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.

Previous Showing 11-18 of 18 results.

A114160 E.g.f. is A(x) = (1-log(B(x)))/B(x), where B(x) = sqrt(1-2*x).

Original entry on oeis.org

1, 2, 7, 38, 281, 2634, 29919, 399342, 6125265, 106156530, 2051433495, 43734832470, 1019650457385, 25807495577850, 704708234182575, 20649996837971550, 646340185330747425, 21521124899877175650, 759572031366463998375, 28325808256035867711750, 1112907316518036732317625
Offset: 0

Views

Author

Creighton Dement, Nov 14 2005

Keywords

Comments

From John M. Campbell, May 20 2011: (Start)
a(n) is the determinant of the n X n matrix of the form:
|2 1 1 1 ... 1 |
|1 4 1 1 ... 1 |
|1 1 6 1 ... 1 |
|1 1 1 8 ... 1 |
|... ... 1 |
|1 1 1 1 2n-2 1 |
|1 1 1 1 1 2n |
See examples. (End)

Examples

			From _John M. Campbell_, May 20 2011: (Start)
Det[{
{2,1,1,1,1,1},
{1,4,1,1,1,1},
{1,1,6,1,1,1},
{1,1,1,8,1,1},
{1,1,1,1,10,1},
{1,1,1,1,1,12}}] = 29919 = a(6), and
Det[{
{2,1,1,1,1,1,1},
{1,4,1,1,1,1,1},
{1,1,6,1,1,1,1},
{1,1,1,8,1,1,1},
{1,1,1,1,10,1,1},
{1,1,1,1,1,12,1},
{1,1,1,1,1,1,14}}] = 399342 = a(7).
(End)

References

  • C. Dement, Floretion Integer Sequences (work in progress)

Links

Crossrefs

Cf. A114161.

Programs

  • Mathematica
    Range[0, 18]! CoefficientList[ Series[(1 - Log[Sqrt[1 - 2x]])/Sqrt[(1 - 2x)], {x, 0, 18}], x] (* or *)
f[n_] := FullSimplify[ 2^(n-1)*Gamma[n + 1/2]/Sqrt[Pi]*(PolyGamma[n + 1/2] + EulerGamma + Log[4] + 2)]; Table[f[n], {n, 0, 18}] (* Robert G. Wilson v *)
twox[x_, y_] := If[x == y, 2*x, 1]; a[n_] := Det[Array[twox[#1, #2] &, {n, n}]]; Join[{1}, Table[a[n], {n, 1, 10}]] (* John M. Campbell, May 20 2011 *)
  • PARI
    my(x='x + O('x^50)); Vec(serlaplace((1 - log(sqrt(1 - 2*x)))/sqrt(1 - 2*x))) \\ G. C. Greubel, Feb 08 2017

Formula

a(n) = A001147(n) + A004041(n-1) = 2^n*Gamma(n+1/2)/Pi^(1/2)*(1/2*Psi(n+1/2)+1/2*gamma+log(2)+1). - Vladeta Jovovic

Extensions

E.g.f. given by Vladeta Jovovic
More terms from Robert G. Wilson v, Nov 15 2005

A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 10, 23, 4, 0, 1, 28, 259, 176, 5, 0, 1, 82, 3527, 12916, 1689, 6, 0, 1, 244, 51331, 1213136, 1057221, 19524, 7, 0, 1, 730, 762743, 123296356, 885533769, 128816766, 264207, 8, 0, 1, 2188, 11406979, 12820180976, 809068942341, 1179489355164, 21878089479, 4098240, 9
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2017

Keywords

Examples

			Square array begins:
  0,   0,     0,       0,         0, ...
  1,   1,     1,       1,         1, ...
  2,   4,    10,      28,        82, ...
  3,  23,   259,    3527,     51331, ...
  4, 176, 12916, 1213136, 123296356, ...

Links

Crossrefs

Columns k=0-5 give: A001477, A004041(n+1), A001824(n+1), A291585, A291586, A291587.
Rows n=0-2 give: A000004, A000012, A034472.
Main diagonal gives A291676.
Cf. A291556.

Formula

T(0,k) = 0, T(1,k) = 1 and T(n+1, k) = ((2*n-1)^k+(2*n+1)^k) * T(n, k) - (2*n-1)^(2*k) * T(n-1, k).

A383231 Expansion of e.g.f. f(x) * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

Original entry on oeis.org

0, 1, 7, 83, 1394, 30330, 810756, 25710012, 943434288, 39324264624, 1835297984160, 94813760519136, 5371462318747392, 331125138305434368, 22065681276731119104, 1580617232453691210240, 121117633854691036502016, 9885823380533972300470272, 856279708828545483688808448
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2025

Keywords

Crossrefs

Cf. A383232, A383233, A383234.
Cf. A004041, A024216, A024382.

