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 1-8 of 8 results.

A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

Original entry on oeis.org

1, 8, 46, 230, 1066, 4718, 20266, 85310, 354106, 1455278, 5938186, 24104990, 97478746, 393095438, 1581931306, 6356390270, 25511588986, 102304505198, 409992599626, 1642294397150, 6576150108826, 26325519044558, 105364834103146, 421647614381630, 1687155299822266
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{3,n}. - Vincent Pilaud, Sep 15 2020

Links

Crossrefs

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027651.
First differences of A016269.
Row 3 of array A099594.

Programs

  • Magma
    [6*4^n-6*3^n+2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011
  • Maple
    a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n, j)/(j+1)^k, j=0..n);
seq(a(n, -3), n = 0..30);
  • Mathematica
    Table[6*4^n-6*3^n+2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
  • PARI
    Vec((1-x)/((1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015
  • SageMath
    [2^n -6*3^n +6*4^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

Formula

a(n) = 6*4^n - 6*3^n + 2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1-x)/((1-2*x)*(1-3*x)*(1-4*x)).
E.g.f.: exp(2*x) - 6*exp(3*x) + 6*exp(4*x). - G. C. Greubel, Aug 02 2022

A027651 Poly-Bernoulli numbers B_n^(k) with k=-4.

Original entry on oeis.org

1, 16, 146, 1066, 6902, 41506, 237686, 1315666, 7107302, 37712866, 197451926, 1023358066, 5262831302, 26903268226, 136887643766, 693968021266, 3508093140902, 17693879415586, 89084256837206, 447884338361266, 2249284754708102, 11285908565322946, 56587579617416246
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{4,n}. - Vincent Pilaud, Sep 16 2020

Links

Crossrefs

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027650.
Row n=4 of array A099594.

Programs

  • Magma
    [24*5^n-36*4^n+14*3^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011
  • Maple
    a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n,j)/(j+1)^k, j=0..n);
seq(a(n, -4), n=0..30);
  • Mathematica
    Table[24*5^n -36*4^n +14*3^n -2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{14,-71,154,-120},{1,16,146,1066},30] (* Harvey P. Dale, Nov 20 2019 *)
  • PARI
    Vec((1+4*x)*((1-x)^2)/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015
  • SageMath
    [24*5^n -36*4^n +14*3^n -2^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

Formula

a(n) = 24*5^n -36*4^n +14*3^n -2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1+4*x)*(1-x)^2/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: 24*exp(5*x) - 36*exp(4*x) + 14*exp(3*x) - exp(2*x). - G. C. Greubel, Feb 07 2018

A027644 Denominators of poly-Bernoulli numbers B_n^(k) with k=2.

Original entry on oeis.org

1, 4, 36, 24, 450, 40, 2205, 168, 350, 120, 38115, 88, 40990950, 10920, 5005, 24, 130180050, 136, 1935088155, 3192, 177827650, 1320, 1539340803, 184, 304521767550, 10920, 37182145, 24, 2814316555050, 1160
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A027643.

Programs

  • Magma
    A027644:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027644(n,2): n in [0..30]]; // G. C. Greubel, Aug 02 2022
  • Maple
    a := n -> denom(add((-1)^(n-m)*m!*Stirling2(n, m)/(m+1)^2, m=0..n)):
seq(a(n), n = 0..29);
  • Mathematica
    f[n_]:= (-1)^n*Sum[(-1)^m*m!*StirlingS2[n, m]/(m + 1)^2, {m, 0, n}];
Table[Denominator[f[n]], {n, 0, 30}] (* Robert G. Wilson v, Oct 28 2004 *)
  • SageMath
    def A027644(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027644(n,2) for n in (0..30)] # G. C. Greubel, Aug 02 2022

Formula

a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 2.

A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

Original entry on oeis.org

1, 16, 1296, 3456, 3240000, 144000, 1555848000, 59270400, 5000940000, 1587600, 9762501672000, 11269843200, 221794053611130000, 39390663312000, 5849513501832000, 519437318400, 407131014322092060000, 1063331477208000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027649, A027650, A027651.

Programs

  • Magma
    A027648:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027648(n,4): n in [0..20]]; // G. C. Greubel, Aug 02 2022
  • Maple
    a:= (n, k) -> denom( (-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m = 0..n) );
seq(a(n, 4), n = 0..30);
  • Mathematica
    With[{k = 4}, Table[Denominator@ Sum[((-1)^(m + n))*m!*StirlingS2[n, m]*(m + 1)^(-k), {m, 0, n}], {n, 0, 17}]] (* Michael De Vlieger, Mar 18 2017 *)
  • SageMath
    def A027648(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027648(n,4) for n in (0..20)] # G. C. Greubel, Aug 02 2022

Formula

a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 4.

A283926 Denominators of poly-Bernoulli numbers B_n^(k) with k=7.

Original entry on oeis.org

1, 128, 279936, 5971968, 699840000000, 93312000000, 115269666624000000, 35129803161600000, 160060165655040000000, 1016255020032000000, 103970660613603049728000000, 240047701272387993600000, 41516393959179372527058885120000000
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2017

Keywords

Examples

			B_0^(7) = 1, B_1^(7) = 1/128, B_2^(7) = -1931/279936, B_3^(7) = 32459/5971968, ...

