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 1-5 of 5 results.

A228911 a(n) = 9^n - 8*8^n + 28*7^n - 56*6^n + 70*5^n - 56*4^n + 28*3^n - 8*2^n + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 40320, 1814400, 46569600, 898128000, 14495120640, 207048441600, 2706620716800, 33094020960000, 384202115256960, 4280991956841600, 46150861752777600, 484294916235312000, 4970346251077025280, 50075960398487654400, 496745174491651008000
Offset: 0

Views

Author

Richard V. Scholtz, III, Sep 07 2013

Keywords

Comments

Calculates the ninth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].
Essentially 40320 * A049447. - Joerg Arndt, Sep 24 2016

Links

Crossrefs

The ninth column of results of A163626.
Cf. A228910 (also for more crossrefs).

Programs

  • Magma
    [9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1: n in [0..32]]; // Vincenzo Librandi, Oct 11 2017
  • Mathematica
    Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[ Derivative[n][y][x], y[x]] // Rest; Join[{0, 0, 0, 0, 0, 0, 0, 0}, Table[row[n], {n, 8, 22}] [[All, 9]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[8!*StirlingS2[n + 1, 9], {n, 0, 22}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1, {n, 0, 22}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[-40320*x^8/ ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)
lst={}; Do[f= 40320 StirlingS2[n, 9];  AppendTo[lst, f], {n, 1, 5!}]; lst (* Vincenzo Librandi, Oct 11 2017 *)
  • PARI
    a(n)=9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1

Formula

G.f.: -40320*x^8/ ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - Colin Barker, Sep 20 2013
E.g.f.: Sum_{k=1..9} (-1)^(9-k)*binomial(9-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Extensions

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 55, 1705, 39325, 752752, 12662650, 193754990, 2758334150, 37112163803, 477297033785, 5917584964655, 71187132291275, 835143799377954, 9593401297313460, 108254081784931500, 1203163392175387500, 13199555372846848005, 143197070509423605675
Offset: 10

Views

Author

Wolfdieter Lang

Keywords

References

Links

Crossrefs

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434, A049447. a(n) = A008277(n, 10).

Programs

Formula

G.f.: x^10/Product_{k=1..10} (1-k*x).
E.g.f.: ((exp(x)-1)^10)/10!.
a(n) = det(|s(i+10,j+9)|, 1 <= i,j <= n-10), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A133360 Number of surjections from an n-element set to a nine-element set.

Original entry on oeis.org

362880, 16329600, 419126400, 8083152000, 130456085760, 1863435974400, 24359586451200, 297846188640000, 3457819037312640, 38528927611574400, 415357755774998400, 4358654246117808000, 44733116259693227520
Offset: 9

Views

Author

Mohamed Bouhamida, Dec 21 2007

Keywords

Links

Crossrefs

Column k=9 of A131689.
Cf. A000918, A000919, A000920, A001117, A001118, A049447, A135456.

Formula

a(n) = Sum_{k=1..9} (-1)^(9-k)*binomial(9,k)*k^n.
a(n) = A049447(n) * 9!. - Max Alekseyev, Nov 12 2009
G.f.: -362880*x^9/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^9. - Ilya Gutkovskiy, Jun 19 2018

Extensions

More terms from Max Alekseyev, Nov 12 2009

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800
Offset: 0

Views

Author

Paul Curtz, Dec 15 2014

Keywords

Comments

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1, 2,
1, 12,
1, 50, 24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1, 0, 0, 0,
1, 0, 0, 0,
1, 2, 0, 0,
1, 12, 0, 0,
1, 50, 24, 0,
1, 180, 360, 0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Crossrefs

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Programs

  • Mathematica
    Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A373173 Triangle read by rows: the exponential almost-Riordan array ( exp(exp(x)-1) | exp(x), exp(x)-1 ).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 1, 3, 1, 15, 1, 7, 6, 1, 52, 1, 15, 25, 10, 1, 203, 1, 31, 90, 65, 15, 1, 877, 1, 63, 301, 350, 140, 21, 1, 4140, 1, 127, 966, 1701, 1050, 266, 28, 1, 21147, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 115975, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1
Offset: 0

Views

Author

Stefano Spezia, May 26 2024

Keywords

Examples

			The triangle begins:
    1;
    1, 1;
    2, 1,  1;
    5, 1,  3,  1;
   15, 1,  7,  6,  1;
   52, 1, 15, 25, 10,  1;
  203, 1, 31, 90, 65, 15, 1;
  ...

Links

Crossrefs

Cf. A000012 (k=1), A000225, A000392 (k=3), A000453 (k=4), A000481 (k=5), A000770 (k=6), A000771 (k=7), A049394 (k=8), A049435 (k=10), A049447 (k=9).
Triangle A008277 with 1st column A000110.

Programs

  • Mathematica
    T[n_,0]:=n!SeriesCoefficient[Exp[Exp[x]-1],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[Exp[x](Exp[x]-1)^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

Formula

T(n,0) = n! * [x^n] exp(exp(x)-1); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] exp(x)*(exp(x)-1)^(k-1).
T(n,2) = A000225(n-1) for n > 1.
Showing 1-5 of 5 results.

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