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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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

A246884 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(4*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 10, 17, 26, 38, 59, 101, 182, 326, 564, 945, 1566, 2622, 4476, 7750, 13455, 23231, 39837, 68101, 116611, 200526, 346137, 598438, 1034227, 1785400, 3080418, 5317009, 9187567, 15893830, 27515434, 47647774, 82513447, 142902640, 247553410, 429020710, 743846284
Offset: 0

Views

Author

Paul D. Hanna, Sep 06 2014

Keywords

Comments

Limit a(n)/a(n+1) = t^2 = 0.569840290998053... where t = A075778 is the positive real root of 1 - x - x^5 = 0.
Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^5). - Seiichi Manyama, Apr 29 2025

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 10*x^7 + 17*x^8 +...
where, by definition,
A(x) = 1 + x*(1 + x^4) + x^2*(1 + 2^2*x^4 + x^8)
+ x^3*(1 + 3^2*x^4 + 3^2*x^8 + x^12)
+ x^4*(1 + 4^2*x^4 + 6^2*x^8 + 4^2*x^12 + x^16)
+ x^5*(1 + 5^2*x^4 + 10^2*x^8 + 10^2*x^12 + 5^2*x^16 + x^20) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^5) + 2*x^5/(1-x+x^5)^3 + 6*x^10/(1-x+x^5)^5 + 20*x^15/(1-x+x^5)^7 + 70*x^20/(1-x+x^5)^9 + 252*x^25/(1-x+x^5)^11 + 924*x^30/(1-x+x^5)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^4) + x^2*(1 + 6*x^4 + x^8)/2
+ x^3*(1 + 15*x^4 + 15*x^8 + x^12)/3
+ x^4*(1 + 28*x^4 + 70*x^8 + 28*x^12 + x^16)/4
+ x^5*(1 + 45*x^4 + 210*x^8 + 210*x^12 + 45*x^16 + x^20)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + 19*x^6/6 + 36*x^7/7 + 57*x^8/8 + 82*x^9/9 + 116*x^10/10 + 199*x^11/11 + 391*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+5*x^4+6*x^5-5*x^9)/((1+x+x^2)*(1-2*x+x^2-x^3)*(1-x+2*x^3-x^5)).

Links

Crossrefs

Cf. A075778, A181665, A246840, A246883, A248193.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[(1 - x + x^5)^2 - 4 x^5], {x, 0, 41}], 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^(4*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^5)^2 - 4*x^6 +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^(5*m) / (1 - x + x^5 +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^4)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(4*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\5, x^(5*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\5, x^(5*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^(4*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)^(2*m) + (1-x^2)^(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\4, binomial(n-4*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(5*n) / (1 - x + x^5)^(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^(4*k)] * (1-x^4)^(2*n+1).
G.f.: Sum_{n>=0} x^(5*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(5*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^(4*k) ).
G.f.: exp( Sum_{n>=1} (x^n/n) * ((1+x^2)^(2*n) + (1-x^2)^(2*n))/2 ).
G.f.: 1 / sqrt((1 - x - 2*x^3 - x^5)*(1 - x + 2*x^3 - x^5)).
G.f.: 1 / sqrt((1 - x - x^5)^2 - 4*x^6).
G.f.: 1 / sqrt((1 - x + x^5)^2 - 4*x^5).
a(n) = Sum_{k=0..[n/4]} C(n-4*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + (2*n-5)*a(n-5) + 2*(n-3)*a(n-6) - (n-5)*a(n-10). - 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

A183876 G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^2 *x^k* A(x)^(2k)].

Original entry on oeis.org

1, 1, 2, 7, 24, 86, 328, 1289, 5180, 21232, 88384, 372582, 1587442, 6825092, 29573380, 129014039, 566183860, 2497841196, 11071594936, 49281430216, 220193658876, 987234942328, 4440142628200, 20027079949202, 90569211556534
Offset: 0

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Author

Paul D. Hanna, Feb 12 2011

Keywords

Comments

Compare g.f. to a g.f. B(x) of Catalan numbers (A000108):
B(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k) *x^k* B(x)^(2k)].

