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.

A118789 Row sums of triangle A118788.

Original entry on oeis.org

1, 2, 9, 71, 800, 11659, 208173, 4398148, 107293711, 2967800711, 91777098006, 3137581240925, 117499040544197, 4783424590188490, 210333509575901445, 9934472399437068811, 501615620424564184408, 26963169913347131361647
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2006

Keywords

Comments

A032188 equals the main diagonal of triangle A118788; A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.

Examples

			E.g.f.: A(x) = 1 + 1*x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! + ... =
exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +... + A032188(n)*x^n/n! +...).

Links

Crossrefs

Cf. A118788, A118790, A032188.

Programs

  • Mathematica
    Table[Sum[n!/(n-k)! * SeriesCoefficient[(x/(2*x + Log[1-x]))^(n + 1), {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 01 2025 *)
  • PARI
    {a(n)=local(x=X+X^2*O(X^n));sum(k=0,n,n!/(n-k)!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X))}

Formula

E.g.f.: A(x) = exp( Sum_{n>=1} A032188(n)*x^n/n! ). As row sums of A118788, a(n) = Sum_{k=0..n} n!/(n-k)!*[x^k]{ x/(2*x + log(1-x)) }^(n+1).
a(n) ~ n^n / (2 * exp(n - 1/2) * (1 - log(2))^(n + 1/2)). - Vaclav Kotesovec, Sep 01 2025

A118790 Secondary diagonal of triangle A118788.

Original entry on oeis.org

1, 3, 23, 255, 3679, 65247, 1371887, 33347535, 919848991, 28382683263, 968565339983, 36217270235055, 1472548026160639, 64679673007670367, 3052084360057720367, 153980839859926277775, 8270901171633696369631
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2006

Keywords

Links

Crossrefs

Cf. A118788, A118789.

Programs

  • Mathematica
    Table[(n+1)! * SeriesCoefficient[(x/(2*x + Log[1-x]))^(n+2),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 01 2025 *)
  • PARI
    {a(n)=local(x=X+X^2*O(X^(n)));(n+1)!*polcoeff((x/(2*x+log(1-x)))^(n+2),n,X)}

Formula

a(n) = (n+1)!*[x^n] ( x/(2*x + log(1-x)) )^(n+2).
a(n) ~ n^(n+1) / (4 * exp(n) * (1 - log(2))^(n + 3/2)). - Vaclav Kotesovec, Sep 01 2025

A241765 a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.

Original entry on oeis.org

0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448
Offset: 0

Views

Author

Bruno Berselli, Apr 28 2014

Keywords

Comments

Equivalently, Sum_{i=0..n} (i+4)*A000217(i).
Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
k = 0, A001296: 0, 1, 7, 25, 65, 140, 266, 462, ...
k = 1, A000914: 0, 2, 11, 35, 85, 175, 322, 546, ...
k = 2, A050534: 0, 3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
k = 3, A215862: 0, 4, 19, 55, 125, 245, 434, 714, ...
k = 4, a(n): 0, 5, 23, 65, 145, 280, 490, 798, ...
k = 5, A239568: 0, 6, 27, 75, 165, 315, 546, 882, ...
Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
After 0, partial sums of A212343 and third column of A118788.
This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).

Examples

			a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.

Links

Crossrefs

Cf. A000217, A005286, A118788, A212343, A227342.
Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).

Programs

  • Magma
    /* By first comment: */ k:=4; A000217:=func; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]];
  • Maple
    A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014
  • Mathematica
    Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40]
CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
  • Maxima
    makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);
  • PARI
    a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015
  • PARI
    x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016
  • Sage
    [n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]

Formula

G.f.: x*(5 - 2*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A227342(A055998(n+1)).
a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - Peter Luschny, Apr 10 2016

A118787 Triangle where T(n,k) = n!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 12, 23, 41, 24, 60, 130, 255, 469, 120, 360, 870, 1860, 3679, 6889, 720, 2520, 6720, 15540, 32858, 65247, 123605, 5040, 20160, 58800, 146160, 328734, 689388, 1371887, 2620169, 40320, 181440, 574560, 1527120, 3638376, 8029980
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2006

Keywords

Comments

Row sums are A112487. Main diagonal is A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.

Examples

			Triangle begins:
1;
1, 1;
2, 3, 5;
6, 12, 23, 41;
24, 60, 130, 255, 469;
120, 360, 870, 1860, 3679, 6889;
720, 2520, 6720, 15540, 32858, 65247, 123605;
5040, 20160, 58800, 146160, 328734, 689388, 1371887, 2620169; ...
Triangle is formed from powers of F(x) = x/(2*x + log(1-x)):
F(x)^1 = (1) + 1/2*x + 7/12*x^2 + 17/24*x^3 + 629/720*x^4 +...
F(x)^2 = (1 + x)/1! +17/12*x^2 + 2*x^3 + 671/240*x^4 ...
F(x)^3 = (2 + 3*x + 5*x^2)/2! + 4*x^3 + 1489/240*x^4 +...
F(x)^4 = (6 + 12*x + 23*x^2 + 41/6*x^3)/3! + 8351/720*x^4 +...
F(x)^5 = (24 + 60*x + 130*x^2 + 255*x^3 + 469*x^4)/4! +...

Crossrefs

Cf. A118788, A112487, A032188.

Programs

  • PARI
    {T(n,k)=local(x=X+X^2*O(X^(k+2)));n!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X)}

Formula

Main diagonal has e.g.f.: series_reversion[2*x+log(1-x)].
Showing 1-4 of 4 results.

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