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.

A325580 G.f.: A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n, where A(0) = 0, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 24, 15, 4, 1, 57, 98, 67, 26, 5, 1, 231, 430, 336, 144, 40, 6, 1, 1023, 2062, 1767, 861, 265, 57, 7, 1, 4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1, 25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1, 140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1, 822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1, 5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1
Offset: 0

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Author

Paul D. Hanna, May 11 2019

Keywords

Examples

			G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k begins:
A(x,y) = 1 + (y + 1)*x + (y^2 + 2*y + 2)*x^2 + (y^3 + 3*y^2 + 7*y + 5)*x^3 + (y^4 + 4*y^3 + 15*y^2 + 24*y + 16)*x^4 + (y^5 + 5*y^4 + 26*y^3 + 67*y^2 + 98*y + 57)*x^5 + (y^6 + 6*y^5 + 40*y^4 + 144*y^3 + 336*y^2 + 430*y + 231)*x^6 + (y^7 + 7*y^6 + 57*y^5 + 265*y^4 + 861*y^3 + 1767*y^2 + 2062*y + 1023)*x^7 + (y^8 + 8*y^7 + 77*y^6 + 440*y^5 + 1845*y^4 + 5300*y^3 + 9873*y^2 + 10610*y + 4926)*x^8 + (y^9 + 9*y^8 + 100*y^7 + 679*y^6 + 3501*y^5 + 13041*y^4 + 33974*y^3 + 58221*y^2 + 58240*y + 25483)*x^9 + (y^10 + 10*y^9 + 126*y^8 + 992*y^7 + 6083*y^6 + 27978*y^5 + 94580*y^4 + 226716*y^3 + 360930*y^2 + 338984*y + 140601)*x^10 + ...
where, by definition,
A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
1;
1, 1;
2, 2, 1;
5, 7, 3, 1;
16, 24, 15, 4, 1;
57, 98, 67, 26, 5, 1;
231, 430, 336, 144, 40, 6, 1;
1023, 2062, 1767, 861, 265, 57, 7, 1;
4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1;
25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1;
140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1;
822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1;
5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1; ...
the leftmost column in which yields A121689:
[1, 1, 2, 5, 16, 57, 231, 1023, 4926, 25483, 140601, ..., A121689, ...]
and has g.f.: Sum_{n>=0} x^n * (1+x)^(n^2).
Column 1 equals
[1, 2, 7, 24, 98, 430, 2062, 10610, 58240, 338984, ..., A325581(n), ...]
and has g.f.: Sum_{n>=0} (n+1) * x^n * (1+x)^(n*(n+1)).
Column 2 equals
[1, 3, 15, 67, 336, 1767, 9873, 58221, 360930, ..., A325586(n), ...]
and has g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1+x)^(n*(n+2)).
The row sums of this triangle begin
[1, 2, 5, 16, 60, 254, 1188, 6043, 33080, 193249, ..., A301306(n), ...]
and has g.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.

Links

Crossrefs

Cf. A121689 (column 0), A301306 (row sums), A325581 (column 1), A325586 (column 2), A325587 (column 3).

Programs

  • PARI
    {T(n,k) = my(Axy = sum(m=0,n, x^m * ((1+x +x*O(x^n))^m + y)^m ) );
polcoeff( polcoeff( Axy,n,x),k,y)}
for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

Formula

G.f.: A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k equals the following.
(1) A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.
(2) A(x,y) = Sum_{n>=0} x^n * (1+x)^(n^2) / (1 - x*y*(1+x)^n)^(n+1).
(3) A(x,y) = Sum_{k>=0} y^k * Sum_{n>=0} binomial(n+k,n) * (x*(1+x)^n)^(n+k).
G.f. of column k: Sum_{n>=0} binomial(n+k,n) * x^n * (1+x)^(n*(n+k)).

A301465 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2*x)^n )^n * x^n.

