)^2*binomial(j,k)^2*(-1)^(k-j)*2^k))}
for(n=0, 20, print1(a(4*n), ", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0,2*n,sum(j=0,4*n-2*k,binomial(4*n-k,k+j)^2*binomial(k+j,k)^2*(-1)^j*2^k))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0, 2*n, binomial(2*k, k)*binomial(2*n+k, 2*n-k)^2*(-2)^(2*n-k))}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1+x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2*(-x)^k] * [Sum_{k=0..n} C(n,k)^2*(2*x)^k].
a(n) = Sum_{k=0..2*n} Sum_{j=0..4*n-2*k} C(4*n-k, k+j)^2 * C(k+j, k)^2 * (-1)^j * 2^k.
a(n) = Sum_{k=0..2*n} C(2*k, k) * C(2*n+k, 2*n-k)^2 * (-2)^(2*n-k).
Recurrence: n^2*(4*n-5)*a(n) = -2*(4*n-3)*(68*n^2 - 102*n + 21)*a(n-1) - 4*(2*n-3)^2 * (4*n-1)*a(n-2). - Vaclav Kotesovec, Aug 15 2014
a(n) ~ sqrt((4+sqrt(18))/16) * (-1)^n * (68+48*sqrt(2))^n / (Pi*n). - Vaclav Kotesovec, Aug 15 2014

A246423 G.f.: Sum_{n>=0} x^n / (1-3*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k].

Original entry on oeis.org

1, 4, 24, 168, 1286, 10440, 88112, 764368, 6766278, 60828024, 553529808, 5086837680, 47127896444, 439608960656, 4124536224864, 38891699480992, 368326082421446, 3501654020899800, 33403335855108368, 319612386771594608, 3066480362268978804, 29493401582426082032, 284301304326376855200
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2014

Keywords

Comments

a(n) == 2 (mod 4) iff n = 2^k for k>=2, and a(n) == 0 (mod 4) elsewhere except at a(0)=1 (conjecture).

Examples

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 168*x^3 + 1286*x^4 + 10440*x^5 +...
where the g.f. is given by the binomial series identity:
A(x) = 1/(1-3*x) + x/(1-3*x)^3 * (1 + 2*x) * (1 + 3*x)
+ x^2/(1-3*x)^5 * (1 + 2^2*2*x + 4*x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-3*x)^7 * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3) * (1 + 3^2*3*x + 3^2*9*x^2 + 27*x^3)
+ x^4/(1-3*x)^9 * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4) * (1 + 4^2*3*x + 6^2*9*x^2 + 4^2*27*x^3 + 81*x^4)
+ x^5/(1-3*x)^11 * (1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5) * (1 + 5^2*3*x + 10^2*9*x^2 + 10^2*27*x^3 + 5^2*81*x^4 + 243*x^5) +...
equals the series
A(x) = 1/(1-x) + x/(1-x)^3 * (1 + x) * (3+2*x)
+ x^2/(1-x)^5 * (1 + 2^2*x + x^2) * (9+2^2*3*2*x+4*x^2)
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (27+3^2*9*2*x+3^2*4*3*x^2+8*x^3)
+ x^4/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4)
+ x^5/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (243+5^2*81*2*x+10^2*27*4*x^2+10^2*9*16*x^3+5^2*3*18*x^4+32*x^5) +...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(3 + (1+2*x)) + x^2*(9 + 2^2*3*(1+2*x) + (1+2^2*2*x+4*x^2))
+ x^3*(27 + 3^2*9*(1+2*x) + 3^2*3*(1+2^2*2*x+4*x^2) + (1+3^2*2*x+3^2*4*x^2+8*x^3))
+ x^4*(81 + 4^2*27*(1+2*x) + 6^2*9*(1+2^2*2*x+4*x^2) + 4^2*3*(1+3^2*2*x+3^2*4*x^2+8*x^3) + (1+4^2*2*x+6^2*4*x^2+4^2*8*x^3+16*x^4))
+ x^5*(243 + 5^2*81*(1+2*x) + 10^2*27*(1+2^2*2*x+4*x^2) + 10^2*9*(1+3^2*2*x+3^2*4*x^2+8*x^3) + 5^2*3*(1+4^2*2*x+6^2*4*x^2+4^2*8*x^3+16*x^4) + (1+5^2*2*x+10^2*4*x^2+10^2*8*x^3+5^2*16*x^4+32*x^5)) +...
equals the series
A(x) = 1 + x*(1 + (3+2*x)) + x^2*(1 + 2^2*(3+2*x) + (9+2^2*3*2*x+4*x^2))
+ x^3*(1 + 3^2*(3+2*x) + 3^2*(9+2^2*3*2*x+4*x^2) + (27+3^2*9*2*x+3^2*4*3*x^2+8*x^3))
+ x^4*(1 + 4^2*(3+2*x) + 6^2*(9+2^2*3*2*x+4*x^2) + 4^2*(27+3^2*9*2*x+3^2*4*3*x^2+8*x^3) + (81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4))
+ x^5*(1 + 5^2*(3+2*x) + 10^2*(9+2^2*3*2*x+4*x^2) + 10^2*(27+3^2*9*2*x+3^2*4*3*x^2+8*x^3) + 5^2*(81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4) + (243+5^2*81*2*x+10^2*27*4*x^2+10^2*9*16*x^3+5^2*3*18*x^4+32*x^5)) +...

