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 1-8 of 8 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

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

A375283 Expansion of 1/((1 - x - x^4)^2 - 4*x^5).

Original entry on oeis.org

1, 2, 3, 4, 7, 16, 35, 68, 122, 220, 417, 816, 1588, 3028, 5707, 10784, 20547, 39322, 75150, 143144, 272212, 517990, 987005, 1881824, 3586808, 6832874, 13013780, 24789200, 47229672, 89991518, 171459667, 326651952, 622295173, 1185547900, 2258689217, 4303264572
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A182890, A375278, A375285.
Cf. A246883, A375282.

Programs

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

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) + 2*a(n-5) - a(n-8).
a(n) = (1/2) * Sum_{k=0..floor(n/4)} binomial(2*n-6*k+2,2*k+1).

A375293 Expansion of 1/sqrt((1 - x + x^4)^2 + 4*x^5).

Original entry on oeis.org

1, 1, 1, 1, 0, -3, -8, -15, -23, -26, -12, 37, 144, 326, 564, 753, 633, -281, -2699, -7346, -14333, -21858, -24097, -8635, 45094, 162928, 362513, 620686, 813906, 633510, -495381, -3408175, -8939865, -17141831, -25663802, -27145201, -6079518, 62953931
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2024

Keywords

Crossrefs

Cf. A375021, A375292.
Cf. A246883.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x+x^4)^2+4*x^5))
  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)^2);

Formula

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).
a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k)^2.

A376721 Expansion of 1/sqrt((1 - x^3 - x^4)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 4, 1, 1, 9, 9, 2, 16, 36, 17, 26, 100, 101, 61, 226, 401, 274, 477, 1227, 1289, 1225, 3186, 4982, 4432, 7841, 16040, 17902, 21457, 45517, 66610, 71327, 123444, 219825, 261945, 354095, 660573, 938598, 1138806, 1909676, 3125553
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Crossrefs

Cf. A051286, A298567, A376722.
Cf. A246883.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k)^2.

A387508 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(n-3*k,k)^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 33, 55, 109, 243, 529, 1071, 2093, 4179, 8673, 18255, 37981, 77923, 159649, 329935, 687117, 1432403, 2977505, 6179215, 12841597, 26757059, 55840033, 116551119, 243209325, 507658803, 1060551137, 2217515151, 4639042909, 9707403811
Offset: 0

Views

Author

Seiichi Manyama, Aug 31 2025

Keywords

Links

Crossrefs

Cf. A001850, A108488, A387507.
Cf. A246883, A375293.

Programs

  • Magma
    [(&+[2^k * Binomial(n-3*k, k)^2: k in [0..Floor(n/4)]]): n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^k*Binomial[n-3*k, k]^2,{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(n-3*k, k)^2);

Formula

G.f.: 1/sqrt((1-x-2*x^4)^2 - 8*x^5).

A376735 a(n) = Sum_{k=0..floor(n/4)} (n-3*k+1) * binomial(n-3*k,k)^2.

Original entry on oeis.org

1, 2, 3, 4, 7, 18, 43, 88, 162, 298, 583, 1188, 2402, 4722, 9123, 17648, 34463, 67632, 132382, 257748, 500244, 970790, 1885815, 3663816, 7110990, 13783264, 26692422, 51672484, 100007876, 193487262, 374149235, 723110880, 1396927383, 2697694410, 5208058825
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Cf. A182884, A375565.
Cf. A246883.

Programs

  • PARI
    a(n) = sum(k=0, n\4, (n-3*k+1)*binomial(n-3*k, k)^2);

Formula

G.f.: (1-x-x^4)/((1-x-x^4)^2 - 4*x^5)^(3/2).
D-finite with recurrence 4*n*(2*n-3)*a(n) +(-22*n^2+43*n-13)*a(n-1) +2*(10*n^2-26*n+15)*a(n-2) -3*(n-1)*(2*n-5)*a(n-3) +8*(-2*n^2-n+16)*a(n-4) +2*(-2*n^2-23*n-15)*a(n-5) +12*(n-1)^2*a(n-6) +4*n*(2*n+5)*a(n-8) -3*(2*n+1)*(n-1)*a(n-9)=0. - R. J. Mathar, Oct 17 2024
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