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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A246423 G.f.: Sum_{n>=0} x^n / (1-3*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k].

Original entry on oeis.org

1, 4, 24, 168, 1286, 10440, 88112, 764368, 6766278, 60828024, 553529808, 5086837680, 47127896444, 439608960656, 4124536224864, 38891699480992, 368326082421446, 3501654020899800, 33403335855108368, 319612386771594608, 3066480362268978804, 29493401582426082032, 284301304326376855200
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2014

Keywords

Comments

a(n) == 2 (mod 4) iff n = 2^k for k>=2, and a(n) == 0 (mod 4) elsewhere except at a(0)=1 (conjecture).

Examples

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 168*x^3 + 1286*x^4 + 10440*x^5 +...
where the g.f. is given by the binomial series identity:
A(x) = 1/(1-3*x) + x/(1-3*x)^3 * (1 + 2*x) * (1 + 3*x)
+ x^2/(1-3*x)^5 * (1 + 2^2*2*x + 4*x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-3*x)^7 * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3) * (1 + 3^2*3*x + 3^2*9*x^2 + 27*x^3)
+ x^4/(1-3*x)^9 * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*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*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*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) +...
equals the series
A(x) = 1/(1-x) + x/(1-x)^3 * (1 + x) * (3+2*x)
+ x^2/(1-x)^5 * (1 + 2^2*x + x^2) * (9+2^2*3*2*x+4*x^2)
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (27+3^2*9*2*x+3^2*4*3*x^2+8*x^3)
+ x^4/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*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) * (243+5^2*81*2*x+10^2*27*4*x^2+10^2*9*16*x^3+5^2*3*18*x^4+32*x^5) +...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(3 + (1+2*x)) + x^2*(9 + 2^2*3*(1+2*x) + (1+2^2*2*x+4*x^2))
+ x^3*(27 + 3^2*9*(1+2*x) + 3^2*3*(1+2^2*2*x+4*x^2) + (1+3^2*2*x+3^2*4*x^2+8*x^3))
+ x^4*(81 + 4^2*27*(1+2*x) + 6^2*9*(1+2^2*2*x+4*x^2) + 4^2*3*(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*(243 + 5^2*81*(1+2*x) + 10^2*27*(1+2^2*2*x+4*x^2) + 10^2*9*(1+3^2*2*x+3^2*4*x^2+8*x^3) + 5^2*3*(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 + (3+2*x)) + x^2*(1 + 2^2*(3+2*x) + (9+2^2*3*2*x+4*x^2))
+ x^3*(1 + 3^2*(3+2*x) + 3^2*(9+2^2*3*2*x+4*x^2) + (27+3^2*9*2*x+3^2*4*3*x^2+8*x^3))
+ x^4*(1 + 4^2*(3+2*x) + 6^2*(9+2^2*3*2*x+4*x^2) + 4^2*(27+3^2*9*2*x+3^2*4*3*x^2+8*x^3) + (81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4))
+ x^5*(1 + 5^2*(3+2*x) + 10^2*(9+2^2*3*2*x+4*x^2) + 10^2*(27+3^2*9*2*x+3^2*4*3*x^2+8*x^3) + 5^2*(81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4) + (243+5^2*81*2*x+10^2*27*4*x^2+10^2*9*16*x^3+5^2*3*18*x^4+32*x^5)) +...

Links

Crossrefs

Cf. A246056 (dual), A243948, A245929, A227845, A245925.

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 * 2^k * 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)=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) * 2^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 * 3^(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 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) * 2^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, 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 / (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) * 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 * 3^(k-j) * 2^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 * 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 * 3^j.
Recurrence: (n-5)*(n-4)*(n-2)*n^2*a(n) = 4*(n-5)*(n-4)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 16*(n-5)*(n-4)*(n-1)*(2*n-3)^2*a(n-2) + 8*(n-5)*(n-2)*(4*n^3 - 28*n^2 + 54*n - 27)*a(n-3) + 24*(n-3)*(5*n^4 - 60*n^3 + 248*n^2 - 408*n + 216)*a(n-4) + 16*(n-4)*(n-1)*(4*n^3 - 44*n^2 + 150*n - 153)*a(n-5) - 64*(n-5)*(n-2)*(n-1)*(2*n-9)^2*a(n-6) + 32*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - 16*(n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Aug 26 2014
a(n) ~ c * d^n / n, where d = 10.094399065494857710014687346... is the root of the equation 16 - 128*d + 256*d^2 - 64*d^3 - 120*d^4 - 32*d^5 + 64*d^6 - 16*d^7 + d^8 = 0, and c = 0.5132545324612697424702223429844481717... . - Vaclav Kotesovec, Aug 26 2014

A246510 G.f.: Sum_{n>=0} x^n / (1-4*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, 5, 36, 305, 2821, 27690, 282699, 2967285, 31785786, 345815975, 3808549531, 42360017130, 474990254821, 5362633500755, 60897115958286, 695012481567465, 7966829676299139, 91674042449673960, 1058486539560201051, 12258669983923625475, 142359286920427682046, 1657287004720545992505
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2014

Keywords

Comments

a(n) == 1 (mod 3) iff n = A074939(k) for k>=0, where A074939 gives even numbers such that base 3 representation contains no 2.
a(n) == 2 (mod 3) iff n = A074938(k) for k>=0, where A074938 gives odd numbers such that base 3 representation contains no 2.

Examples

			G.f.: A(x) = 1 + 5*x + 36*x^2 + 305*x^3 + 2821*x^4 + 27690*x^5 +...

Crossrefs

Cf. A246509, A246056, A074938, A074939.

