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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A317799 G.f.: Sum_{n>=0} (4*(1+x)^n - 1)^n / 4^(n+1).

Original entry on oeis.org

1, 28, 2644, 418108, 92624756, 26388012380, 9189259388052, 3782063138596476, 1796136011427955636, 966755321167565129372, 581573928178258915024596, 386690499153558305585430460, 281600848152507182372274325492, 222904650325844057584524049181660, 190559248618061561787517993382005012
Offset: 0

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Author

Paul D. Hanna, Aug 14 2018

Keywords

Examples

			G.f.: A(x) = 1 + 28*x + 2644*x^2 + 418108*x^3 + 92624756*x^4 + 26388012380*x^5 + 9189259388052*x^6 + 3782063138596476*x^7 + 1796136011427955636*x^8 + ...
such that
A(x) = 1/4  +  (4*(1+x) - 1)/4^2  +  (4*(1+x)^2 - 1)^3/4^3  +  (4*(1+x)^3 - 1)^4/4^4  +  (4*(1+x)^4 - 1)^4/4^5  +  (4*(1+x)^5 - 1)^5/4^6  + ...
Also,
A(x) = 1/5  +  4*(1+x)/(4 + (1+x))^2  +  4^2*(1+x)^4/(4 + (1+x)^2)^4  +  4^3*(1+x)^9/(4 + (1+x)^3)^4  +  4^4*(1+x)^16/(4 + (1+x)^4)^5  +  4^5*(1+x)^25/(4 + (1+x)^5)^6  +  4^6*(1+x)^36/(4 + (1+x)^6)^7  + ...

Crossrefs

Cf. A122400, A301463, A317798, A301583.

Formula

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

A326000 G.f.: Sum_{n>=0} (n+1) * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 2, 12, 120, 1607, 26862, 536816, 12466468, 329648274, 9774030812, 321057111308, 11570735358300, 453874209520951, 19248243764760562, 877497573254643438, 42791783608096161848, 2222646606788322292656, 122500263059540271947448, 7140154262067048381368062, 438819217371889984410077532, 28360033818941846664929891481, 1922734355204851243123303962324
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 = 2 and p = -1, q = 1+x, r = 1.

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 120*x^3 + 1607*x^4 + 26862*x^5 + 536816*x^6 + 12466468*x^7 + 329648274*x^8 + 9774030812*x^9 + 321057111308*x^10 + ...
such that
A(x) = 1 + 2*((1+x)-1) + 3*((1+x)^2-1)^2 + 4*((1+x)^3-1)^3 + 5*((1+x)^4-1)^4 + 6*((1+x)^5-1)^5  + 7*((1+x)^6-1)^6 + 8*((1+x)^7-1)^7 + 9*((1+x)^8-1)^8 + 10*((1+x)^9-1)^9 +...
is equal to
A(x) = 1/2^2 + 2*(1+x)/(1+(1+x))^3 + 3*(1+x)^4/(1+(1+x)^2)^4 + 4*(1+x)^9/(1+(1+x)^3)^5 + 5*(1+x)^16/(1+(1+x)^4)^6 + 6*(1+x)^25/(1+(1+x)^5)^7 + 7*(1+x)^36/(1+(1+x)^6)^8 + 8*(1+x)^49/(1+(1+x)^7)^9 + ...

Crossrefs

Cf. A122400, A326001.

Programs

  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1) * ((1+x)^m - 1 +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

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

A326001 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 3, 24, 294, 4656, 89745, 2030628, 52649478, 1537164243, 49869371362, 1778978945148, 69186794933664, 2912819826915180, 131960308762527981, 6400097287173710814, 330837181757021507028, 18156728772490730152533, 1054301496336544498961490, 64575610215886231406950048, 4160633168635583272785917712, 281297401484498036336048099574, 19912063300892223213096919150032
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 = 1+x, r = 1.

Examples

			G.f.: A(x) = 1 + 3*x + 24*x^2 + 294*x^3 + 4656*x^4 + 89745*x^5 + 2030628*x^6 + 52649478*x^7 + 1537164243*x^8 + 49869371362*x^9 + 1778978945148*x^10 + ...
such that
A(x) = 1 + 3*((1+x)-1) + 6*((1+x)^2-1)^2 + 10*((1+x)^3-1)^3 + 15*((1+x)^4-1)^4 + 21*((1+x)^5-1)^5  + 28*((1+x)^6-1)^6 + 36*((1+x)^7-1)^7 + 45*((1+x)^8-1)^8 + 55*((1+x)^9-1)^9 +...
is equal to
A(x) = 1/2^3 + 3*(1+x)/(1+(1+x))^4 + 6*(1+x)^4/(1+(1+x)^2)^5 + 10*(1+x)^9/(1+(1+x)^3)^6 + 15*(1+x)^16/(1+(1+x)^4)^7 + 21*(1+x)^25/(1+(1+x)^5)^8 + 28*(1+x)^36/(1+(1+x)^6)^9 + 36*(1+x)^49/(1+(1+x)^7)^10 + ...

Crossrefs

Cf. A122400, A326000.

Programs

  • PARI
    {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * ((1+x)^m - 1 +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

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

A326088 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2) * x^n = Sum_{n>=0} ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 3, 18, 184, 2742, 51650, 1148054, 29089167, 823981958, 25773170170, 882457387327, 32850733172032, 1322072236117388, 57235976215014221, 2653750194041974871, 131246449114771366495, 6898798056206294405352, 384135589510998920667366, 22590113301696105008398833, 1399232160794278896505982537, 91058022280629233139175411272
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2019

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 184*x^3 + 2742*x^4 + 51650*x^5 + 1148054*x^6 + 29089167*x^7 + 823981958*x^8 + 25773170170*x^9 + 882457387327*x^10 + ...
such that the following sum
B(x) = 1 + A(x)*x + A(x)^4*x^2 + A(x)^9*x^3 + A(x)^16*x^4 + A(x)^25*x^5 + A(x)^36*x^6 + A(x)^49*x^7 + A(x)^64*x^8 + ... + A(x)^(n^2)*x^n + ...
equals
B(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)^2 + ((1+x)^3 - 1)^3 + ((1+x)^4 - 1)^4 + ((1+x)^5 - 1)^5 + ((1+x)^6 - 1)^6 + ... + ((1+x)^n - 1)^n + ...
as well as
B(x) = 1/2 + (1+x)/(1 + (1+x))^2 + (1+x)^4/(1 + (1+x)^2)^3 + (1+x)^9/(1 + (1+x)^3)^4 + (1+x)^16/(1 + (1+x)^4)^5 + ... + (1+x)^(n^2)/(1 + (1+x)^n)^(n+1) + ...
where
B(x) = 1 + x + 4*x^2 + 31*x^3 + 338*x^4 + 4769*x^5 + 82467*x^6 + 1687989*x^7 + 39905269*x^8 + 1069863695*x^9 + ... + A122400(n)*x^n + ...

Crossrefs

Cf. A122400.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=0,#A, ((1+x)^m - 1  +x*O(x^#A))^m - x^m*Ser(A)^(m^2) ), #A)); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

The g.f. A(x) allows the following series to be equal:
(1) B(x) = Sum_{n>=0} A(x)^(n^2) * x^n.
(2) B(x) = Sum_{n>=0} ((1+x)^n - 1)^n.
(3) B(x) = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1).
Previous Showing 21-24 of 24 results.

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