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.

A246455 G.f.: Sum_{n>=0} x^n / (1-2*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, 3, 14, 75, 439, 2718, 17483, 115575, 779716, 5342937, 37065029, 259694334, 1834494025, 13048232853, 93352065596, 671245767795, 4847769647701, 35145881098272, 255674861620439, 1865618971630305, 13650358024120804, 100123485522075447, 736036670999528711
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2014

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 75*x^3 + 439*x^4 + 2718*x^5 + 17483*x^6 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-2*x) + x/(1-2*x)^3 * (1+x) * (1+2*x)
+ x^2/(1-2*x)^5 * (1 + 2^2*x + x^2) * (1 + 2^2*2*x + 4*x^2)
+ x^3/(1-2*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/(1-2*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/(1-2*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:
A(x) = 1 + x*(1 + (2+x)) + x^2*(1 + 2^2*(2+x) + (4+2^2*2*x+x^2))
+ x^3*(1 + 3^2*(2+x) + 3^2*(4+2^2*2*x+x^2) + (8+3^2*4*x+3^2*2*x^2+x^3))
+ x^4*(1 + 4^2*(2+x) + 6^2*(4+2^2*2*x+x^2) + 4^2*(8+3^2*4*x+3^2*2*x^2+x^3) + (16+4^2*8*x+6^2*4*x^2+4^2*2*x^3+x^4))
+ x^5*(1 + 5^2*(2+x) + 10^2*(4+2^2*2*x+x^2) + 10^2*(8+3^2*4*x+3^2*2*x^2+x^3) + 5^2*(16+4^2*8*x+6^2*4*x^2+4^2*2*x^3+x^4) + (32+5^2*16*x+10^2*8*x^2+10^2*4*x^3+5^2*2*x^4+x^5)) +...

Crossrefs

Cf. A243948 (dual), A246538.

Programs

  • Mathematica
    Table[Sum[Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, k]^2 * 2^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Aug 27 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*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^(k-j)*x^j)+x*O(x^n))), 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 * 2^(m-k) * sum(j=0, k, binomial(k, j)^2*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^j))}
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^(k-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 * 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^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)*(34*n^2 - 102*n + 77)*a(n-2) - 3*(n-5)*(n-2)*(2*n-5)*a(n-3) + (n-3)*(53*n^4 - 636*n^3 + 2617*n^2 - 4254*n + 2226)*a(n-4) - 3*(n-4)*(n-1)*(2*n-7)*a(n-5) - (n-5)*(n-2)*(n-1)*(34*n^2 - 306*n + 689)*a(n-6) + 3*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - (n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Aug 27 2014
a(n) ~ c * d^n / n, where d = 5/2 + sqrt(2) + sqrt(29+20*sqrt(2))/2 = 7.698532234454699125... is the root of the equation d^4 - 10*d^3 + 19*d^2 - 10*d + 1 = 0, and c = 0.51515814369432019442929480754730417101213524575391074352368... . - Vaclav Kotesovec, Aug 27 2014

Extensions

Minor edits by Vaclav Kotesovec, Nov 05 2014

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

A246540 G.f.: Sum_{n>=0} 4^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, 5, 37, 325, 3125, 31925, 339077, 3700645, 41200981, 465736725, 5328229797, 61552244485, 716791570549, 8403794763125, 99096946864325, 1174370518273125, 13977636401394069, 167001257979441365, 2002052157653251557, 24073717683854557125, 290261630170911545525, 3508332484300450371125
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2014

Keywords

Comments

a(n) == 5 (mod 16) for n>=1.

Examples

			G.f.: A(x) = 1 + 5*x + 37*x^2 + 325*x^3 + 3125*x^4 + 31925*x^5 +...
where
A(x) = 1/(1-x) + 4*x/(1-x)^3*(1+x)^2
+ 4^2*x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
+ 4^3*x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2
+ 4^4*x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
+ 4^5*x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
+ 4^6*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, A246539, A246056, A246423, A243948, A245929, A227845, A245925.

Programs

  • Mathematica
    Table[Sum[4^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 4^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, 4^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/(1-4*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 4^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 * 4^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 * 4^(m-k) * sum(j=0, k, binomial(k, j)^2 * 4^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, 4^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 4^j))}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k]^2.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 4^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 * 4^(n-k) * Sum_{j=0..k} C(k,j)^2 * 4^j * x^j.
a(n) = Sum_{k=0..[n/2]} 4^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 4^j.
Recurrence: (n-3)*n^2*a(n) = 5*(n-3)*(3*n^2 - 3*n + 1)*a(n-1) - (n-1)*(23*n^2 - 92*n + 65)*a(n-2) - 5*(n-2)*(15*n^2 - 60*n + 53)*a(n-3) - 4*(n-3)*(23*n^2 - 92*n + 65)*a(n-4) + 80*(n-1)*(3*n^2 - 21*n + 37)*a(n-5) - 64*(n-4)^2*(n-1)*a(n-6). - Vaclav Kotesovec, Nov 05 2014
a(n) ~ ((13+3*sqrt(17))/2)^(n+1) / (8*Pi*n). - Vaclav Kotesovec, Nov 05 2014
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