Programs

  • Mathematica
    Table[Sum[3^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)=local(A=1); A=sum(m=0, n, x^m/(1-4*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 * 4^(m-k) * 3^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 * 4^(k-j) * 3^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 * 4^(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
    /* 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 * 4^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 * 4^(n-k) * 3^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) * 3^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 * 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 * 4^j.
a(n) ~ sqrt(36 + 29*sqrt(3) + 3*sqrt(423 + 232*sqrt(3))) * (9/2 + sqrt(3) + 3/2*sqrt(9 + 4*sqrt(3)))^n / (8*Pi*n). - Vaclav Kotesovec, Oct 04 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

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

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

A249891 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 * (-x)^k].

Original entry on oeis.org

1, -1, -11, 59, 241, -3361, -419, 172451, -575399, -7443481, 58900909, 216416771, -4229184959, 2577683471, 244556409709, -1057605544621, -11063790445559, 106391247022391, 291287267857021, -7745138594921101, 10707792104722681, 449533741924068119, -2373288757544551451
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2014

Keywords

Comments

The signs of the terms form a seemingly unpredictable pattern.
The sequence begins [1, 0, -1, 0, -11, 0, 59, 0, ...]. As is standard practice in the database, the zeros have been omitted.
Even order terms of the sequence (i.e., the nonzero ones) gives the diagonal of rational function 1/(1 - (x + y + x*z - y*z - x*y*z)). - Gheorghe Coserea, Aug 29 2018

Examples

			G.f.: A(x) = 1 - x^2 - 11*x^4 + 59*x^6 + 241*x^8 - 3361*x^10 - 419*x^12 + ...
where the g.f. is given by the series:
A(x) = 1/(1+x) + x/(1+x)^3 * (1+x) * (1-x)
+ x^2/(1+x)^5 * (1 + 2^2*x + x^2) * (1 - 2^2*x + x^2)
+ x^3/(1+x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (1 - 3^2*x + 3^2*x^2 - x^3)
+ x^4/(1+x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + 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*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5) + ...
in which the coefficients of the odd powers of x vanish.
We can also express the g.f. by the series:
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)) + ...

Links

Crossrefs

Cf. A245925, A246056.

Programs

  • Maple
    seq(add((-1)^k*binomial(2*k,k)*binomial(n+k,n-k)^2, k = 0..n), n = 0..20); # Peter Bala, Mar 13 2018
  • Mathematica
    a[n_] := HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1/2, 1, 1}, -1/4];Table[a[n], {n, 0, 22}] (* Peter Luschny, Mar 14 2018 *)
  • 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 * (-x)^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(2*n), ", "))
  • PARI
    /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, (-x)^m*sum(k=0, m, (-1)^k * binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * x^j) +x*O(x^n))), n)}
for(n=0, 25, print1(a(2*n), ", "))
  • PARI
    /* Formula for a(2*n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, (-1)^j * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 ))}
for(n=0, 25, print1(a(2*n), ", "))
  • PARI
    seq(N) = {
  my(a=vector(N), t1, t2); a[1]=-1; a[2]=-11; a[3]=59; a[4]=241;
  for (n=5, N,
    t1 = (2*n-5)*(2*n-1)^2*a[n-1] + (2*n-3)*(70*n^2-210*n+87)*a[n-2];
    t2 = (2*n-1)*(2*n-5)^2*a[n-3] + (2*n-1)*(n-3)^2*a[n-4];
    a[n] = -(t1 + t2)/(n^2*(2*n-5)));
  concat(1,a);
};
seq(22)
\\ test: y=Ser(seq(303), 'x); 0 == 2*x*(x^4 + 4*x^3 + 70*x^2 + 4*x + 1)*y''' - (3*x^6 + 12*x^5 + 190*x^4 - 52*x^3 - 837*x^2 - 32*x - 4)*y'' - (9*x^5 + 24*x^4 + 166*x^3 - 106*x^2 - 947*x - 18)*y' - (3*x^4 + 3*x^3 - 67*x^2 - 29*x - 106)*y
\\ Gheorghe Coserea, Aug 29 2018

Formula

G.f.: Sum_{n>=0} (-x)^n * Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j = 1 - x^2 - 11*x^4 + 59*x^6 + 241*x^8 - ....
a(n) = Sum_{k=0..n} (-1)^k * Sum_{j=0..2*n-2*k} C(2*n-k, k+j)^2 * C(k+j, j)^2.
From Peter Bala, Mar 13 2018: (Start)
a(n) = Sum_{k = 0..n} (-1)^k*C(2*k,k)*C(n+k,n-k)^2.
n^2*(2*n-5)*a(n) = -( (2*n-5)*(2*n-1)^2*a(n-1) + (2*n-3)*(70*n^2-210*n+87)*a(n-2) + (2*n-1)*(2*n-5)^2*a(n-3) + (2*n-1)*(n-3)^2*a(n-4) ). (End)
a(n) = hypergeom([-n, -n, n + 1, n + 1], [1/2, 1, 1], -1/4). - Peter Luschny, Mar 14 2018
y = Sum_{n>=0} a(n)*x^n satisfies: 0 = 2*x*(x^4 + 4*x^3 + 70*x^2 + 4*x + 1)*y''' - (3*x^6 + 12*x^5 + 190*x^4 - 52*x^3 - 837*x^2 - 32*x - 4)*y'' - (9*x^5 + 24*x^4 + 166*x^3 - 106*x^2 - 947*x - 18)*y' - (3*x^4 + 3*x^3 - 67*x^2 - 29*x - 106)*y. - Gheorghe Coserea, Aug 29 2018
From Mark van Hoeij, Nov 04 2023: (Start)
a(n) = LegendreP(n, 2+sqrt(5)) * LegendreP(n, 2-sqrt(5)).
G.f.: hypergeom([1/4, 3/4],[1],-64*x^2/(x+1)^4)/(x+1). (End)

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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