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 1-7 of 7 results.

A246840 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(2*k).

Original entry on oeis.org

1, 1, 1, 2, 5, 10, 18, 35, 73, 151, 306, 623, 1286, 2668, 5531, 11477, 23889, 49852, 104175, 217936, 456534, 957609, 2010839, 4226417, 8891022, 18719637, 39443860, 83170162, 175484915, 370491775, 782648333, 1654197568, 3498049053, 7400639286, 15664103420, 33168342557, 70260909811
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2014

Keywords

Comments

Compare to the g.f. of Narayana's cows sequence A000930:
Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * x^(2*k) = 1/(1-x-x^3).
Compare to the g.f. of Whitney numbers sequence A051286:
Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k = 1/sqrt((1+x+x^2)*(1-3*x+x^2)).
...
Lim_{n->infinity} a(n)/a(n+1) = t^2 = 0.465571231876768... (A088559) where t = ((sqrt(93)+9)/18)^(1/3) - ((sqrt(93)-9)/18)^(1/3) is the positive real root of 1 - x - x^3 = 0.
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - (x*y)^3). - Ilya Gutkovskiy, Apr 23 2025

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 35*x^7 + ...
where, by definition,
A(x) = 1 + x*(1 + x^2) + x^2*(1 + 2^2*x^2 + x^4)
+ x^3*(1 + 3^2*x^2 + 3^2*x^4 + x^6)
+ x^4*(1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8)
+ x^5*(1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10) + ...
which is also given by the series identity:
A(x) = 1/(1-x+x^3) + 2*x^3/(1-x+x^3)^3 + 6*x^6/(1-x+x^3)^5 + 20*x^9/(1-x+x^3)^7 + 70*x^12/(1-x+x^3)^9 + 252*x^15/(1-x+x^3)^11 + 924*x^18/(1-x+x^3)^13 + ...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^2) + x^2*(1 + 6*x^2 + x^4)/2
+ x^3*(1 + 15*x^2 + 15*x^4 + x^6)/3
+ x^4*(1 + 28*x^2 + 70*x^4 + 28*x^6 + x^8)/4
+ x^5*(1 + 45*x^2 + 210*x^4 + 210*x^6 + 45*x^8 + x^10)/5 + ...
more explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 26*x^5/5 + 46*x^6/6 + 99*x^7/7 + 229*x^8/8 + 499*x^9/9 + 1046*x^10/10 + 2223*x^11/11 + 4810*x^12/12 + ...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+3*x^2+4*x^3-3*x^5)/((1-x+2*x^2-x^3)*(1-x-2*x^2-x^3)).

Links

Crossrefs

Cf. A088559, A246861, A181665, A246883, A246884, A248193.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[(1 - x - x^3)^2 - 4*x^4], {x,0,50}], x] (* G. C. Greubel, Apr 27 2017 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1);A=sum(m=0,n,x^m*sum(k=0,m,binomial(m,k)^2*x^(2*k)) +x*O(x^n));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    /* From closed formula: */
{a(n)=local(A=1);A= 1/sqrt((1 - x - x^3)^2 - 4*x^4 +x*O(x^n));polcoeff(A, n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    /* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(3*m) / (1 - x + x^3 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^2)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\3, x^(3*m)*sum(k=0, n-3*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\3, x^(3*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From exponential formula: */
{a(n)=local(A=1);A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(2*k)) * x^m/m) +x*O(x^n));polcoeff(A, n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    /* From exponential formula: */
{a(n)=local(A=1);A=exp(sum(m=1, n, ((1+x)^(2*m) + (1-x)^(2*m))/2 * x^m/m) +x*O(x^n));polcoeff(A, n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0,n\2,binomial(n-2*k,k)^2)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(3*n) / (1 - x + x^3)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(2*k)] * (1-x^2)^(2*n+1).
G.f.: Sum_{n>=0} x^(3*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(3*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(2*k) ).
G.f.: exp( Sum_{n>=1} (x^n/n) * ((1+x)^(2*n) + (1-x)^(2*n))/2 ).
G.f.: 1 / sqrt((1 - x + 2*x^2 - x^3)*(1 - x - 2*x^2 - x^3)).
G.f.: 1 / sqrt((1 - x - x^3)^2 - 4*x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-2*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + (2*n-3)*a(n-3) + 2*(n-2)*a(n-4) - (n-3)*a(n-6). - Seiichi Manyama, Aug 10 2024

A246883 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(3*k).

