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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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.

A249921 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 * 4^k * x^k].

Original entry on oeis.org

1, 3, 17, 111, 805, 6147, 48641, 394863, 3266629, 27421395, 232867889, 1996302447, 17248208485, 150013649955, 1312111499105, 11532737017839, 101799869875717, 901975446062451, 8018470050567953, 71496291428776815, 639204721160345509, 5728606469731066947, 51453397357702434497
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2014

Keywords

Comments

Compare this sequence to its dual, A248053.

Examples

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 111*x^3 + 805*x^4 + 6147*x^5 + 48641*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 + 4*x)
+ x^2/(1-2*x)^5 * (1 + 2^2*2*x + 4*x^2) * (1 + 2^2*4*x + 16*x^2)
+ x^3/(1-2*x)^7 * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3) * (1 + 3^2*4*x + 3^2*16*x^2 + 64*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*4*x + 6^2*16*x^2 + 4^2*64*x^3 + 2561*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*4*x + 10^2*16*x^2 + 10^2*64*x^3 + 5^2*256*x^4 + 1024*x^5) +...
equals the series
A(x) = 1/(1-x) + x/(1-x)^3 * (1 + x) * (2 + 4*x)
+ x^2/(1-x)^5 * (1 + 2^2*x + x^2) * (4 + 2^2*2*4*x + 16*x^2)
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (8 + 3^2*4*4*x + 3^2*2*16*x^2 + 64*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*4*x + 6^2*4*16*x^2 + 4^2*2*64*x^3 + 256*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*4*x + 10^2*8*16*x^2 + 10^2*4*64*x^3 + 5^2*2*256*x^4 + 1024*x^5) +...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(2 + (1+4*x)) + x^2*(4 + 2^2*2*(1+4*x) + (1+2^2*4*x+16*x^2))
+ x^3*(8 + 3^2*4*(1+4*x) + 3^2*2*(1+2^2*4*x+16*x^2) + (1+3^2*4*x+3^2*16*x^2+64*x^3))
+ x^4*(16 + 4^2*8*(1+4*x) + 6^2*4*(1+2^2*4*x+16*x^2) + 4^2*2*(1+3^2*4*x+3^2*16*x^2+64*x^3) + (1+4^2*4*x+6^2*16*x^2+4^2*64*x^3+256*x^4))
+ x^5*(32 + 5^2*16*(1+4*x) + 10^2*8*(1+2^2*4*x+16*x^2) + 10^2*4*(1+3^2*4*x+3^2*16*x^2+64*x^3) + 5^2*2*(1+4^2*4*x+6^2*16*x^2+4^2*64*x^3+256*x^4) + (1+5^2*4*x+10^2*16*x^2+10^2*64*x^3+5^2*256*x^4+1024*x^5)) +...
equals the series
A(x) = 1 + x*(1 + (2+4*x)) + x^2*(1 + 2^2*(2+4*x) + (4+2^2*2*4*x+16*x^2))
+ x^3*(1 + 3^2*(2+4*x) + 3^2*(4+2^2*2*4*x+16*x^2) + (8+3^2*4*4*x+3^2*2*16*x^2+64*x^3))
+ x^4*(1 + 4^2*(2+4*x) + 6^2*(4+2^2*2*4*x+16*x^2) + 4^2*(8+3^2*4*4*x+3^2*2*16*x^2+64*x^3) + (16+4^2*8*4*x+6^2*4*16*x^2+4^2*2*64*x^3+256*x^4))
+ x^5*(1 + 5^2*(2+4*x) + 10^2*(4+2^2*2*4*x+16*x^2) + 10^2*(8+3^2*4*4*x+3^2*2*16*x^2+64*x^3) + 5^2*(16+4^2*8*4*x+6^2*4*16*x^2+4^2*2*64*x^3+256*x^4) + (32+5^2*16*4*x+10^2*8*26*x^2+10^2*4*64*x^3+5^2*2*256*x^4+1024*x^5)) +...

Crossrefs

Cf. A248053, A243948, A245929, A227845, A245925, A005836, A246510, A246423, A246455, A246056.

Programs

  • Mathematica
    Table[Sum[4^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 09 2014 *)
  • PARI
    /* By definition: */
{a(n,p,q)=local(A=1); A=sum(m=0, n, x^m/(1-p*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * p^k * x^k) * sum(k=0, m, binomial(m, k)^2 * q^k *x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n,2,4), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p,q)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*p^(m-k)*q^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,2,4), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p,q)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * p^(m-k)* sum(j=0, k, binomial(k, j)^2 * q^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n,2,4), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p,q)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * p^(k-j) * q^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n,2,4), ", "))
  • PARI
    /* Formula for a(n): */
{a(n,p,q)=sum(k=0, n\2, sum(j=0, n-2*k, q^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * p^j))}
for(n=0, 25, print1(a(n,2,4), ", "))

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) * 4^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) * 4^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 * 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 * 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)*(22*n^2 - 66*n + 53)*a(n-2) - 12*(n-5)*(n-2)*(3*n^3 - 21*n^2 + 44*n - 29)*a(n-3) + (n-3)*(143*n^4 - 1716*n^3 + 7111*n^2 - 11778*n + 6336)*a(n-4) - 48*(n-4)*(n-1)*(3*n^3 - 33*n^2 + 116*n - 127)*a(n-5) - 16*(n-5)*(n-2)*(n-1)*(22*n^2 - 198*n + 449)*a(n-6) + 192*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - 256*(n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Nov 09 2014
a(n) ~ sqrt((56 + 49*sqrt(2) + sqrt(2*(3905+2744*sqrt(2))))/2) * ((7 + 2*sqrt(2) + sqrt(41 + 28*sqrt(2)))/2)^n / (8*Pi*n). - Vaclav Kotesovec, Nov 09 2014
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