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.

A248053 G.f.: Sum_{n>=0} x^n / (1-4*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, 5, 35, 285, 2519, 23545, 228715, 2284365, 23294551, 241366025, 2532599675, 26845322925, 286946021495, 3088651368025, 33443864305675, 363983410742925, 3979005075583255, 43667580737050025, 480884378835323675, 5311978947724802925, 58839469859529979319, 653372409529941364345
Offset: 0

Views

Author

Paul D. Hanna, Sep 30 2014

Keywords

Comments

Compare this sequence to its dual, A249921.

Examples

			G.f.: A(x) = 1 + 5*x + 35*x^2 + 285*x^3 + 2519*x^4 + 23545*x^5 +...
where the g.f. is given by the binomial series identity:
A(x) = 1/(1-4*x) + x/(1-4*x)^3 * (1 + 2*x) * (1 + 4*x)
+ x^2/(1-4*x)^5 * (1 + 2^2*2*x + 4*x^2) * (1 + 2^2*4*x + 16*x^2)
+ x^3/(1-4*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/(1-4*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/(1-4*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) * (4 + 2*x)
+ x^2/(1-x)^5 * (1 + 2^2*x + x^2) * (16 + 2^2*4*2*x + 4*x^2)
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (64 + 3^2*16*2*x + 3^2*4*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) * (256 + 4^2*64*2*x + 6^2*16*4*x^2 + 4^2*4*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) * (1024 + 5^2*256*2*x + 10^2*64*4*x^2 + 10^2*16*8*x^3 + 5^2*4*16*x^4 + 32*x^5) +...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(4 + (1+2*x)) + x^2*(16 + 2^2*4*(1+2*x) + (1+2^2*2*x+4*x^2))
+ x^3*(64 + 3^2*16*(1+2*x) + 3^2*4*(1+2^2*2*x+4*x^2) + (1+3^2*2*x+3^2*4*x^2+8*x^3))
+ x^4*(256 + 4^2*64*(1+2*x) + 6^2*16*(1+2^2*2*x+4*x^2) + 4^2*4*(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*(1024 + 5^2*256*(1+2*x) + 10^2*64*(1+2^2*2*x+4*x^2) + 10^2*16*(1+3^2*2*x+3^2*4*x^2+8*x^3) + 5^2*4*(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 + (4+2*x)) + x^2*(1 + 2^2*(4+2*x) + (16+2^2*4*2*x+4*x^2))
+ x^3*(1 + 3^2*(4+2*x) + 3^2*(16+2^2*4*2*x+4*x^2) + (64+3^2*16*2*x+3^2*4*4*x^2+8*x^3))
+ x^4*(1 + 4^2*(4+2*x) + 6^2*(16+2^2*4*2*x+4*x^2) + 4^2*(64+3^2*16*2*x+3^2*4*4*x^2+8*x^3) + (256+4^2*64*2*x+6^2*16*4*x^2+4^2*8*4*x^3+16*x^4))
+ x^5*(1 + 5^2*(4+2*x) + 10^2*(16+2^2*4*2*x+4*x^2) + 10^2*(64+3^2*16*2*x+3^2*4*4*x^2+8*x^3) + 5^2*(256+4^2*64*2*x+6^2*16*4*x^2+4^2*8*4*x^3+16*x^4) + (1024+5^2*256*2*x+10^2*64*4*x^2+10^2*16*8*x^3+5^2*4*16*x^4+32*x^5)) +...

Crossrefs

Cf. A249921, A246510, A246423, A246455, A246056.

Programs

  • Mathematica
    Table[Sum[2^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, Oct 04 2014 *)
  • PARI
    /* By definition: */
{a(n,p=4,q=2)=local(A=1); A=sum(m=0, n, x^m/(1-p*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * q^k * x^k) * sum(k=0, m, binomial(m,k)^2 * p^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n,4,2), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p=4,q=2)=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,4,2), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p=4,q=2)=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,4,2), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p=4,q=2)=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,4,2), ", "))
  • PARI
    /* Formula for a(n): */
{a(n,p=4,q=2)=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,4,2), ", "))

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 * 4^(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 * 4^(k-j) * 2^j * 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 * 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 * 4^j.
a(n) ~ sqrt(24+17*sqrt(2)) * (6+4*sqrt(2))^n / (4*Pi*n). - Vaclav Kotesovec, Oct 04 2014

A246509 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] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].

Original entry on oeis.org

1, 4, 26, 200, 1694, 15224, 141972, 1359120, 13264246, 131384536, 1316769196, 13323636208, 135885868780, 1395157192624, 14406117404584, 149489132177440, 1557898906160806, 16297193704008856, 171058624529373116, 1800860588158214960, 19010179617892702404, 201164103801453466896
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 200*x^3 + 1694*x^4 + 15224*x^5 +...

Crossrefs

Cf. A246510, A246423.

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 * 3^k * x^k) * sum(k=0, m, binomial(m, k)^2 * 4^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) * 4^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 * sum(j=0, k, binomial(k, j)^2 * 3^(k-j) * 4^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 * 3^(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
    /* 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 * 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) * 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 * 3^(k-j) * 4^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 * 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 * 3^j.

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

A248055 G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].

Original entry on oeis.org

1, 5, 34, 265, 2219, 19490, 177119, 1651405, 15707416, 151791375, 1485814989, 14697965770, 146673432721, 1474490991635, 14915914368896, 151701887367585, 1550083118902041, 15903333300738320, 163749905809635219, 1691449817705302875, 17521670544878571584, 181972459046153912945
Offset: 0

Views

Author

Paul D. Hanna, Sep 30 2014

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 34*x^2 + 265*x^3 + 2219*x^4 + 19490*x^5 +...

Crossrefs

Cf. A246510, A248053, A246423, A246455, A246056, A246813.

Programs

  • Mathematica
    Table[Sum[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, Oct 04 2014 *)
  • PARI
    /* By definition: */
{a(n,p=4,q=1)=local(A=1); A=sum(m=0, n, x^m/(1-p*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * q^k * x^k) * sum(k=0, m, binomial(m,k)^2 * p^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n,4,1), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p=4,q=1)=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,4,1), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p=4,q=1)=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,4,1), ", "))
  • PARI
    /* By a binomial identity: */
{a(n,p=4,q=1)=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,4,1), ", "))
  • PARI
    /* Formula for a(n): */
{a(n,p=4,q=1)=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,4,1), ", "))

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 * 4^(n-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 * 4^(k-j) * 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 * 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 * 4^j.
a(n) ~ (11+3*sqrt(13))^(n+1) / (Pi * n * 2^(n+7/2)). - Vaclav Kotesovec, Oct 04 2014
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.