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.

A228910 a(n) = 8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 181440, 3780000, 59875200, 801496080, 9574044480, 105398092800, 1091804313600, 10794490827120, 102896614941120, 952741767650400, 8617145057539200, 76461500619902160, 667855517349303360, 5757691363157764800, 49099453300298016000, 414884142077935345200
Offset: 0

Views

Author

Richard V. Scholtz, III, Sep 07 2013

Keywords

Comments

Calculates the eighth 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. A008277, A049434.
The eighth column of results of A163626.
Cf. A000225, A028243, A028244, A028245, A032180, A228909.

Programs

  • Magma
    [8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*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, 0}, Table[ -row[n], {n, 7, 23}] [[All, 8]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[7!*StirlingS2[n + 1, 8], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)
  • PARI
    a(n)=8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1.
  • PARI
    for(n=0,30, print1(5040*stirling(n+1,8,2), ", ")) \\ G. C. Greubel, Nov 19 2017

Formula

a(n) = 5040 * S2(n+1,8), n>=0.
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)*(8*x-1)). - Colin Barker, Sep 20 2013
E.g.f.: Sum_{k=1..8} (-1)^(8-k)*binomial(8-1,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

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

A133068 Number of surjections from an n-element set to an eight-element set.

Original entry on oeis.org

40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960, 823172919528960, 7621934141203200, 68937160460313600, 611692004959217280, 5342844138794426880, 46061530905262118400, 392795626402384128000
Offset: 8

Views

Author

Mohamed Bouhamida, Dec 16 2007

Keywords

Links

Crossrefs

Column k=8 of A019538 and A131689.
Cf. A000918, A000919, A000920, A001117, A001118, A135456.

Programs

  • Magma
    [&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // Vincenzo Librandi, Oct 21 2017
  • Mathematica
    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)), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* G. C. Greubel, Oct 21 2017 *)
  • PARI
    x='x+O('x^50); Vec(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))) \\ G. C. Greubel, Oct 20 2017

Formula

a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).
a(n) = A049434(n) * 8!. - Max Alekseyev, Nov 13 2009
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)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^8. - Ilya Gutkovskiy, Jun 19 2018

Extensions

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar
More terms from Max Alekseyev, Nov 13 2009

A245602 Triangle read by rows: the negative terms of A163626.

Original entry on oeis.org

-1, -3, -7, -6, -15, -60, -31, -390, -120, -63, -2100, -2520, -127, -10206, -31920, -5040, -255, -46620, -317520, -181440, -511, -204630, -2739240, -3780000, -362880, -1023, -874500, -21538440, -59875200, -19958400, -2047
Offset: 0

Views

Author

Paul Curtz, Dec 17 2014

Keywords

Comments

These numbers a(n) are the companion of A249163(n).
Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):
1,
1, -1/2,
1, -3/2, +2/3,
1, -7/2, +12/3, -6/4,
etc.
From the second row on, the sum of the numerators is 0.
The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).
a(n) triangle is shifted. It starts from second row and second column of triangle above.
-1,
-3,
-7, -6,
-15, -60,
-31, -390, -120,
-63, -2100, -2520,
-127, -10206, -31920, -5040,
-255, -46620, -317520, -181440,
etc.
Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).
Successive columns: A000225, A028244, from the Stirling numbers of second kind S(n,2), S(n,4), S(n,6), S(n,8), S(n,10), ... . See A000770, A032180, A049434, A228910, A049435, A228912, A008277.
Thanks to Jean-François Alcover.

Crossrefs

Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.

Programs

  • Mathematica
    Select[ Table[ (-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten, Negative] (* Jean-François Alcover, Dec 26 2014 *)
Showing 1-5 of 5 results.

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

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