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 1-6 of 6 results.

A100872 a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.

Original entry on oeis.org

1, 13, 421, 25453, 2473141, 352444093, 69251478661, 17943523153933, 5927841361456981, 2431910546406522973, 1212989379862721528101, 722875495525684291639213, 507275965883448333971692021, 414031618935013558427928710653, 388884101194230308462039862028741
Offset: 1

Views

Author

Benoit Cloitre, Jan 08 2005

Keywords

Comments

A bisection of "Stirling-Bernoulli transform" of Fibonacci numbers.

Links

Crossrefs

Cf. A001622, A050946, A100868, A241171.

Programs

  • Mathematica
    FullSimplify[Table[PolyLog[-2k, GoldenRatio^(-2)]/Sqrt[5], {k, 1, 10}]] (* Vladimir Reshetnikov, Feb 16 2011 *)
T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[, 1] = 1; T[, ] = 0; a[n] := Sum[2^(k-1) T[n, k], {k, 1, n}]; Array[a, 15] (* Jean-François Alcover, Jul 03 2019 *)
  • PARI
    a(n)=round(1/sqrt(5)*sum(k=1,500,k^(2*n)/((1+sqrt(5))/2)^(2*k)))

Formula

a(n) = A050946(2*n).
From Peter Bala, Aug 20 2014: (Start)
E.g.f.: -1/2 + (1/2)*exp(z)/(3*exp(z) - exp(2*z) - 1) = z^2/2! + 13*z^4/4! + 421*z^6/6! + ....
a(n) = Sum_{k = 1..n} 2^(k-1)*A241171(n,k).
a(n) = Sum_{1 <= j <= k <= n} (-1)^(k-j)*binomial(2*k,k+j)*j^(2*n).
(End)

A100868 a(n) = Sum_{k>0} k^(2n-1)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.

Original entry on oeis.org

1, 7, 151, 6847, 532231, 63206287, 10645162711, 2413453999327, 708721089607591, 261679010699505967, 118654880542567722871, 64819182599591545006207, 41987713702382161714004551, 31821948327041297758906340047, 27896532358791207565357448388631
Offset: 1

Views

Author

Benoit Cloitre, Jan 08 2005

Keywords

Comments

A bisection of "Stirling-Bernoulli transform" of Fibonacci numbers.

Links

Crossrefs

Cf. A001622, A050946, A100872.
Row sums of A303675.

Programs

  • Mathematica
    FullSimplify[Table[PolyLog[1 - 2k, GoldenRatio^(-2)], {k, 1, 10}]] (* Vladimir Reshetnikov, Feb 16 2011 *)
  • PARI
    a(n)=round(sum(k=1,500,k^(2*n-1)/((1+sqrt(5))/2)^(2*k)))

Formula

a(n) = A050946(2*n-1).

A230324 a(n) = A226158(n) - 2*A226158(n+1).

Original entry on oeis.org

2, 1, -1, -2, 1, 6, -3, -34, 17, 310, -155, -4146, 2073, 76454, -38227, -1859138, 929569, 57641238, -28820619, -2219305810, 1109652905, 103886563462, -51943281731, -5810302084962, 2905151042481, 382659344967926
Offset: 0

Views

Author

Paul Curtz, Oct 16 2013

Keywords

Comments

The array A(n,k) = A(n-1,k+1) - A(n-1,k) of the sequence in the first row and higher-order sequences in followup rows starts:
2, 1, -1, -2, 1, 6, -3, ...
-1, -2, -1, 3, 5, -9, -31, ...
-1, 1, 4, 2, -14, -22, 82, ...
2, 3, -2, -16, -8, 104, 160, ...
1, -5, -14, 8, 112, 56, -1160, ...
-6, -9, 22, 104, -56, -1216, -608, ...
-3, 31, 82, -160, -1160, 608, 18880, ...
etc.
a(n) is an autosequence: Its inverse binomial transform is the sequence (up to a sign), which means top row and left column in the difference array have the same absolute values.
The main diagonal is the double of the first upper diagonal: A(n,n) = 2*A(n,n+1).
A(n,n+1) = (-1)^n*A005439(n), which also appears as the first upper diagonal of the difference array of A226158(n).

Examples

			a(0) =  0 - 2 * (-1) =  2,
a(1) = -1 - 2 * (-1) =  1,
a(2) = -1 - 2 *   0  = -1,
a(3) =  0 - 2 *   1  = -2,
a(4) =  1 - 2 *   0  =  1,
a(5) =  0 - 2 * (-3) =  6.