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 24*x^4 + 86*x^5 + 328*x^6 + ...
where g.f. A(x) satisfies:
(1) A(x) = 1 + x*(1 + x*A(x)^2) + x^2*(1 + 4*x*A(x)^2 + x^2*A(x)^4) + x^3*(1 + 9*x*A(x)^2 + 9*x^2*A(x)^4 + x^3*A(x)^6) + x^4*(1 + 16*x*A(x)^2 + 36*x^2*A(x)^4 + 16*x^3*A(x)^6 + x^4*A(x)^8) + ...;
(2) A(x) = 1/(1-x) + x^2*A(x)^2*(1+x)/(1-x)^3 + x^4*A(x)^4*(1+4*x+x^2)/(1-x)^5  + x^6*A(x)^6*(1+9*x+9*x^2+x^3)/(1-x)^7 + ...

Links

Crossrefs

Cf. A181665.

Programs

  • Mathematica
    Table[Sum[Binomial[n+k, 2*k]*Binomial[n+1, n-2*k]/(n+1),{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Mar 07 2014 *)
  • PARI
    {a(n)=sum(k=0, n\2, binomial(n+k, 2*k)*binomial(n+1, n-2*k))/(n+1)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(A^2+x*O(x^n))^k))); polcoeff(A, n)}
  • PARI
    {a(n)=polcoeff((1/x)*serreverse(x*(1-x^2)^2/(sqrt((1-x^2)^3+x^2*(1+x^2)^2+x*O(x^n))+x*(1+x^2))),n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1/sqrt((1-x-x^2*A^2)^2-4*x^3*A^2+x*O(x^n)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1-x*A^2)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^k*(A^2+x^2*O(x^n))^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*(A^2+x*O(x^n))^m*sum(k=0, n, binomial(m+k, k)^2*x^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*A^(2*m)/(1-x+x*O(x^n))^(2*m+1)*sum(k=0, m, binomial(m, k)^2*x^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3,(2*m)!/m!^2*x^(3*m)*A^(2*m)/(1-x-x^2*A^2+x*O(x^n))^(2*m+1))); polcoeff(A, n)}

Formula

a(n) = Sum_{k=0..[n/2]} C(n+k, 2k)*C(n+1, n-2k)/(n+1).
G.f. A(x) satisfies:
(1) A(x) = 1/sqrt{ [1-x - x^2*A(x)^2]^2 - 4*x^3*A(x)^2 }.
(2) A(x) = (1/x)*Series_Reversion{ x*(1-x^2)^2/[sqrt((1-x^2)^3 + x^2*(1+x^2)^2) + x*(1+x^2)] }.
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) satisfies: 1-x^2 = (1-x^2)^2*G(x)^2 - 2*x*(1+x^2)*G(x).
(4) A(x) = Sum_{n>=0} x^n*(1 - x*A(x)^2)^(2*n+1) * [Sum_{k>=0} C(n+k,k)^2 *x^k*A(x)^(2k)].
(5) A(x) = Sum_{n>=0} x^(2n)*A(x)^(2n)*[Sum_{k>=0} C(n+k,k)^2*x^k].
(6) A(x) = Sum_{n>=0} x^(2n)*A(x)^(2n)*[Sum_{k=0..n} C(n,k)^2*x^k] /(1-x)^(2n+1).
(7) A(x) = Sum_{n>=0} (2n)!/n!^2 * x^(3n)*A(x)^(2n)/(1-x-x^2*A(x)^2)^(2n+1).
Recurrence: 8*n*(n+1)*(2*n+1)*(5221*n^4 - 34247*n^3 + 79127*n^2 - 74851*n + 23310)*a(n) = 12*n*(26105*n^6 - 171235*n^5 + 398694*n^4 - 384054*n^3 + 116038*n^2 + 18484*n - 9792)*a(n-1) + 6*(31326*n^7 - 252471*n^6 + 759709*n^5 - 1008483*n^4 + 452035*n^3 + 190224*n^2 - 218420*n + 50400)*a(n-2) + (n-2)*(1164283*n^6 - 8801364*n^5 + 23941663*n^4 - 27089928*n^3 + 8937814*n^2 + 3724572*n - 2026080)*a(n-3) - 10*(n-3)*(n-2)*(2*n-7)*(5221*n^4 - 13363*n^3 + 7712*n^2 + 1546*n - 1440)*a(n-4). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ (r^(1/2-n) * sqrt((1 + 7*r^6*s^6 - 2*r^7*s^8 - 3*r^2*(-1+s^2) - 5*r^4*s^2*(-1+s^2) + r*(-3+2*s^2) - r^3*(1 + 4*s^2 + 6*s^4) + r^5*(-9*s^4 + 6*s^6))/(r^2*(1 - r - 5*r^5*s^4 + 3*r^6*s^6 + r^2*(-1+s^2) + r^3*(1+14*s^2) + r^4*(s^2 - 5*s^4))))) / (2*n^(3/2)*sqrt(Pi)), where r = 0.2079338501416944274..., s = 1.815065347470593612... are roots of the system of equations (2*r^2*s*(1 + r - r^2*s^2))/(1 - 2*r - 2*r^3*s^2 + r^4*s^4 + r^2*(1 - 2*s^2))^(3/2) = 1, 1/sqrt(-4*r^3*s^2 + (-1 + r + r^2*s^2)^2) = s. - Vaclav Kotesovec, Mar 07 2014