Links

Crossrefs

Cf. A027643-A027648, A283925.

Programs

  • Mathematica
    B[n_]:= Sum[((-1)^(m + n))*m!*StirlingS2[n, m] * (m + 1)^(-7), {m, 0, n}]; Table[Denominator[B[n]], {n, 0, 15}] (* Indranil Ghosh, Mar 18 2017 *)
  • PARI
    B(n) = sum(m=0, n, ((-1)^(m + n)) * m! * stirling(n, m, 2) * (m + 1)^(-7));
for(n=0, 15, print1(numerator(B(n)),", ")) \\ Indranil Ghosh, Mar 18 2017

A120282 Numerator of the coefficients of k^2 term at Sum[Sum[(i-j)^(2n),{i,1,k}],{j,1,k}].

Original entry on oeis.org

-1, 1, -5, 7, -15, 7601, -91, 3617, -745739, 3317609, -5981591, 5436374093, -213827575, 213745149261, -249859397004145, 238988952277727, -28354566442037, 26315271553053477373, -108409774812137683, 3394075340453838586663
Offset: 1

Views

Author

Alexander Adamchuk, Jul 06 2006

Keywords

Crossrefs

Cf. A027643.

Programs

  • Mathematica
    Numerator[Coefficient[Table[Sum[Sum[(i-j)^(2n),{i,1,k}],{j,1,k}],{n,1,20}],k,2]]

Formula

a(n) = numerator[Coefficient[Sum[Sum[(i-j)^(2n),{i,1,k}],{j,1,k}],k,2]]. a(n) = A027643(2n+3) - bisection of numerators of poly-Bernoulli numbers B_n^(k) with k=2.

A120283 Numerator of the coefficients of the k^2 terms of Sum[Sum[(i+j)^(2n),{i,1,k}],{j,1,k}].

Original entry on oeis.org

5, 1, -5, 7, -15, 7601, -91, 3617, -745739, 3317609, -5981591, 5436374093, -213827575, 213745149261, -249859397004145, 238988952277727, -28354566442037, 26315271553053477373, -108409774812137683, 3394075340453838586663
Offset: 1

Views

Author

Alexander Adamchuk, Jul 06 2006

Keywords

Crossrefs

Cf. A100615, A027643.

Programs

  • Mathematica
    Numerator[Coefficient[Table[Sum[Sum[(i+j)^(2n),{i,1,k}],{j,1,k}],{n,1,20}],k,2]]

Formula

a(n) = numerator[Coefficient[Sum[Sum[(i+j)^(2n),{i,1,k}],{j,1,k}],k,2]]. a(n) = A100615(2n) - bisection of the numerators of B(n)(2).

A160035 Clausen-normalized numerators of the Bernoulli numbers of order 2.

Original entry on oeis.org

1, 0, -1, 0, 3, 0, -5, 0, 7, 0, -45, 0, 7601, 0, -91, 0, 54255, 0, -745739, 0, 3317609, 0, -17944773, 0, 5436374093, 0, -213827575, 0, 641235447783, 0, -249859397004145, 0, 238988952277727, 0, -85063699326111, 0, 921034504356871708055, 0, -108409774812137683
Offset: 0

Views

Author

Peter Luschny, Apr 30 2009

Keywords

Comments

Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function
(t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!
Bernoulli numbers of order 1 (defined as B_n{^(1)}(1)) can be regarded as a pair of sequences B1_n = N1_n / D1_n with
N1_n = A027641, D1_n = A141056 (Clausen).
Similarly Bernoulli numbers of order 2 (defined as B_n{^(2)}(1)) can be regarded as a pair of sequences B2_n = N2_n / D2_n with
N2_n = this sequence, D2_n = A141056 (Clausen).

Examples

			The Clausen-normalized Bernoulli polynomials of order 2 are:
1
2 x - 2
6 x^2 - 12 x + 5
2 x^3 - 6 x^2 + 5 x - 1
30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
42 x^6 - 252 x^5 + 525 x^4 - 420 x^3 + 63 x^2 + 42 x - 5
The value of these polynomials at x = 1 gives the sequence.

References

  • L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
  • C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
  • N. E. Nørlund, Vorlesungen über Differenzenrechnung, Berlin, Springer-Verlag, 1924.

Crossrefs

Cf. A120282, A132094, A100615 and A027643.

Programs

  • Maple
    aList := proc(n) local g,c,i; g := k -> (t/(exp(t)-1))^k*exp(x*t): c := proc(n) local i; mul(i,i=select(isprime, map(i->i+1,numtheory[divisors](n)))) end: convert(series(g(2),t,n+8),polynom): seq(i!*c(i)*subs(x=1,coeff(%,t,i)),i=0..n) end: aList(38);
  • Mathematica
    aList[n_] := Module[{g, c, s},
   g[k_] := (t/(Exp[t]-1))^k*Exp[x*t];
   c[k_] := Times @@ Select[Divisors[k]+1, PrimeQ];
   s = Series[g[2], {t, 0, n + 8}] // Normal;
Join[{1}, Table[i!*c[i]*Coefficient[s, t, i] /. x -> 1, {i, 1, n}]]];
aList[38] (* Jean-François Alcover, May 28 2024, after Peter Luschny *)
Showing 1-8 of 8 results.

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

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