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 1-5 of 5 results.

A217668 G.f.: Sum_{n>=0} x^n*(1 + x^n)^n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 5, 4, 6, 1, 14, 1, 8, 11, 13, 1, 25, 1, 22, 22, 12, 1, 61, 6, 14, 37, 50, 1, 77, 1, 73, 56, 18, 36, 175, 1, 20, 79, 211, 1, 135, 1, 188, 232, 24, 1, 421, 8, 236, 137, 313, 1, 307, 331, 422, 172, 30, 1, 1423, 1, 32, 295, 601, 716, 727, 1
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 +...
where we have the following series identity:
A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5  + x^6*(1+x^6)^6 + x^7*(1+x^7)^7 + x^8*(1+x^8)^8 + x^9*(1+x^9)^9 +...
A(x) = 1/(1-x) + x^2/(1-x^2)^2 + x^6/(1-x^3)^3 + x^12/(1-x^4)^4 + x^20/(1-x^5)^5 + x^30/(1-x^6)^6 + x^42/(1-x^7)^7 + x^56/(1-x^8)^8 +...

Links

Crossrefs

Cf. A217667, A217669, A217670, A143862.

Programs

  • Mathematica
    terms = 100; Sum[x^n*(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
  • PARI
    {a(n,t=1)=polcoeff(sum(m=0,n,x^m*(t+x^m +x*O(x^n))^m),n)}
for(n=0,100,print1(a(n),", "))
  • PARI
    {a(n,t=1)=local(A=1+x); A=sum(k=0, sqrtint(n+1), x^(k*(k+1))/(1 - t*x^(k+1) +x*O(x^n))^(k+1) ); polcoeff(A, n)}
for(n=0,100,print1(a(n),", ")) \\ Paul D. Hanna, Sep 13 2014
  • PARI
    {a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d,d-1)) )}
for(n=0,100,print1(a(n),", ")) \\ Paul D. Hanna, Apr 25 2018

Formula

G.f.: Sum_{n>=0} x^(n*(n+1)) / (1 - x^(n+1))^(n+1). - Paul D. Hanna, Sep 13 2014
a(n) = Sum_{d|n} binomial(n/d, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018

A325997 G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.

Original entry on oeis.org

1, 4, 3, 10, 8, 18, 7, 40, 9, 44, 41, 54, 13, 150, 15, 88, 127, 168, 19, 298, 21, 324, 275, 180, 25, 854, 132, 238, 524, 774, 31, 1286, 33, 1180, 893, 378, 674, 2998, 39, 460, 1406, 3744, 43, 2790, 45, 3458, 4397, 648, 49, 8420, 303, 4714, 2960, 6270, 55, 7060, 6492, 9120, 4049, 990, 61, 30748, 63, 1120, 7697, 13788, 15082, 17626, 69, 16834, 6971, 28788, 73, 48088, 75, 1558, 39792, 25578, 12091, 41578, 81, 77874, 11540, 1890, 85, 121650, 58227
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 2 and p = x, q = x, r = 1.

Examples

			G.f.: A(x) = 1 + 4*x + 3*x^2 + 10*x^3 + 8*x^4 + 18*x^5 + 7*x^6 + 40*x^7 + 9*x^8 + 44*x^9 + 41*x^10 + 54*x^11 + 13*x^12 + 150*x^13 + 15*x^14 + 88*x^15 +...
where
A(x) = 1 + 2*(x + x) + 3*(x + x^2)^2 + 4*(x + x^3)^3 + 5*(x + x^4)^4 + 6*(x + x^5)^5 + 7*(x + x^6)^6 + 8*(x + x^7)^7 + 9*(x + x^8)^8 + 10*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^2 + 2*x/(1 - x^2)^3 + 3*x^4/(1 - x^3)^4 + 4*x^9/(1 - x^4)^5 + 5*x^16/(1 - x^5)^6 + 6*x^25/(1 - x^6)^7 + 7*x^36/(1 - x^7)^8 + 8*x^49/(1 - x^8)^9 + 9*x^64/(1 - x^9)^10 + 10*x^81/(1 - x^10)^11 + ...

Links

Crossrefs

Cf. A217669, A325998, A325999.

Programs

  • Maple
    G:= add((n+1)*(x+x^n)^n,n=0..100):
S:= series(G,x,101):
seq(coeff(S,x,n),n=0..100); # Robert Israel, Jun 02 2019
  • PARI
    {a(n)=polcoeff(sum(m=0, n, (m+1) * (x + x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(m=0, n, (m+1) * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+2)), n)}
for(n=0, 100, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.
G.f.: Sum_{n>=0} (n+1) * x^(n^2) / (1 - x^(n+1))^(n+2).

A217667 G.f.: Sum_{n>=0} (x + x^(2*n))^n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 5, 1, 1, 5, 1, 4, 6, 1, 1, 7, 8, 1, 8, 1, 1, 19, 1, 5, 10, 1, 16, 11, 1, 1, 23, 22, 1, 13, 1, 1, 42, 21, 1, 20, 1, 37, 16, 1, 36, 17, 46, 1, 34, 1, 1, 130, 1, 1, 20, 1, 67, 56, 85, 7, 22, 79, 1, 23, 1, 121, 185, 1, 1, 25, 23, 106, 191, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2012

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),
for any fixed integer k; here, k = 1 and y = 1, F = x, G = x^2.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 + x^7 + 5*x^8 + x^9 +...
where
A(x) = 1 + (x + x^2) + (x + x^4)^2 + (x + x^6)^3 + (x + x^8)^4 + (x + x^10)^5 +...
Also
A(x) = 1/(1-x) + x/(1 - x^3)^2 + x^4/(1 - x^5)^3 + x^9/(1 - x^7)^4 + x^16/(1 - x^9)^5 + x^25/(1 - x^11)^6 + x^36/(1 - x^13)^7 + x^49/(1 - x^15)^8 + ...