Links

Crossrefs

Cf. A246056 (dual), A243948, A245929, A227845, A245925.

Programs

  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-3*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 2^k * x^k) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 3^(m-k) * 2^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 3^(m-k) * sum(j=0, k, binomial(k, j)^2 * 2^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * 3^(k-j) * 2^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 2^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 3^j))}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * 2^k * x^k].
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 3^(k-j) * 2^j * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * Sum_{j=0..k} C(k,j)^2 * 2^j * x^j.
a(n) = Sum_{k=0..[n/2]} 2^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 3^j.
Recurrence: (n-5)*(n-4)*(n-2)*n^2*a(n) = 4*(n-5)*(n-4)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 16*(n-5)*(n-4)*(n-1)*(2*n-3)^2*a(n-2) + 8*(n-5)*(n-2)*(4*n^3 - 28*n^2 + 54*n - 27)*a(n-3) + 24*(n-3)*(5*n^4 - 60*n^3 + 248*n^2 - 408*n + 216)*a(n-4) + 16*(n-4)*(n-1)*(4*n^3 - 44*n^2 + 150*n - 153)*a(n-5) - 64*(n-5)*(n-2)*(n-1)*(2*n-9)^2*a(n-6) + 32*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - 16*(n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Aug 26 2014
a(n) ~ c * d^n / n, where d = 10.094399065494857710014687346... is the root of the equation 16 - 128*d + 256*d^2 - 64*d^3 - 120*d^4 - 32*d^5 + 64*d^6 - 16*d^7 + d^8 = 0, and c = 0.5132545324612697424702223429844481717... . - Vaclav Kotesovec, Aug 26 2014

A243948 G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k].

Original entry on oeis.org

1, 2, 8, 36, 182, 964, 5296, 29832, 171238, 997244, 5874992, 34937400, 209392796, 1263258760, 7664233696, 46726270992, 286089651718, 1758215706476, 10841476837424, 67049791851672, 415784950498964, 2584585251386296, 16101542183281312, 100511325748165488, 628579719997550044
Offset: 0

Views

Author

Paul D. Hanna, Aug 16 2014

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 182*x^4 + 964*x^5 + 5296*x^6 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-x) + x/(1-x)^3*(1+x) * (1+2*x)
+ x^2/(1-x)^5*(1 + 2^2*x + x^2) * (1 + 2^2*2*x + 4*x^2)
+ x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3)
+ x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4)
+ x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5) +...
We can also express the g.f. by the binomial series identity:
A(x) = 1 + x*(1 + (1+2*x)) + x^2*(1 + 2^2*(1+2*x) + (1+2^2*2*x+4*x^2))
+ x^3*(1 + 3^2*(1+2*x) + 3^2*(1+2^2*2*x+4*x^2) + (1+3^2*2*x+3^2*4*x^2+8*x^3))
+ x^4*(1 + 4^2*(1+2*x) + 6^2*(1+2^2*2*x+4*x^2) + 4^2*(1+3^2*2*x+3^2*4*x^2+8*x^3) + (1+4^2*2*x+6^2*4*x^2+4^2*8*x^3+16*x^4))
+ x^5*(1 + 5^2*(1+2*x) + 10^2*(1+2^2*2*x+4*x^2) + 10^2*(1+3^2*2*x+3^2*4*x^2+8*x^3) + 5^2*(1+4^2*2*x+6^2*4*x^2+4^2*8*x^3+16*x^4) + (1+5^2*2*x+10^2*4*x^2+10^2*8*x^3+5^2*16*x^4+32*x^5)) +...

