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.

Showing 1-3 of 3 results.

A227845 G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [ Sum_{k=0..n} C(n,k)^2*x^k ]^2.

Original entry on oeis.org

1, 2, 7, 28, 125, 590, 2891, 14536, 74497, 387450, 2038743, 10830148, 57986773, 312542678, 1694166275, 9228580464, 50486521785, 277239830210, 1527533993871, 8441627856300, 46776754474709, 259830443968046, 1446468759734131, 8068688342238328, 45091854560015025, 252423540736438890
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2013

Keywords

Comments

Equals antidiagonal sums of table A143007.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 28*x^3 + 125*x^4 + 590*x^5 + 2891*x^6 +...
where
A(x) = 1/(1-x) + x/(1-x)^3 * (1+x)^2 + x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3)^2
+ x^4/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
+ x^5/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
+ 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 +...
We can also express the g.f. by the binomial series identity:
A(x) = 1 + x*(1 + (1+x)) + x^2*(1 + 2^2*(1+x) + (1+2^2*x+x^2))
+ x^3*(1 + 3^2*(1+x) + 3^2*(1+2^2*x+x^2) + (1+3^2*x+3^2*x^2+x^3))
+ x^4*(1 + 4^2*(1+x) + 6^2*(1+2^2*x+x^2) + 4^2*(1+3^2*x+3^2*x^2+x^3) + (1+4^2*x+6^2*x^2+4^2*x^3+x^4))
+ x^5*(1 + 5^2*(1+x) + 10^2*(1+2^2*x+x^2) + 10^2*(1+3^2*x+3^2*x^2+x^3) + 5^2*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) + (1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5)) +...
The square-root of the g.f. is an integer series:
A(x)^(1/2) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 215*x^5 + 1029*x^6 +...+ A227846(n)*x^n +...
The g.f. also satisfies A(x) = F(x*A(x))^2 and F(x)^2 = A(x/F(x)^2) where
F(x) = 1 + x + x^2 + x^4 - 2*x^5 - 4*x^6 - 7*x^8 + 20*x^9 + 42*x^10 + 84*x^12 - 272*x^13 - 584*x^14 - 1239*x^16 +...+ A258053(n)*x^n +...
such that A258053(4*n+3) = 0 for n>=0.

Links

Crossrefs

Cf. A227846, A245930, A143007.
Cf. A258053.

Programs

  • Maple
    U := proc(n) options remember;
        if n < 1 then 1
        elif n = 1 then 2
        elif n = 2 then 7/2
        else
                (2*(3*n^2-3*n+1)*U(n-2) - (n-1)^2*U(n-4))/n^2
        fi
end:
seq(U(n)*U(n-1), n=0..25); # Mark van Hoeij, Jul 10 2024
  • Mathematica
    Table[Sum[Sum[Binomial[n-k,j]^2*Binomial[j,k]^2,{j,k,n-k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Jul 05 2014 *)
  • PARI
    /* From 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)^2+x*O(x^n)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From alternate g.f.: */
{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*x^j)+x*O(x^n))),n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0,n\2,sum(j=k,n-k,binomial(n-k,j)^2*binomial(j,k)^2))}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From g.f.: 1/AGM((1+x)^2, 1-6*x+x^2) */
{a(n)=local(A);A = 1 / agm((1+x)^2, 1-6*x+x^2 +x*O(x^n));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * Sum_{j=0..k} binomial(k,j)^2 * x^j.
a(n) = Sum_{k=0..[n/2]} Sum_{j=k..n-k} binomial(n-k,j)^2 * binomial(j,k)^2.
Recurrence: n^2*a(n) = 2*(3*n^2 - 3*n + 1)*a(n-1) - 2*(3*n^2 - 9*n + 7)*a(n-3) + (n-2)^2*a(n-4). - Vaclav Kotesovec, Jul 05 2014
a(n) ~ (3+2*sqrt(2))^(n+1) / (4*Pi*n). - Vaclav Kotesovec, Jul 05 2014
G.f.: 1 / AGM((1+x)^2, 1 - 6*x + x^2), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) denotes the arithmetic-geometric mean. - Paul D. Hanna, Jul 31 2014
G.f. satisfies: A(x) = F(x*A(x))^2, where F(x) is the g.f. of A258053. - Paul D. Hanna, May 17 2015
G.f.: hypergeom([1/2, 1/2], [1], -16*x^2/((x+1)^2*(x^2-6*x+1)))/((x+1)*sqrt(x^2-6*x+1)). - Mark van Hoeij, Jul 08 2024
a(n) = U(n)*U(n-1) where the sequences U(-1),U(1),U(3),... and U(0),U(2),U(4),... satisfy a second order recurrence n^2*U(n) = 2*(3*n^2-3*n+1)*U(n-2) - (n-1)^2*U(n-4) with initial terms U(-1), U(1)=2 and U(0)=1, U(2)=7/2. - Mark van Hoeij, Jul 10 2024

Extensions

Name changed by Paul D. Hanna, Sep 07 2014

A158100 G.f. satisfies: A(x) = 1/AGM(1, 1 - 8*x/A(x) ).