Links

Crossrefs

Cf. A217668, A217669.

Programs

  • Mathematica
    terms = 100; Sum[(x + x^(2*n))^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(x+x^(2*m) +x*O(x^n))^m),n)}
for(n=0,100,print1(a(n),", "))

Formula

Generating functions.
(1) Sum_{n>=0} (x + x^(2*n))^n.
(2) Sum_{n>=0} x^(n^2) / (1 - x^(2*n+1))^(n+1). - Paul D. Hanna, Jun 02 2019

A325998 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.

Original entry on oeis.org

1, 6, 6, 22, 21, 51, 28, 126, 45, 170, 156, 246, 91, 627, 120, 496, 588, 876, 190, 1626, 231, 1776, 1536, 1392, 325, 4977, 798, 2086, 3405, 5025, 496, 8694, 561, 8122, 6636, 4086, 3881, 21597, 780, 5440, 11781, 26016, 946, 24114, 1035, 28001, 33348, 8976, 1225, 70302, 2586, 36946, 30501, 56127, 1540, 66318, 46698, 82056, 45660, 16710, 1891, 268242, 2016, 20032, 79806, 140106, 122398, 171738, 2415, 180835, 92256, 249612, 2701, 482532, 2850, 32566
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 3 and q = x, p = x, r = 1.

Examples

			G.f.: A(x) = 1 + 6*x + 6*x^2 + 22*x^3 + 21*x^4 + 51*x^5 + 28*x^6 + 126*x^7 + 45*x^8 + 170*x^9 + 156*x^10 + 246*x^11 + 91*x^12 + 627*x^13 + 120*x^14 +...
where
A(x) = 1 + 3*(x + x) + 6*(x + x^2)^2 + 10*(x + x^3)^3 + 15*(x + x^4)^4 + 21*(x + x^5)^5 + 28*(x + x^6)^6 + 36*(x + x^7)^7 + 45*(x + x^8)^8 + 55*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^3 + 3*x/(1 - x^2)^4 + 6*x^4/(1 - x^3)^5 + 10*x^9/(1 - x^4)^6 + 15*x^16/(1 - x^5)^7 + 21*x^25/(1 - x^6)^8 + 28*x^36/(1 - x^7)^9 + 36*x^49/(1 - x^8)^10 + 45*x^64/(1 - x^9)^11 + 55*x^81/(1 - x^10)^12 + ...

Crossrefs

Cf. A217669, A325997, A325999.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2 * (x + x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2 * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3)), n)}
for(n=0, 100, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.
G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^(n^2) / (1 - x^(n+1))^(n+3).

A325999 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * (x + x^n)^n.

Original entry on oeis.org

1, 8, 10, 40, 45, 116, 84, 320, 165, 520, 496, 868, 455, 2100, 680, 2136, 2264, 3680, 1330, 6920, 1771, 7988, 6920, 8060, 2925, 22732, 4914, 13580, 17365, 26440, 5456, 46212, 6545, 45000, 37800, 32376, 20773, 119660, 10660, 46900, 74221, 143528, 14190, 161540, 16215, 177196, 194764, 89800, 20825, 447040, 28046, 239928, 229725, 384860, 29260, 492128, 257734, 569140, 372480, 201500, 39711, 1763416, 43680, 255200, 639430, 1068856, 733074, 1337080
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 4 and q = x, p = x, r = 1.

Examples

			G.f.: A(x) = 1 + 8*x + 10*x^2 + 40*x^3 + 45*x^4 + 116*x^5 + 84*x^6 + 320*x^7 + 165*x^8 + 520*x^9 + 496*x^10 + 868*x^11 + 455*x^12 + 2100*x^13 + 680*x^14 +...
where
A(x) = 1 + 4*(x + x) + 10*(x + x^2)^2 + 20*(x + x^3)^3 + 35*(x + x^4)^4 + 56*(x + x^5)^5 + 84*(x + x^6)^6 + 120*(x + x^7)^7 + 165*(x + x^8)^8 + 220*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^4 + 4*x/(1 - x^2)^5 + 10*x^4/(1 - x^3)^6 + 20*x^9/(1 - x^4)^7 + 35*x^16/(1 - x^5)^8 + 56*x^25/(1 - x^6)^9 + 84*x^36/(1 - x^7)^10 + 120*x^49/(1 - x^8)^11 + 165*x^64/(1 - x^9)^12 + 220*x^81/(1 - x^10)^13 + ...

Crossrefs

Cf. A217669, A325997, A325998.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)*(m+3)/3! * (x + x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)*(m+3)/3! * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+4)), n)}
for(n=0, 100, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * (x + x^n)^n.
G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^(n^2) / (1 - x^(n+1))^(n+4).
FORMULAS FOR TERMS.
a(5*n + 2) = 0 (mod 5),
a(5*n + 3) = 0 (mod 5),
a(5*n + 4) = 0 (mod 5), for n >= 0.
Showing 1-5 of 5 results.

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