Links

Crossrefs

Cf. A246455 (dual), A245929, A227845, A245925.

Programs

  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*x^k) * sum(k=0, m, binomial(m, k)^2*2^k*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*2^j*x^j)+x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, binomial(n-k, k+j)^2*binomial(k+j, k)^2*2^k))}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 2^j * x^j.
a(n) = Sum_{k=0..[n/2]} Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, k)^2 * 2^k.
Recurrence: (n-3)*(n-2)*n^2*a(n) = 2*(n-3)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 2*(n-1)*(5*n^3 - 30*n^2 + 58*n - 38)*a(n-2) + 8*(n-2)*a(n-3) + 4*(n-3)*(5*n^3 - 30*n^2 + 58*n - 34)*a(n-4) - 8*(n-1)*(4*n^3 - 36*n^2 + 106*n - 101)*a(n-5) + 8*(n-4)^2*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Aug 17 2014
a(n) ~ sqrt(c) * d^n / (2^(3/2)*Pi*n), where d = 2 + sqrt(2) + 2*sqrt(1+sqrt(2)) = 6.52176151043316966349... is the root of the equation 4 - 16*d + 12*d^2 - 8*d^3 + d^4 = 0, and c = 4 + 5/sqrt(2) + 2*sqrt(7+5*sqrt(2)) = 15.0378183078640521... is the root of the equation 1 - 32*c + 60*c^2 - 64*c^3 + 4*c^4 = 0. - Vaclav Kotesovec, Aug 17 2014

Extensions

Minor edits by Vaclav Kotesovec, Nov 05 2014

A245926 Expansion of g.f. sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) ).

Original entry on oeis.org

1, 5, 51, 587, 7123, 89055, 1135005, 14660805, 191253843, 2513963567, 33244446601, 441772827105, 5894323986301, 78912561223553, 1059543126891027, 14261959492731387, 192392702881384275, 2600355510685245087, 35206018016510388345, 477377227987055971905
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2014

Keywords

Comments

Square each term to form a bisection of A245925.
Limit a(n+1)/a(n) = 7 + 4*sqrt(3).

Examples

			G.f.: A(x) = 1 + 5*x + 51*x^2 + 587*x^3 + 7123*x^4 + 89055*x^5 +...
where
A(x)^2 = (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)).
Explicitly,
A(x)^2 = 1 + 10*x + 127*x^2 + 1684*x^3 + 22717*x^4 + 309214*x^5 + 4231675*x^6 + 58117672*x^7 + 800173945*x^8 +...+ A245923(n)*x^n +...

Links

Crossrefs

Column k=3 of A337389.
Cf. A245925, A245927, A245923, A243946, A337422.