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 10, 17, 27, 46, 86, 165, 308, 558, 1006, 1841, 3421, 6383, 11863, 21966, 40697, 75662, 141099, 263429, 491778, 918104, 1715259, 3208078, 6005818, 11250198, 21082487, 39524241, 74135187, 139128897, 261228200, 490682127, 922015964, 1733127107, 3258939997, 6130162494, 11534742080
Offset: 0

Views

Author

Paul D. Hanna, Sep 06 2014

Keywords

Comments

Limit a(n)/a(n+1) = t^2 = 0.524888598656404... (A072223) where t is the positive real root of 1 - x - x^4 = 0.
Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^4). - Seiichi Manyama, Apr 29 2025

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 17*x^7 +...
where, by definition,
A(x) = 1 + x*(1 + x^3) + x^2*(1 + 2^2*x^3 + x^6)
+ x^3*(1 + 3^2*x^3 + 3^2*x^6 + x^9)
+ x^4*(1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)
+ x^5*(1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^4) + 2*x^4/(1-x+x^4)^3 + 6*x^8/(1-x+x^4)^5 + 20*x^12/(1-x+x^4)^7 + 70*x^16/(1-x+x^4)^9 + 252*x^20/(1-x+x^4)^11 + 924*x^24/(1-x+x^4)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^3) + x^2*(1 + 6*x^3 + x^6)/2
+ x^3*(1 + 15*x^3 + 15*x^6 + x^9)/3
+ x^4*(1 + 28*x^3 + 70*x^6 + 28*x^9 + x^12)/4
+ x^5*(1 + 45*x^3 + 210*x^6 + 210*x^9 + 45*x^12 + x^15)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + 5*x^4/4 + 16*x^5/5 + 31*x^6/6 + 50*x^7/7 + 77*x^8/8 + 145*x^9/9 + 306*x^10/10 + 628*x^11/11 + 1199*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+4*x^3+5*x^4-4*x^7)/((1-x+2*x^2+x^4)*(1-x-2*x^2+x^4)).

Links

Crossrefs

Cf. A072223, A181665, A246840, A246884, A248193.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[(1 - x + x^4)^2 - 4 x^4], {x, 0, 40}], x] (* Michael De Vlieger, Sep 10 2021 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x + x^4)^2 - 4*x^4 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(4*m) / (1 - x + x^4 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^3)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m)*sum(k=0, n-4*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(3*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0, n\3, binomial(n-3*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(4*n) / (1 - x + x^4)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(3*k)] * (1-x^3)^(2*n+1).
G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(3*k) ).
G.f.: 1 / sqrt((1 - x + 2*x^2 + x^4)*(1 - x - 2*x^2 + x^4)).
G.f.: 1 / sqrt((1 - x + x^4)^2 - 4*x^4).
G.f.: 1 / sqrt((1 - x - x^4)^2 - 4*x^5).
a(n) = Sum_{k=0..[n/3]} C(n-3*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 2*(n-2)*a(n-4) + (2*n-5)*a(n-5) - (n-4)*a(n-8). - Seiichi Manyama, Aug 10 2024

A248193 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(5*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 10, 17, 26, 37, 51, 74, 118, 201, 347, 586, 955, 1509, 2351, 3682, 5871, 9545, 15700, 25851, 42292, 68606, 110635, 178190, 287852, 467313, 761957, 1245011, 2033856, 3317230, 5401332, 8787539, 14301168, 23301005, 38016585, 62090615, 101457357, 165778774
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2014

Keywords

Comments

Limit a(n)/a(n+1) = t^2 = 0.6054234235718265... where t is the positive real root of 1 - x - x^6 = 0.
Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^6). - Seiichi Manyama, Apr 29 2025

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 5*x^7 + 10*x^8 +...
where, by definition,
A(x) = 1 + x*(1 + x^5) + x^2*(1 + 2^2*x^5 + x^10)
+ x^3*(1 + 3^2*x^5 + 3^2*x^10 + x^15)
+ x^4*(1 + 4^2*x^5 + 6^2*x^10 + 4^2*x^15 + x^20)
+ x^5*(1 + 5^2*x^5 + 10^2*x^10 + 10^2*x^15 + 5^2*x^20 + x^25) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^6) + 2*x^6/(1-x+x^6)^3 + 6*x^12/(1-x+x^6)^5 + 20*x^18/(1-x+x^6)^7 + 70*x^24/(1-x+x^6)^9 + 252*x^30/(1-x+x^6)^11 + 924*x^36/(1-x+x^6)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^5) + x^2*(1 + 6*x^5 + x^10)/2
+ x^3*(1 + 15*x^5 + 15*x^10 + x^15)/3
+ x^4*(1 + 28*x^5 + 70*x^10 + 28*x^15 + x^20)/4
+ x^5*(1 + 45*x^5 + 210*x^10 + 210*x^15 + 45*x^20 + x^25)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + 7*x^6/6 + 22*x^7/7 + 41*x^8/8 + 64*x^9/9 + 91*x^10/10 + 122*x^11/11 + 163*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1 - x + 6*x^5 + 7*x^6 - 6*x^11) / ((1 - x + 2*x^3 + x^6)*(1 - x - 2*x^3 + x^6)).

