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 1-4 of 4 results.

A179431 a(n) = binomial(3^(n-1), n).

Original entry on oeis.org

1, 1, 3, 84, 17550, 25621596, 268715232324, 21091830512086620, 12814543323816738705045, 61742372998425082372103866380, 2399699340005498870742886195375900380, 761689137813999393167583510790986701377432464, 1992997938492157367948224731863936229108552184201415196
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2010

Keywords

Comments

Equals column 0 of triangle T=A179430 where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 84*x^3 + 17550*x^4 + 25621596*x^5 +...
A(x) = 1 + log(1+3*x)/3 + log(1+3^2*x)^2/(3^2*2!) + log(1+3^3*x)^3/(3^3*3!) + log(1+3^4*x)^4/(3^4*4!) +...

Links

Crossrefs

Cf. A179430, A179432, A136393, A179433, A179434.

Programs

  • Mathematica
    Table[Binomial[3^(n-1),n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    a(n)=binomial(3^(n-1), n)
  • PARI
    /* G.f. A(x) as Sum of Series: */
{a(n)=polcoeff(sum(k=0, n, (1/3)^k*log(1+3^k*x +x*O(x^n))^k/k!), n)}

Formula

G.f.: A(x) = Sum_{n>=0} (1/3)^n * log(1 + 3^n*x)^n / n!.
a(n) ~ 3^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

Extensions

Terms a(11) and beyond from Andrew Howroyd, Apr 13 2021

A179430 Triangular matrix T where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Original entry on oeis.org

1, 1, 1, 3, 9, 1, 84, 405, 81, 1, 17550, 121500, 32805, 729, 1, 25621596, 247203171, 82255257, 2539107, 6561, 1, 268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1, 21091830512086620, 373203783345533355
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2010

Keywords

Examples

			Triangle T begins:
1;
1, 1;
3, 9, 1;
84, 405, 81, 1;
17550, 121500, 32805, 729, 1;
25621596, 247203171, 82255257, 2539107, 6561, 1;
268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1;
21091830512086620, 373203783345533355, 165018275857291311, 7607829219099993, 36456526295226, 15884240049, 531441, 1; ...
where column 0 of T equals A179431(n) = C(3^(n-1), n):
[1, 1, 3, 84, 17550, 25621596, 268715232324, ...]. ...
Illustrate row n in column 0 of T^m equals C(m*3^(n-1), n) as follows.
Matrix square T^2 begins:
1;
2, 1;
15, 18, 1;
816, 1539, 162, 1;
316251, 833490, 124659, 1458, 1;
873642672, 3060203490, 585411786, 9861183, 13122, 1; ...
where column 0 of T^2 equals A179432(n) = C(2*3^(n-1), n):
[1, 2, 15, 816, 316251, 873642672, 17743125256857, ...]. ...
Matrix cube T^3 begins:
1;
3, 1;
36, 27, 1;
2925, 3402, 243, 1;
1663740, 2667411, 275562, 2187, 1;
6774333588, 14164214850, 1896890076, 21966228, 19683, 1; ...
where column 0 of T^3 equals A136393(n) = C(3^n, n):
[1, 3, 36, 2925, 1663740, 6774333588, 204208594169580, ...].

Crossrefs

Cf. A179431, A179432, A136393, A179433, A179434, variant: A136467.

Programs

  • PARI
    {T(n, k)=local(M=matrix(n+1, n+1, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+1, n+1, r, c, binomial((r+1)*3^(c-2), c-1)); (P~*M~^-1)[n+1, k+1]}

A179432 a(n) = C(2*3^(n-1), n).

Original entry on oeis.org

1, 2, 15, 816, 316251, 873642672, 17743125256857, 2739097835911193328, 3301626910467952067341626, 31698997711344336177849363574320, 2460103385023594223069956382123378560008
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2010

Keywords

Comments

Equals column 0 in the matrix square of triangle T=A179430 where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Examples

			G.f.: A(x) = 1 + 2*x + 15*x^2 + 816*x^3 + 316251*x^4 +...
A(x) = 1 + 2*log(1+3*x)/3 + 2^2*log(1+3^2*x)^2/(3^2*2!) + 2^3*log(1+3^3*x)^3/(3^3*3!) + 2^4*log(1+3^4*x)^4/(3^4*4!) +...

Crossrefs

Cf. A179430, A179431, A136393, A179433, A179434.

Programs

  • Mathematica
    Table[Binomial[2*3^(n-1),n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    {a(n)=binomial(2*3^(n-1), n)}
  • PARI
    /* G.f. A(x) as Sum of Series: */
{a(n)=polcoeff(sum(k=0, n, (2/3)^k*log(1+3^k*x +x*O(x^n))^k/k!), n)}

Formula

G.f.: A(x) = Sum_{n>=0} (2/3)^n * log(1 + 3^n*x)^n / n!.
a(n) ~ 2^n * 3^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

A179434 Row sums of triangle A179430.

Original entry on oeis.org

1, 2, 13, 571, 172585, 357625693, 5248165593907, 566958191345077996, 465798195439736703244606, 2982999334066325867630228374270, 151658307264909973462110073089257457502
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2010

Keywords

Comments

T=A179430 is a Triangular matrix where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Crossrefs

Cf. A179430, A179431, A136393, A179432, A179433.

Programs

  • PARI
    {a(n)=local(M=matrix(n+2, n+2, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+2, n+2, r, c, binomial((r+1)*3^(c-2), c-1)); sum(k=0,n,(P~*M~^-1)[n+1, k+1])}
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.