Programs

  • Maple
    A245926 := n -> sqrt(add(binomial(4*n-2*k, 2*n-k)*binomial(4*n-k, k)^2, k=0..2*n)); seq(A245926(n), n=0..20); # Peter Luschny, Aug 17 2014
  • Mathematica
    CoefficientList[Series[Sqrt[(1 - x + Sqrt[1 - 14*x + x^2])/(2*(1 - 14*x + x^2))], {x,0,50}], x] (* G. C. Greubel, Jan 29 2017 *)
a[n_] := (-1)^n Hypergeometric2F1[-n, n + 1/2, 1, 4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 16 2018 *)
  • PARI
    /* From definition: */
{a(n)=polcoeff( sqrt( (1-x + sqrt(1-14*x+x^2 +x*O(x^n))) / (2*(1-14*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* From formula for a(n)^2: */
{a(n)=sqrtint(sum(k=0, 2*n, sum(j=0, 4*n-2*k, (-1)^(j+k)*binomial(4*n-k,j+k)^2*binomial(j+k, k)^2)))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* From formula for a(n)^2: */
{a(n) = sqrtint( sum(k=0, 2*n, binomial(2*k, k)^2*binomial(2*n+k, 2*n-k)*(-1)^k) )}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n)^2 = Sum_{k=0..2*n} Sum_{j=0..4*n-2*k} (-1)^(j+k) * C(4*n-k,j+k)^2 * C(j+k,k)^2.
a(n) ~ (3*sqrt(3)-5) * (7+4*sqrt(3))^(n+1) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 16 2014
a(n)^2 = C(4*n,2*n)*hyper4F3([-2*n,-2*n,-2*n,-2*n+1/2],[1,-4*n,-4*n],4). - Peter Luschny, Aug 17 2014
a(n)^2 = Sum_{k=0..2*n} binomial(4*n-2*k, 2*n-k)*binomial(4*n-k, k)^2. - Peter Luschny, Aug 17 2014
a(n)^2 = Sum_{k=0..2*n} (-1)^k * C(2*k, k)^2 * C(2*n+k, 2*n-k). - Paul D. Hanna, Aug 17 2014
From Peter Bala, Mar 14 2018: (Start)
a(n) = (-1)^n*P(2*n,sqrt(-3)), where P(n,x) denotes the n-th Legendre polynomial. See A008316.
a(n) = (1/C(2*n,n))*Sum_{k = 0..n} (-1)^(n+k)*C(n,k)*C(n+k,k)* C(2*n+2*k,n+k) = Sum_{k = 0..n} (-1)^(n+k)*C(2*k,k)*C(n,k) *C(2*n+2*k,2*n)/C(n+k,n). In general, P(2*n,sqrt(1+4*x)) = (1/C(2*n,n))*Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k) *x^k.
a(n) = Sum_{k = 0..2*n} C(2*n,k)^2 * u^(n-k), where u = (1 - sqrt(-3))/2 is a primitive sixth root of unity.
a(n) = (-1)^n*Sum_{k = 0..2*n} C(2*n,k)*C(2*n+k,k)*u^(2*k).
(End)
a(n) = (-1)^n*hypergeom([-n, n + 1/2], [1], 4). - Peter Luschny, Mar 16 2018
a(0) = 1, a(1) = 5 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (28*n^2-42*n+9) * a(n-1) - (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020
From Peter Bala, May 03 2022: (Start)
Conjecture: a(n) = [x^n] ( (1 + x + x^2)*(1 + x)^2/(1 - x)^2 )^n. Equivalently, a(n) = Sum_{k = 0..n} Sum_{i = 0..k} Sum_{j = 0..n-k-i} C(n, k)*C(k, i)*C(2*n, j)*C(3*n-k-i-j-1, n-k-i-j).
If the conjecture is true then the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Calculation suggests that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

A245927 G.f.: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ).

Original entry on oeis.org

1, 9, 99, 1175, 14499, 183195, 2351805, 30539241, 400000275, 5274560891, 69929215641, 931226954949, 12446852889901, 166888293332805, 2243683808486451, 30235162687458327, 408274269493595283, 5523024440001832875, 74834275541765522505, 1015429462194625633125
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2014

Keywords

Comments

Multiply the square of each term by -3 to form a bisection of A245925.
Limit a(n+1)/a(n) = 7 + 4*sqrt(3).

Examples

			G.f.: A(x) = 1 + 9*x + 99*x^2 + 1175*x^3 + 14499*x^4 + 183195*x^5 +... where
A(x)^2 = (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)). Explicitly,
A(x)^2 = 1 + 18*x + 279*x^2 + 4132*x^3 + 59949*x^4 + 860022*x^5 + 12252547*x^6 + 173756232*x^7 + 2456093529*x^8 +...+ A245924(n)*x^n +...

Links

Crossrefs

Cf. A245925, A245926, A245924, A243947.