Links

Crossrefs

Cf. A181665, A246840, A246883, A246884.

Programs

  • Mathematica
    CoefficientList[Series[1 / Sqrt[(1-x+x^6)^2 - 4*x^6], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2014 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(5*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x + x^6)^2 - 4*x^6 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(6*m) / (1 - x + x^6 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^5)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(5*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\6, x^(6*m)*sum(k=0, n-5*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\6, x^(6*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(5*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0, n\5, binomial(n-5*k, k)^2)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(6*n) / (1 - x + x^6)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(5*k)] * (1-x^5)^(2*n+1).
G.f.: Sum_{n>=0} x^(6*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(6*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(5*k) ).
G.f.: 1 / sqrt((1 - x + 2*x^3 + x^6)*(1 - x - 2*x^3 + x^6)).
G.f.: 1 / sqrt((1 - x + x^6)^2 - 4*x^6).
G.f.: 1 / sqrt((1 - x - x^6)^2 - 4*x^7).
a(n) = Sum_{k=0..[n/5]} C(n-5*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 2*(n-3)*a(n-6) + (2*n-7)*a(n-7) - (n-6)*a(n-12). - Seiichi Manyama, Aug 10 2024

A376722 Expansion of 1/sqrt((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 4, 1, 0, 1, 9, 9, 1, 1, 16, 36, 16, 2, 25, 100, 100, 26, 37, 225, 400, 226, 85, 442, 1225, 1226, 505, 833, 3137, 4901, 3217, 2080, 7120, 15878, 15976, 9081, 15696, 44182, 63626, 47125, 41625, 110926, 213688, 217801, 157300, 272251, 630458
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Crossrefs

Cf. A051286, A298567, A376721.
Cf. A246884.

Programs

  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/sqrt((1-x^4-x^5)^2-4*x^9))
  • PARI
    a(n) = sum(k=0, n\4, binomial(k, n-4*k)^2);

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k)^2.

A375285 Expansion of 1/((1 - x - x^5)^2 - 4*x^6).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 17, 36, 69, 120, 196, 320, 547, 980, 1786, 3216, 5661, 9804, 16932, 29472, 51820, 91602, 161767, 284424, 498103, 871150, 1525380, 2676544, 4703158, 8265354, 14514236, 25464576, 44656997, 78324398, 137430720, 241225072, 423451668, 743244866
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A182890, A375278, A375283.
Cf. A246884.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/((1-x-x^5)^2-4*x^6))
  • PARI
    a(n) = sum(k=0, n\5, binomial(2*n-8*k+2, 2*k+1))/2;

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-5) + 2*a(n-6) - a(n-10).
a(n) = (1/2) * Sum_{k=0..floor(n/5)} binomial(2*n-8*k+2,2*k+1).

A376783 Expansion of 1/sqrt((1 - x^2 - x^5)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 4, 1, 9, 2, 16, 10, 25, 37, 37, 101, 65, 226, 164, 443, 481, 810, 1325, 1522, 3258, 3251, 7236, 7926, 15010, 20234, 30557, 50234, 64501, 117966, 145557, 263107, 346293, 569726, 835909, 1233943, 1984730, 2740492, 4579704, 6288323, 10311571
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Crossrefs

Cf. A246884, A376722, A376784.
Cf. A001687.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/sqrt((1-x^2-x^5)^2-4*x^7))
  • PARI
    a(n) = sum(k=0, n\5, ((n-3*k)%2==0)*binomial((n-3*k)/2, k)^2);

Formula

G.f.: 1/sqrt((1 - x^2 + x^5)^2 - 4*x^5) = 1/sqrt((1 + x^2 - x^5)^2 - 4*x^2).

A376784 Expansion of 1/sqrt((1 - x^3 - x^5)^2 - 4*x^8).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 4, 1, 1, 9, 1, 9, 16, 2, 36, 25, 17, 100, 37, 101, 225, 74, 401, 442, 289, 1226, 820, 1306, 3138, 1737, 5000, 7106, 5161, 15998, 15185, 18901, 44308, 34282, 67861, 110365, 92501, 219609, 259621, 295242, 637570, 620350, 986373, 1694619
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Crossrefs

Cf. A246884, A376722, A376783.
Cf. A052920.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[(1-x^3-x^5)^2-4x^8],{x,0,50}],x] (* Harvey P. Dale, Apr 28 2025 *)
  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/sqrt((1-x^3-x^5)^2-4*x^8))
  • PARI
    a(n) = sum(k=0, n\5, ((n-2*k)%3==0)*binomial((n-2*k)/3, k)^2);

Formula

G.f.: 1/sqrt((1 - x^3 + x^5)^2 - 4*x^5) = 1/sqrt((1 + x^3 - x^5)^2 - 4*x^3).
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