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 1-4 of 4 results.

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

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

A246813 G.f.: Sum_{n>=0} x^n / (1-3*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k].

Original entry on oeis.org

1, 4, 23, 152, 1085, 8156, 63579, 509136, 4161649, 34566580, 290798551, 2471871784, 21191824645, 182984610220, 1589620392835, 13881368684128, 121767703088377, 1072382299895428, 9477296423786207, 84017470425706040, 746903374745524629, 6656552616997851036, 59459592374756968323
Offset: 0

Views

Author

Paul D. Hanna, Sep 03 2014

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1085*x^4 + 8156*x^5 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-3*x) + x/(1-3*x)^3 * (1+x) * (1+3*x)
+ x^2/(1-3*x)^5 * (1 + 2^2*x + x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-3*x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*3*x + 3^2*9*x^2 + 27*x^3)
+ x^4/(1-3*x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + 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*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + 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) +...
We can also express the g.f. by the binomial series:
A(x) = 1 + x*(1 + (3+x)) + x^2*(1 + 2^2*(3+x) + (9+2^2*3*x+x^2))
+ x^3*(1 + 3^2*(3+x) + 3^2*(9+2^2*3*x+x^2) + (27+3^2*9*x+3^2*3*x^2+x^3))
+ x^4*(1 + 4^2*(3+x) + 6^2*(9+2^2*3*x+x^2) + 4^2*(27+3^2*9*x+3^2*3*x^2+x^3) + (81+4^2*27*x+6^2*9*x^2+4^2*3*x^3+x^4))
+ x^5*(1 + 5^2*(3+x) + 10^2*(9+2^2*3*x+x^2) + 10^2*(27+3^2*9*x+3^2*3*x^2+x^3) + 5^2*(81+4^2*27*x+6^2*9*x^2+4^2*3*x^3+x^4) + (243+5^2*81*x+10^2*27*x^2+10^2*9*x^3+5^2*3*x^4+x^5)) +...

Crossrefs

Cf. A246812 (dual), A246455, A243948, A246056, A246423, A246539, A245929, A227845, A245925.

Programs

  • Mathematica
    Table[Sum[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, Oct 04 2014 *)
  • 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 * 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)=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 * 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, 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 * Sum_{k=0..n} C(n,k)^2 * 3^(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, j)^2 * 3^j.
a(n) ~ sqrt(12 + 23/sqrt(3) + 2*sqrt(80 + 46*sqrt(3))) * (3 + sqrt(3) + sqrt(11 + 6*sqrt(3)))^n / (4*Pi*n). - Vaclav Kotesovec, Oct 04 2014

A246812 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^k * x^k].

Original entry on oeis.org

1, 2, 9, 44, 241, 1374, 8145, 49512, 306729, 1927802, 12256753, 78661620, 508786129, 3312561638, 21688815729, 142699137072, 942873631497, 6253352120322, 41611854129585, 277723513754364, 1858529465302329, 12467403845702526, 83817799189753785, 564633483609422808, 3810607016379076521
Offset: 0

Views

Author

Paul D. Hanna, Sep 03 2014

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 44*x^3 + 241*x^4 + 1374*x^5 + 8145*x^6 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-x) + x/(1-x)^3*(1+x) * (1+3*x)
+ x^2/(1-x)^5*(1 + 2^2*x + x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*3*x + 3^2*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) * (1 + 4^2*3*x + 6^2*9*x^2 + 4^2*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) * (1 + 5^2*3*x + 10^2*9*x^2 + 10^2*27*x^3 + 5^2*81*x^4 + 243*x^5) +...
We can also express the g.f. by the binomial series identity:
A(x) = 1 + x*(1 + (1+3*x)) + x^2*(1 + 2^2*(1+3*x) + (1+2^2*3*x+9*x^2))
+ x^3*(1 + 3^2*(1+3*x) + 3^2*(1+2^2*3*x+9*x^2) + (1+3^2*3*x+3^2*9*x^2+27*x^3))
+ x^4*(1 + 4^2*(1+3*x) + 6^2*(1+2^2*3*x+9*x^2) + 4^2*(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*(1 + 5^2*(1+3*x) + 10^2*(1+2^2*3*x+9*x^2) + 10^2*(1+3^2*3*x+3^2*9*x^2+27*x^3) + 5^2*(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)) +...

Crossrefs

Cf. A246813 (dual), A243948, A246455, A246056, A246423, A246539, 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 * 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)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 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
    /* Formula for a(n): */
{a(n)=sum(k=0, n\2, 3^k * sum(j=0, n-2*k, binomial(n-k, k+j)^2 * binomial(k+j, j)^2))}
for(n=0, 25, 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 * 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.
Showing 1-4 of 4 results.

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