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 1-7 of 7 results.

A228909 a(n) = 7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 20160, 332640, 4233600, 46070640, 451725120, 4115105280, 35517081600, 294293759760, 2362955474880, 18509835445920, 142172988048000, 1074905737084080, 8023358912869440, 59263889194762560, 433988913576556800, 3155502239364459600, 22807773973299268800
Offset: 0

Views

Author

Richard V. Scholtz, III, Sep 07 2013

Keywords

Comments

Essentially Stirling Numbers of the Second Kind, with an offset index, and multiplied by 720.
Calculates the seventh 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)].

Links

Crossrefs

Cf. A000771, A008277.
Represents the seventh column of results of A163626.

Programs

  • Magma
    [7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1: n in [0..30]]; // G. C. Greubel, Nov 19 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}, Table[row[n], {n, 6, 23}] [[All, 7]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[6!*StirlingS2[n + 1, 7], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
  • PARI
    a(n)=7^(n)-6*6^(n)+15*5^(n)-20*4^(n)+15*3^(n)-6*2^(n)+1
  • PARI
    concat([0,0,0,0,0,0], Vec(-720*x^6/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)) + O(x^100))) \\ Colin Barker, Dec 16 2014

Formula

a(n) = 720 * S(n+1,7), n>=0.
G.f.: -720*x^6 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Dec 16 2014
E.g.f.: Sum_{k=1..7} (-1)^(7-k)*binomial(7-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Extensions

Offset corrected by Jean-François Alcover, Dec 16 2014
a(20) corrected by Jean-François Alcover, Dec 16 2014
Formula adapted for new offset by Vaclav Kotesovec, Dec 16 2014

A135456 Number of surjections from an n-element set onto a seven-element set.

Original entry on oeis.org

5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960, 248619571200, 2060056318320, 16540688324160, 129568848121440, 995210916336000, 7524340159588560, 56163512390086080, 414847224363337920
Offset: 7

Views

Author

Mohamed Bouhamida, Dec 15 2007

Keywords

Links

Crossrefs

Column k=7 of A019538 and A131689.
Cf. A000918, A000919, A000920, A001117, A001118.

Programs

  • Mathematica
    LinearRecurrence[{28, -322, 1960, -6769, 13132, -13068, 5040}, {5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960}, 25] (* G. C. Greubel, Oct 14 2016 *)

Formula

a(n) = 7^n -C(7,6)*6^n +C(7,5)*5^n -C(7,4)*4^n +C(7,3)*3^n -C(7,2)*2^n +C(7,1) with n>=7.
a(n) = A000771(n) * 7!. - Max Alekseyev, Nov 12 2009
G.f.: -5040*x^7/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^7. - Ilya Gutkovskiy, Jun 19 2018

Extensions

More terms from Max Alekseyev, Nov 12 2009

A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053, 20415995028, 189036065010, 1709751003480, 15170932662679, 132511015347084, 1142399079991620, 9741955019900400, 82318282158320505, 690223721118368580, 5749622251945664950
Offset: 8

Views

Author

Wolfdieter Lang

Keywords

References

Crossrefs

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049435. a(n)= A008277(n, 8).

Programs

Formula

G.f.: x^8/product_{k=1..8} (1-k*x).
E.g.f.: ((exp(x)-1)^8)/8!.
a(n) = det(|s(i+8,j+7)|, 1 <= i,j <= n-8), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

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

A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 45, 1155, 22275, 359502, 5135130, 67128490, 820784250, 9528822303, 106175395755, 1144614626805, 12011282644725, 123272476465204, 1241963303533920, 12320068811796900, 120622574326072500, 1167921451092973005, 11201516780955125625, 106563273280541795575
Offset: 9

Views

Author

Wolfdieter Lang

Keywords

References

Crossrefs

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434.

Programs

Formula

a(n)= A008277(n, 9).
G.f.: x^9/product_{k=1..9} (1-k*x).
E.g.f.: ((exp(x)-1)^9)/9!.
a(n) = det(|s(i+9,j+8)|, 1 <= i,j <= n-9), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

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 1-7 of 7 results.

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

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