Programs

  • Maple
    seq(add( binomial(2*n+1,2*k)*binomial(2*k,k)*3^(n-k), k = 0..n),n = 0..20); # Peter Bala, Mar 17 2018
  • Mathematica
    CoefficientList[Series[Sqrt[(1-x-Sqrt[1-14x+x^2])/(6x(1-14x+x^2))],{x,0,20}],x] (* Harvey P. Dale, Oct 23 2015 *)
a[n_] := (-1)^n Hypergeometric2F1[-n, n + 3/2, 1, 4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 17 2018 *)
  • PARI
    /* From definition: */
{a(n)=polcoeff( sqrt( (1-x - sqrt(1-14*x+x^2 +x^2*O(x^n))) / (6*x*(1-14*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* From formula for a(n)^2: */
{a(n)=sqrtint((-1/3)*sum(k=0, 2*n+1, sum(j=0, 4*n-2*k+2, (-1)^(j+k)*binomial(4*n-k+2,j+k)^2*binomial(j+k, k)^2)))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n)^2 = (-1/3)*Sum_{k=0..2*n+1} Sum_{j=0..4*n-2*k+2} (-1)^(j+k) * C(4*n-k+2,j+k)^2 * C(j+k,k)^2.
a(n) ~ (3-sqrt(3)) * (7+4*sqrt(3))^(n+1) / (12*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 16 2014
From Peter Bala, Mar 17 2018: (Start)
a(n) = Sum_{k = 0..n} C(2*n+1,2*k)*C(2*k,k)*3^(n-k).
a(n) = 3^n*hypergeom([-n, -n-1/2], [1], 4/3).
n*(4*n-3)*(2*n+1)*a(n) = (4*n-1)*(28*n^2-14*n-5)*a(n-1) - (n-1)*(2*n-1)*(4*n+1)*a(n-2). (End)
a(n) = (-1)^n*hypergeom([-n, n + 3/2], [1], 4). Peter Luschny, Mar 17 2018
a(n) = (-1)^n/sqrt(-3) * P(2*n+1, sqrt(-3)), where P(n, x) denotes the n-th Legendre polynomial. - Peter Bala, Mar 17 2024

A243945 a(n) = Sum_{k=0..n} C(2*k, k)^2 * C(n+k, n-k).

Original entry on oeis.org

1, 5, 49, 605, 8281, 120125, 1809025, 27966125, 440790025, 7051890125, 114160867129, 1865975723045, 30743797894681, 509948702030045, 8507207970913729, 142626515754330125, 2401552098016698025, 40591712338241826125, 688413807606268692025, 11710401759994742685125
Offset: 0

Views

Author

Paul D. Hanna, Aug 17 2014

Keywords

Comments

The g.f.s formed from a(2*n)^(1/2) and (a(2*n+1)/5)^(1/2) are:
A243946: sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) );
A243947: sqrt( (1+x - sqrt(1-18*x+x^2)) / (10*x*(1-18*x+x^2)) ).
Lim_{n->infinity} a(n+1)/a(n) = 9 + 4*sqrt(5).
Diagonal of rational function 1/(1 - (x + y + x*z + y*z + x*y*z)). - Gheorghe Coserea, Aug 24 2018

Examples

			G.f.: A(x) = 1 + 5*x + 49*x^2 + 605*x^3 + 8281*x^4 + 120125*x^5 + ... where
A(x) = 1/(1-x) + 2^2*x/(1-x)^3 + 6^2*x^2/(1-x)^5 + 20^2*x^3/(1-x)^7 + 70^2*x^4/(1-x)^9 + 252^2*x^5/(1-x)^11 + 924^2*x^6/(1-x)^13 + ... + A000984(n)^2*x^n/(1-x)^(2*n+1) + ...

Links

Crossrefs

Cf. A243946, A243947, A245925.

Programs

  • Magma
    &cat[ [&+[ Binomial(2*k, k)^2 * Binomial(n+k, n-k): k in [0..n]]]: n in [0..30]]; // Vincenzo Librandi, Aug 25 2018
  • Mathematica
    Table[Sum[Binomial[2*k, k]^2 * Binomial[n + k, n - k],{k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 18 2014 *)
a[n_] := HypergeometricPFQ[{1/2, -n, n + 1}, {1, 1}, -4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 14 2018 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(2*k, k)^2*binomial(n+k, n-k))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=sum(m=0, n, binomial(2*m, m)^2 * x^m/(1-x +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff( 1 / agm(1-x, sqrt((1-x)^2 - 16*x +x*O(x^n))), n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} binomial(2*n, n)^2 * x^n / (1-x)^(2*n+1).
G.f.: 1 / AGM(1-x, sqrt(1-18*x+x^2)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
a(2*n) = A243946(n)^2.
a(2*n+1) = 5 * A243947(n)^2.
Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(19*n^2 - 38*n + 14)*a(n-1) - (2*n-3)*(19*n^2 - 38*n + 14)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014
a(n) ~ (2+sqrt(5)) * (9+4*sqrt(5))^n / (4*Pi*n). - Vaclav Kotesovec, Aug 18 2014. Equivalently, a(n) ~ phi^(6*n + 3) / (4*Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -4). - Peter Luschny, Mar 14 2018
G.f. y=A(x) satisfies: 0 = x*(x^2 - 1)*(x^2 - 18*x + 1)*y'' + (3*x^4 - 34*x^3 - 38*x^2 + 38*x - 1)*y' + (x^3 - 3*x^2 - 19*x + 5)*y. - Gheorghe Coserea, Aug 29 2018
From Peter Bala, Feb 07 2022: (Start)
a(n) = P(n,sqrt(5))^2, where P(n,x) denotes the n-th Legendre polynomial.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all primes p and positive integers n and k. (End)