A186097 G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^3 *x^k* A(x)^k].

Original entry on oeis.org

1, 1, 2, 10, 39, 147, 639, 2857, 12725, 58081, 270250, 1268444, 6009439, 28736727, 138401100, 670641714, 3268021317, 16004012529, 78716657052, 388701645264, 1926266491659, 9576792342099, 47753368809171, 238759903786041
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 39*x^4 + 147*x^5 + 639*x^6 +...
where g.f. A(x) satisfies:
* A(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + 8*x*A(x) + x^2*A(x)^2) + x^3*(1 + 27*x*A(x) + 27*x^2*A(x)^2 + x^3*A(x)^3) + x^4*(1 + 64*x*A(x) + 216*x^2*A(x)^2 + 64*x^3*A(x)^3 + x^4*A(x)^4) +...;

Crossrefs

Cf. A181665, A183876, A181543.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^3*x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*(A+x*O(x^n))^m*sum(k=0, n, binomial(m+k, k)^3*x^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3,(3*m)!/m!^3*x^(3*m)*A^m/(1-x-x^2*A+x*O(x^n))^(3*m+1))); polcoeff(A, n)}

Formula

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

A246861 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * (x*A(x))^(2*k).

Original entry on oeis.org

1, 1, 1, 2, 7, 21, 54, 141, 407, 1231, 3691, 10990, 33144, 101674, 314679, 977289, 3047527, 9557503, 30133759, 95390622, 302960929, 965282651, 3085146472, 9888455045, 31774215928, 102334358736, 330298415136, 1068242904256, 3461372341327, 11235251353747, 36527859658661
Offset: 0