Original entry on oeis.org

1, 2, 7, 32, 180, 1142, 8162, 63807, 541202, 4926969, 47799429, 491254999, 5322191407, 60533874755, 720300053902, 8939841040157, 115427887156492, 1546880293065700, 21472465304481214, 308173758866549496, 4565492159926897329, 69713352146246150541, 1095715822838490323551, 17705238547423980566108, 293792275476690070452598
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 32*x^3 + 180*x^4 + 1142*x^5 + 8162*x^6 + 63807*x^7 + 541202*x^8 + 4926969*x^9 + 47799429*x^10 + ...
such that
A(x) = 1 + ((1+x) + (1+2*x))*x + ((1+x)^2 + (1+2*x)^2)^2*x^2 + ((1+x)^3 + (1+2*x)^3)^3*x^3 + ((1+x)^4 + (1+2*x)^4)^4*x^4 + ((1+x)^5 + (1+2*x)^5)^5*x^5 + ...

Crossrefs

Cf. A301306, A301312, A301308.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(k=0, n, ((1+x)^k + (1+2*x)^k +x*O(x^n))^k * x^k ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

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

A326276 G.f.: Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n.

Original entry on oeis.org

1, 2, 6, 21, 85, 382, 1879, 9986, 56818, 343640, 2196596, 14770122, 104063085, 765661874, 5866191429, 46683934520, 385048724001, 3285146877603, 28942067342876, 262882422213165, 2458316711782337, 23637510378534754, 233423898596027454, 2364847720082290621, 24555411743247510317, 261085211212909391915
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 21*x^3 + 85*x^4 + 382*x^5 + 1879*x^6 + 9986*x^7 + 56818*x^8 + 343640*x^9 + 2196596*x^10 + ...
such that
A(x) = 1 + (1 + (1+x)^2)*x + (1 + (1+x)^3)^2*x^2 + (1 + (1+x)^4)^3*x^3 + (1 + (1+x)^5)^4*x^4 + ... + (1 + (1+x)^(n+1))^n*x^n + ...
also
A(x) = 1/(1 - x) + (1+x)^2*x/(1 - x*(1+x))^2 + (1+x)^6*x^2/(1 - x*(1+x)^2)^3 + (1+x)^12*x^3/(1 - x*(1+x)^3)^4 + ... + (1+x)^(n*(n+1))*x^n/(1 - x*(1+x)^n)^(n+1) + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n  =  Sum_{n>=0} (1+x)^(n*(n+1)) * x^n / (1 - x*(1+x)^n)^(n+1).
(1) At x = -1/2, the following sums are equal
S1 = Sum_{n>=0} (-1)^n * (2^(n+1) + 1)^n / 2^(n*(n+2)),
S1 = Sum_{n>=0} (-1)^n * 2 / (2^(n+1) + 1)^(n+1),
where S1 = 0.58938625589631021783349702645576048800172938765646329470992...
(2) At x = -1/3, the following sums are equal
S2 = Sum_{n>=0} (-1)^n * (2^(n+1) + 3^(n+1))^n / 3^(n*(n+2)),
S2 = Sum_{n>=0} (-1)^n * 3 * 2^(n*(n+1)) / (3^(n+1) + 2^n)^(n+1),
where S2 = 0.65707817941052544107009145640756914928885409483935267126701...
(3) At x = -2/3, the following sums are equal
S3 = Sum_{n>=0} (-2)^n * (3^(n+1) + 1)^n / 3^(n*(n+2)),
S3 = Sum_{n>=0} (-2)^n * 3 / (3^(n+1) + 2)^(n+1),
where S3 = 0.55090474258125970373130850821926676214280685554645756713729...

Links

Crossrefs

Cf. A301306.

Programs

  • PARI
    {a(n) = polcoeff( sum(m=0,n, (1 + (1+x)^(m+1) +x*O(x^n) )^m * x^m), n)}
for(n=0,30,print1(a(n),", "))

Formula

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

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