A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

Original entry on oeis.org

1, 9, 169, 3969, 103041, 2832489, 80802121, 2365752321, 70611901441, 2139090528969, 65568745087209, 2029206892664961, 63300531617048961, 1987912809986437161, 62787371136571152009, 1992942254830520803329, 63531842302018973818881, 2033004661359005674887561
Offset: 0

Views

Author

Paul D. Hanna, Aug 17 2014

Keywords

Comments

In general, we have the binomial identity:
if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k), then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k), where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2), and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
Note that the g.f. of A001850 is 1/sqrt(1 - 6*x + x^2).
Limit_{n -> oo} a(n+1)/a(n) = (3 + 2*sqrt(2))^2 = 17 + 12*sqrt(2).
From Gheorghe Coserea, Jul 05 2016: (Start)
Diagonal of the rational function 1/(1 - x - y - z - x*y + x*z - y*z - x*y*z).
Annihilating differential operator: x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2 + (3*x^4-66*x^3-70*x^2+70*x-1)*Dx + x^3-7*x^2-35*x+9.
(End).
The sequence b(n) mentioned above is the sequence of shifted Legendre polynomials P(n,2*t + 1) (see A063007). See Zudilin for a g.f. for the sequence b(n)^2. - Peter Bala, Mar 02 2017

Examples

			G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...

Links

Crossrefs

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), this sequence (m=1), A243943 (m=2), A243944 (m=3), A243007 (m=4).
Related to diagonal of rational functions: A268545 - A268555.
Cf. A001850, A243945, A245925.

Programs

  • Magma
    [Evaluate(LegendrePolynomial(n), 3)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023
  • Mathematica
    Table[Sum[2^k *Binomial[2*k, k]^2 *Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 18 2014 *)
a[n_]:= HypergeometricPFQ[{1/2, -n, n+1}, {1, 1}, -8];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Mar 14 2018 *)
LegendreP[Range[0, 30], 3]^2 (* G. C. Greubel, May 17 2023 *)
  • PARI
    {a(n) = sum(k=0, n, 2^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 36*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • Python
    from math import comb
def A243949(n): return sum(comb(n,k)*comb(n+k,k) for k in range(n+1))**2 # Chai Wah Wu, Mar 23 2023
  • SageMath
    [gen_legendre_P(n,0,3)^2 for n in range(41)] # G. C. Greubel, May 17 2023

Formula

G.f.: 1 / AGM(1-x, sqrt(1-34*x+x^2)). - Paul D. Hanna, Aug 30 2014
a(n) = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k).
a(n)^(1/2) = Sum_{k=0..n} C(2*k, k) * C(n+k, n-k).
Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(35*n^2 - 70*n + 26)*a(n-1) - (2*n-3)*(35*n^2 - 70*n + 26)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014
a(n) ~ (4 + 3*sqrt(2)) * (3 + 2*sqrt(2))^(2*n) / (8*Pi*n). - Vaclav Kotesovec, Aug 18 2014
From Gheorghe Coserea, Jul 05 2016: (Start)
G.f.: hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4).
0 = x*(x-1)*(x+1)*(x^2-34*x+1)*y'' + (3*x^4-66*x^3-70*x^2+70*x-1)*y' + (x^3-7*x^2-35*x+9)*y, where y is g.f.
(End)
a(n) = Sum_{k = 0..n} 4^k*binomial(n+k,2*k)^2*binomial(2*k,k). - Peter Bala, Mar 02 2017
a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -8). - Peter Luschny, Mar 14 2018
G.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)^2 *x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022

A246539 G.f.: Sum_{n>=0} 3^n * x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2.

Original entry on oeis.org

1, 4, 25, 184, 1489, 12796, 114241, 1047568, 9796057, 92989876, 893250193, 8663461000, 84697699297, 833616713164, 8251811812465, 82088310375904, 820140832103881, 8225191769615620, 82768982623011841, 835404195075128536, 8454743911307336857, 85775961307556225596, 872156269935215409577
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2014

Keywords

Comments

a(n) == 1 (mod 3) for n>=0.