Views

Author

Paul D. Hanna, Sep 05 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 21*x^5 + 54*x^6 + 141*x^7 + ...
where the g.f. A = A(x) equals the binomial series:
A(x) = 1 + x*(1 + x^2*A^2) + x^2*(1 + 2^2*x^2*A^2 + x^4*A^4)
+ x^3*(1 + 3^2*x^2*A^2 + 3^2*x^4*A^4 + x^6*A^6)
+ x^4*(1 + 4^2*x^2*A^2 + 6^2*x^4*A^4 + 4^2*x^6*A^6 + x^8*A^8)
+ x^5*(1 + 5^2*x^2*A^2 + 10^2*x^4*A^4 + 10^2*x^6*A^6 + 5^2*x^8*A^8 + x^10*A^10) + ...
Let A = g.f. A(x), then the g.f. satisfies:
log(A(x)) = x*(1 + x^2*A^2) + x^2*(1 + 6*x^2*A^2 + x^4*A^4)/2
+ x^3*(1 + 15*x^2*A^2 + 15*x^4*A^4 + x^6*A^6)/3
+ x^4*(1 + 28*x^2*A^2 + 70*x^4*A^4 + 28*x^6*A^6 + x^8*A^8)/4
+ x^5*(1 + 45*x^2*A^2 + 210*x^4*A^4 + 210*x^6*A^6 + 45*x^8*A^8 + x^10*A^10)/5 + ...
RELATED SERIES:
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 19*x^4 + 60*x^5 + 168*x^6 + ...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 39*x^4 + 126*x^5 + 376*x^6 + ...
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 71*x^4 + 232*x^5 + 726*x^6 + ...
A(x)^6 = 1 + 6*x + 21*x^2 + 62*x^3 + 192*x^4 + 642*x^5 + 2145*x^6 + ...
where 1 = (1-x)^2*A(x)^2 - 2*x^3*(1+x)*A(x)^4 + x^6*A(x)^6.
Let G(x) = x/Series_Reversion(x*A(x)), then G(x*A(x)) = A(x), where
G(x) = x + x^3 + 1 + 2*x^4 - 2*x^8 + 4*x^12 - 10*x^16 + 28*x^20 - 84*x^24 + ...
G(x) = x + x^3 + sqrt(1 + 4*x^4).

Crossrefs

Cf. A246840, A181665.

Programs

  • PARI
    /* By definition: */
{a(n)=local(A=1); for(i=1,n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*(x*A)^(2*k)) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
  • PARI
    /* (1) From a binomial series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * sum(k=0, n, binomial(m+k, k)^2*(x*A +x*O(x^n))^(2*k)) * (1-x^2*A^2)^(2*m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
  • PARI
    /* (2) From a binomial series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(3*m)*(A +x*O(x^n))^(2*m)*sum(k=0, n, binomial(m+k, k)^2*x^k))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
  • PARI
    /* (3) From a binomial series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(3*m)*A^(2*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, 35, print1(a(n), ", "))
  • PARI
    /* (4) From exponential series formula: */
{a(n)=local(A=1); for(i=1,n, A=exp(sum(m=1, n, ((1+x*A)^(2*m) + (1-x*A)^(2*m))/2 * x^m/m) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
  • PARI
    /* (6) From functional equation: */
{a(n)=local(A=1); for(i=1,n, A =  1 / sqrt((1 - x*(1 - x*A)^2) * (1 - x*(1 + x*A)^2)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* (7) From functional equation: */
{a(n)=local(A=1); for(i=1,n, A = x*A + x^3*A^3 + sqrt(1 + 4*x^4*A^4 +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* (8) From explicit formula: */
{a(n)=local(A=1); A= 1/x * serreverse( x / (x + x^3 + sqrt(1 + 4*x^4 +x*O(x^n) )) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 *(x*A(x))^(2*k)] * (1 - x^2*A(x)^2)^(2*n+1).
(2) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(2*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
(3) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(2*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
(4) A(x) = exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * (x*A(x))^(2*k) ).
(5) A(x) = exp( Sum_{n>=1} (x^n/n) * ((1 + x*A(x))^(2*n) + (1 - x*A(x))^(2*n))/2 ).
(6) A(x) = 1 / sqrt((1 - x*(1 - x*A(x))^2) * (1 - x*(1 + x*A(x))^2)).
(7) A(x) = x*A(x) + x^3*A(x)^3 + sqrt(1 + 4*x^4*A(x)^4).
(8) A(x) = 1/x * Series_Reversion( x / (x + x^3 + sqrt(1 + 4*x^4)) ).
Showing 1-7 of 7 results.

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