Links

Crossrefs

Cf. A050946.

Programs

  • Maple
    A226158 := proc(n)
    if n = 0 then
        0;
    else
        Zeta(1-n)*2*n*(2^n-1) ;
    end if;
end proc:
A230324 := proc(n)
    A226158(n)-2*A226158(n+1) ;
end proc: # R. J. Mathar, Oct 28 2013
  • Mathematica
    a[0] = 2; a[1] = 1; a[n_] := n EulerE[n-1, 0] - 2 (n+1) EulerE[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 07 2017 *)

Formula

a(n)/2 + A164555(n)/A027642(n) = 2*A225825(n)/A141056(n).

A105796 "Stirling-Bernoulli transform" of Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 13, 25, 541, 1561, 47293, 181945, 7087261, 34082521, 1622632573, 9363855865, 526858348381, 3547114323481, 230283190977853, 1771884893993785, 130370767029135901, 1128511554418948441, 92801587319328411133, 892562598748128067705, 81124824998504073881821
Offset: 0

Views

Author

Paul Barry, Apr 20 2005

Keywords

Links

Crossrefs

Cf. A050946.

Programs

  • Maple
    a:= n-> -add((-1)^k*k!*Stirling2(n+1, k+1)*(<<0|1>, <2|1>>^k)[1, 2], k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, May 09 2018
  • Mathematica
    CoefficientList[Series[E^x*(1-E^x)/((2-E^x)*(1-2*E^x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)

Formula

E.g.f.: e^x*(1-e^x)/((2-e^x)*(1-2*e^x)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * S2(n,k) * A001045(k).
a(n) ~ n! * (2-(-1)^n)/(6*log(2)^(n+1)). - Vaclav Kotesovec, Sep 26 2013
a(n) = Sum_{k = 0..n} (-1)^(n-k)*A131689(n,k)*A001045(k). - Philippe Deléham, May 25 2015

A105797 "Stirling-Bernoulli transform" of Pell numbers.

Original entry on oeis.org

0, 1, 3, 19, 135, 1291, 14343, 188539, 2815095, 47412811, 886239783, 18231365659, 409053408855, 9943622273131, 260300948527623, 7300927107288379, 218426614502831415, 6943261704033434251, 233692323131307301863
Offset: 0

Views

Author

Paul Barry, Apr 20 2005

Keywords

Crossrefs

Cf. A050946.

Programs

  • Mathematica
    CoefficientList[Series[E^x*(1-E^x)/(1-4*E^x+2*E^(2*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)

Formula

E.g.f.: e^x*(1-e^x)/(1-4*e^x+2*e^(2*x)).
a(n) = Sum_{k = 0..n} (-1)^(n-k) * k! * S2(n, k) * A000129(k).
a(n) ~ n!/(4*log(1+1/sqrt(2))^(n+1)). - Vaclav Kotesovec, Sep 26 2013
a(n) = Sum_{k = 0..n} (-1)^(n-k) * A131689(n,k) * A000129(k). - Philippe Deléham, May 25 2015

A307361 Expansion of e.g.f. (sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x)).

Original entry on oeis.org

0, 1, 5, 7, 65, 151, 2105, 6847, 127265, 532231, 12365705, 63206287, 1762220465, 10645162711, 346257393305, 2413453999327, 89717615769665, 708721089607591, 29639206807284905, 261679010699505967, 12159552732032614865, 118654880542567722871, 6064946899313607640505
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2019

Keywords

Crossrefs

Cf. A000204, A050946, A105796, A105797, A263575.

Programs

  • Maple
    a:=series((sinh(x)+5*cosh(x)-5)/(3-2*cosh(x)),x=0,23):seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Apr 12 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[(Sinh[x] + 5 Cosh[x] - 5)/(3 - 2 Cosh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Sum[j! LucasL[j] x^j/Product[(1 + k x), {k, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) StirlingS2[n, k] k! LucasL[k], {k, 1, n}], {n, 0, 22}]
  • PARI
    my(x = 'x + O('x^30)); concat(0, Vec(serlaplace((sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x))))) \\ Michel Marcus, Apr 05 2019

Formula

G.f.: Sum_{j>=1} j!*Lucas(j)*x^j / Product_{k=1..j} (1 + k*x).
a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*k!*Lucas(k).
a(n) ~ n! * (phi + (-1)^n/phi) / (2*log(phi))^(n+1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Apr 05 2019
Showing 1-6 of 6 results.

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