Examples

			G.f.: A(x) = 1 + 4*x + 25*x^2 + 184*x^3 + 1489*x^4 + 12796*x^5 +...
where
A(x) = 1/(1-x) + 3*x/(1-x)^3*(1+x)^2
+ 3^2*x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
+ 3^3*x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2
+ 3^4*x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
+ 3^5*x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
+ 3^6*x^6/(1-x)^13*(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +...

Crossrefs

Cf. A246538, A246540, A246056, A246423, A243948, A245929, A227845, A245925.

Programs

  • Mathematica
    Table[Sum[3^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 3^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Nov 05 2014 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, 3^m*x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-3*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 3^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 3^(m-k) * sum(j=0, k, binomial(k, j)^2 * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 3^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 3^j))}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1-3*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k]^2.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 3^k * Sum_{j=0..k} C(k,j)^2 * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * Sum_{j=0..k} C(k,j)^2 * 3^j * x^j.
a(n) = Sum_{k=0..[n/2]} 3^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 3^j.
Recurrence: (n-3)*n^2*a(n) = 4*(n-3)*(3*n^2 - 3*n + 1)*a(n-1) - 3*(n-1)*(3*n^2 - 12*n + 8)*a(n-2) - 8*(n-2)*(7*n^2 - 28*n + 24)*a(n-3) - 9*(n-3)*(3*n^2 - 12*n + 8)*a(n-4) + 36*(n-1)*(3*n^2 - 21*n + 37)*a(n-5) - 27*(n-4)^2*(n-1)*a(n-6). - Vaclav Kotesovec, Nov 05 2014
a(n) ~ sqrt(3) * (2 + 2*sqrt(3) + sqrt(13+8*sqrt(3)))^(n+1) / (12*Pi*n). - Vaclav Kotesovec, Nov 05 2014

A246538 G.f.: Sum_{n>=0} 2^n * x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2.

Original entry on oeis.org

1, 3, 15, 87, 559, 3807, 26919, 195399, 1445967, 10859967, 82527687, 633165255, 4896345487, 38117454303, 298435452135, 2348094847047, 18554434810831, 147171478237695, 1171272947140359, 9349653181797063, 74834845484454927, 600441007306747167, 4828337830880795943, 38904633251921442375
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2014

Keywords

Comments

a(n) == 1 (mod 3) iff n = 4*A005836(k) for k>=0, and a(n) == 0 (mod 3) otherwise, where A005836 gives numbers n whose base 3 representation contains no 2.

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 87*x^3 + 559*x^4 + 3807*x^5 + 26919*x^6 +...
where
A(x) = 1/(1-x) + 2*x/(1-x)^3*(1+x)^2
+ 2^2*x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
+ 2^3*x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2
+ 2^4*x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
+ 2^5*x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
+ 2^6*x^6/(1-x)^13*(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +...

Crossrefs

Cf. A246539, A246540, A246056, A246423, A243948, A245929, A227845, A245925, A005836.

Programs

  • Mathematica
    Table[Sum[2^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 2^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Nov 05 2014 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, 2^m*x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-2*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 2^k * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 2^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 2^(m-k) * sum(j=0, k, binomial(k, j)^2 * 2^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* By a formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 2^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 2^j))}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1-2*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k]^2.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 2^k * Sum_{j=0..k} C(k,j)^2 * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 2^(n-k) * Sum_{j=0..k} C(k,j)^2 * 2^j * x^j.
a(n) = Sum_{k=0..[n/2]} 2^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 2^j.
D-finite with recurrence: (n-4)*(n-1)^2*a(n) = 3*(n-4)*(3*n^2 - 9*n + 7)*a(n-1) - (n-2)*(n^2 - 6*n + 6)*a(n-2) - 3*(n-3)*(11*n^2 - 66*n + 92)*a(n-3) - 2*(n-4)*(n^2 - 6*n + 6)*a(n-4) + 12*(n-2)*(3*n^2 - 27*n + 61)*a(n-5) - 8*(n-5)^2*(n-2)*a(n-6). - Vaclav Kotesovec, Nov 05 2014, for offset 1.
a(n) ~ ((3 + 4*sqrt(2) + sqrt(33+24*sqrt(2))))^n / (Pi *n * 2^(n+5/2)). - Vaclav Kotesovec, Nov 05 2014
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