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 1-3 of 3 results.

A326002 G.f.: Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n.

Original entry on oeis.org

1, 2, 5, 4, 11, 6, 22, 8, 29, 22, 41, 12, 89, 14, 71, 76, 109, 18, 214, 20, 196, 190, 155, 24, 573, 56, 209, 388, 519, 30, 877, 32, 809, 694, 341, 316, 2119, 38, 419, 1132, 2411, 42, 2045, 44, 2531, 2986, 599, 48, 6053, 106, 3011, 2500, 4759, 54, 4978, 4016, 6589, 3478, 929, 60, 21468, 62, 1055, 5524, 10713, 10076, 12046, 68, 13499, 6142, 18656, 72, 34474, 74, 1481, 29716, 20939, 5622, 28432, 80, 57921, 10000, 1805, 84, 84155, 42926, 1979, 12268, 41449, 90, 122339, 24116, 44759, 14974, 2351, 77616, 153969
Offset: 0

Views

Author

Paul D. Hanna, Jun 01 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 some fixed integer k; here, k = 2 and p = 1, q = x, r = x.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 4*x^3 + 11*x^4 + 6*x^5 + 22*x^6 + 8*x^7 + 29*x^8 + 22*x^9 + 41*x^10 + 12*x^11 + 89*x^12 + 14*x^13 + 71*x^14 + 76*x^15 + 109*x^16 + 18*x^17 + 214*x^18 + 20*x^19 + 196*x^20 + ...
where we have the following series identity:
A(x) = 1 + 2*x*(1+x) + 3*x^2*(1+x^2)^2 + 4*x^3*(1+x^3)^3 + 5*x^4*(1+x^4)^4 + 6*x^5*(1+x^5)^5  + 7*x^6*(1+x^6)^6 + 8*x^7*(1+x^7)^7 + 9*x^8*(1+x^8)^8 + 10*x^9*(1+x^9)^9 + ...
is equal to
A(x) = 1/(1-x)^2 + 2*x^2/(1-x^2)^3 + 3*x^6/(1-x^3)^4 + 4*x^12/(1-x^4)^5 + 5*x^20/(1-x^5)^6 + 6*x^30/(1-x^6)^7 + 7*x^42/(1-x^7)^8 + 8*x^56/(1-x^8)^9 + ...

Links

Crossrefs

Cf. A217668 (k=1), A326003 (k=3), A326004 (k=4), A326005 (k=5).

Programs

  • Maple
    N:= 100: # for a(0)..a(N)
S:= series(add((n+1)*x^n*(1+x^n)^n,n=0..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 03 2019
  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1) * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))
  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1) * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+2))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Formula

Generating functions.
(1) Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1) * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).

A326003 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.

Original entry on oeis.org

1, 3, 9, 10, 27, 21, 64, 36, 105, 85, 171, 78, 359, 105, 372, 346, 573, 171, 1105, 210, 1116, 1009, 1134, 300, 3237, 456, 1743, 2386, 3375, 465, 5947, 528, 5529, 4885, 3537, 1926, 14917, 741, 4770, 9010, 16551, 903, 17963, 990, 19977, 22291, 8028, 1176, 49925, 1527, 23961, 24634, 41289, 1485, 48502, 27336, 58809, 37621, 15255, 1830, 184218, 1953, 18384, 59830, 106137, 77286, 121705, 2346, 140115, 78385, 159846, 2628, 346846, 2775, 30267, 293866
Offset: 0

Views

Author

Paul D. Hanna, Jun 01 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 p = 1, q = x, r = x.

Examples

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 10*x^3 + 27*x^4 + 21*x^5 + 64*x^6 + 36*x^7 + 105*x^8 + 85*x^9 + 171*x^10 + 78*x^11 + 359*x^12 + 105*x^13 + 372*x^14 + 346*x^15 + 573*x^16 + 171*x^17 + 1105*x^18 + 210*x^19 + 1116*x^20 + ...
where we have the following series identity:
A(x) = 1 + 3*x*(1+x) + 6*x^2*(1+x^2)^2 + 10*x^3*(1+x^3)^3 + 15*x^4*(1+x^4)^4 + 21*x^5*(1+x^5)^5  + 28*x^6*(1+x^6)^6 + 36*x^7*(1+x^7)^7 + 45*x^8*(1+x^8)^8 + 55*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^3 + 3*x^2/(1-x^2)^4 + 6*x^6/(1-x^3)^5 + 10*x^12/(1-x^4)^6 + 15*x^20/(1-x^5)^7 + 21*x^30/(1-x^6)^8 + 28*x^42/(1-x^7)^9 + 36*x^56/(1-x^8)^10 +...

Links

Crossrefs

Cf. A217668 (k=1), A326002 (k=2), A326004 (k=4), A326005 (k=5).

Programs

  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))
  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Formula

Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+3).

A326005 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n.

Original entry on oeis.org

1, 5, 20, 35, 100, 126, 330, 330, 775, 820, 1631, 1365, 3535, 2380, 5370, 5136, 9085, 5985, 16900, 8855, 21966, 19580, 29965, 17550, 60375, 24381, 58345, 57205, 90350, 40920, 152837, 52360, 164145, 141120, 175560, 93801, 404500, 101270, 280175, 309050, 503041, 148995, 714435, 178365, 748705, 708946, 633950, 249900, 1771645, 295135, 1120236, 1155015, 1760500, 395010, 2483110, 905576, 2622545, 2036060, 1744525, 595665, 6962328, 677040, 2343880
Offset: 0

Views

Author

Paul D. Hanna, Jun 01 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 = 5 and p = 1, q = x, r = x.

Examples

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 35*x^3 + 100*x^4 + 126*x^5 + 330*x^6 + 330*x^7 + 775*x^8 + 820*x^9 + 1631*x^10 + 1365*x^11 + 3535*x^12 + 2380*x^13 + 5370*x^14 + 5136*x^15 + 9085*x^16 + 5985*x^17 + 16900*x^18 + 8855*x^19 + 21966*x^20 + ...
where we have the following series identity:
A(x) = 1 + 5*x*(1+x) + 15*x^2*(1+x^2)^2 + 35*x^3*(1+x^3)^3 + 70*x^4*(1+x^4)^4 + 126*x^5*(1+x^5)^5  + 210*x^6*(1+x^6)^6 + 330*x^7*(1+x^7)^7 + 495*x^8*(1+x^8)^8 + 715*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^5 + 5*x^2/(1-x^2)^6 + 15*x^6/(1-x^3)^7 + 35*x^12/(1-x^4)^8 + 70*x^20/(1-x^5)^9 + 126*x^30/(1-x^6)^10 + 210*x^42/(1-x^7)^11 + 330*x^56/(1-x^8)^12 +...

Crossrefs

Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326004 (k=4).

Programs

  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)*(m+4)/4! * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))
  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)*(m+4)/4! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+5))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Formula

Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^(n*(n+1)) / (1 - x^(n+1))^(n+5).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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