Original entry on oeis.org

1, 4, 4, 0, 4, 0, -16, 0, -28, 0, 176, 0, 336, 0, -2496, 0, -4956, 0, 40112, 0, 81488, 0, -694720, 0, -1432688, 0, 12647488, 0, 26360896, 0, -238598400, 0, -501256668, 0, 4623092400, 0, 9772018896, 0, -91458048960, 0, -194263943664, 0, 1839634167360
Offset: 0

Views

Author

Paul D. Hanna, Mar 13 2009

Keywords

Comments

See A060691 for the expansion of AGM(1,1-8x), where AGM denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 +-...
1 - 8*x/A(x) = 1 - 8*x + 32*x^2 - 96*x^3 + 256*x^4 - 608*x^5 +-...
From _Paul D. Hanna_, Mar 14 2009: (Start)
Convolution square root is A158122 and begins:
[1,2,0,0,2,-4,0,0,-16,40,0,0,200,-544,0,0,-3006,8540,0,0,...]
in which the convolution of the quadrisections equals 2:
[1,2,-16,200,-3006,...]*[2,-4,40,-544,8540,...] = 2. (End)

Crossrefs

Cf. A060691, A158101 (bisection), A258053.
Cf. A158122 (sqrt), A158212, A158213.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x/agm(1,1-8*x +x*O(x^n))),n)}

Formula

G.f.: A(x) = x/Series_Reversion( x/AGM(1, 1-8*x) ).
Convolution square-root is A158122, which has two nonzero quadrisections, A158212 and A158213, that are inverse convolutions of each other (by a factor of 2). - Paul D. Hanna, Mar 14 2009

A158122 G.f. A(x) satisfies: A(x)^2 = 1/AGM(1, 1 - 8*x/A(x)^2 ).

Original entry on oeis.org

1, 2, 0, 0, 2, -4, 0, 0, -16, 40, 0, 0, 200, -544, 0, 0, -3006, 8540, 0, 0, 49956, -145720, 0, 0, -884352, 2625648, 0, 0, 16349648, -49161024, 0, 0, -311986480, 947069352, 0, 0, 6098614912, -18650752400, 0, 0, -121497078016, 373773754912, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Mar 13 2009

Keywords

Comments

See A060691 for the expansion of AGM(1,1-8x), where AGM denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 -+...
A(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 + 176*x^10 +...
Contribution from _Paul D. Hanna_, Mar 14 2009: (Start)
G.f. of quadrasection A158212 is:
B(x) = 1 + 2*x - 16*x^2 + 200*x^3 - 3006*x^4 + 49956*x^5 +...;
G.f. of quadrasection A158213 is C(x) = 2/B(x):
C(x) = 2 - 4*x + 40*x^2 - 544*x^3 + 8540*x^4 - 145720*x^5 +...
where g.f. A(x) = B(x^4) + x*C(x^4) = B(x^4) + 2*x/B(x^4). (End)

Crossrefs

Cf. A060691, A158100 (self-convolution), A258053.
Cf. quadrasections: A158212, A158213.

Programs

  • PARI
    {a(n)=polcoeff(sqrt(x/serreverse(x/agm(1,1-8*x +x*O(x^n)))),n)}

Formula

G.f.: A(x) = sqrt( x/Series_Reversion( x/AGM(1,1-8*x) ) ).
Self-convolution equals A158100.
Contribution from Paul D. Hanna, Mar 14 2009: (Start)
Quadrasections are A158212(n) = A158122(4n) and A158213 = A158122(4n+1);
let B(x), C(x), be the g.f.s of A158212 and A158213, respectively,
then C(x) = 2/B(x) so that
A(x) = B(x^4) + x*C(x^4) = B(x^4) + 2*x/B(x^4) = 2/C(x^4) + x*C(x^4). (End)
Showing 1-3 of 3 results.

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