Programs

  • PARI
    a(n) = sum(k=1, n, k*5^(n-k)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} k * 5^(n-k) * |Stirling1(n,k)|.
a(n) = 5^(n-1) * n! * Sum_{k=0..n-1} (-1)^k * binomial(-1/5,k)/(n-k).
a(n) = (10*n-13) * a(n-1) - (5*n-9)^2 * a(n-2) for n > 1.

A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 15, 23, 18, 6, 105, 176, 172, 96, 24, 945, 1689, 1900, 1380, 600, 120, 10395, 19524, 24278, 20880, 12120, 4320, 720, 135135, 264207, 354662, 344274, 241080, 116760, 35280, 5040, 2027025, 4098240, 5848344, 6228096, 4993296, 2956800, 1229760
Offset: 0

Views

Author

Peter Luschny, May 19 2013

Keywords

Comments

The Stirling-Frobenius cycle numbers are defined in A225470.

Examples

			[n\k][ 0,    1,    2,    3,   4,   5]
[0]    1,
[1]    1,    1,
[2]    3,    4,    2,
[3]   15,   23,   18,    6,
[4]  105,  176,  172,   96,  24,
[5]  945, 1689, 1900, 1380, 600, 120.

Links

Crossrefs

Cf. A028338, A225479 (m=1), A048594.

Programs

  • Mathematica
    SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 02 2013, translated from Sage *)
  • Sage
    @CachedFunction
def SF_CO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)
for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

Formula

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, 1) ~ A004041.
T(n, n) ~ A000142; T(n, n-1) ~ A001563.
T(n,k) = A028338(n,k)*A000142(k). - Philippe Deléham, Jun 24 2015

A291587 a(n) = ((2n-1)!!)^5 * Sum_{i=1..n} 1/(2*i-1)^5.

Original entry on oeis.org

0, 1, 244, 762743, 12820180976, 757031629267449, 121921454556651769524, 45268703999809586294371407, 34375967164840303438628549400000, 48808991831991566280900452880679940625, 120855944455445379138034328603009420077012500
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2017

Keywords

Links

Crossrefs

Cf. A001147, A099827.
Cf. A004041, A001824, A291585, A291586.

Programs

  • Mathematica
    Table[(2*n-1)!!^5 * Sum[1/(2*i-1)^5, {i, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Aug 27 2017 *)

Formula

a(0) = 0, a(1) = 1, a(n+1) = ((2*n-1)^5+(2*n+1)^5)*a(n) - (2*n-1)^10*a(n-1) for n > 0.
a(n) ~ 31*Zeta(5) * 2^(5*n-5/2) * n^(5*n) / exp(5*n). - Vaclav Kotesovec, Aug 27 2017

A203159 (n-1)-st elementary symmetric function of {2,4,6,8,...,2n}.

Original entry on oeis.org

1, 6, 44, 400, 4384, 56448, 836352, 14026752, 262803456, 5441863680, 123436892160, 3044235018240, 81112101027840, 2322150583173120, 71092846618214400, 2317820965473484800, 80177108784198451200, 2932996578806543155200
Offset: 1

Views

Author

Clark Kimberling, Dec 29 2011

Keywords

Examples

			(n-1)-st elementary symmetric function of {2,4,6,8,...,2n}.
Let esf abbreviate "elementary symmetric function".  Then
0th esf of {2}:  1
1st esf of {2,4}: 2+4=6
2nd esf of {2,4,6}: 2*4+2*6+4*6=44

Crossrefs

Cf. A004041.

Programs

  • Mathematica
    f[k_] := 2 k; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203159 *)

Formula

Conjecture: a(n) +2*(-2*n+1)*a(n-1) +4*(n-1)^2*a(n-2)=0. - R. J. Mathar, Oct 01 2016

A334000 a(n) = (2*n+1)!! * Sum_{k=0..n} k/(2*k+1).

Original entry on oeis.org

0, 1, 11, 122, 1518, 21423, 340869, 6058980, 119218860, 2575293165, 60628447215, 1545696702270, 42437227275450, 1248581232985275, 39197268410049225, 1307969571015966600, 46233376386927067800, 1725823391345415833625, 67845041198360981737875
Offset: 0

Views

Author

Greg Huber, Apr 11 2020

Keywords

Examples

			a(3) = 122 since 0/1 + 1/3 + 2/5 + 3/7 = 122/105 = 122/(7!!).

Crossrefs

Cf. A004041.

Programs

  • Mathematica
    Table[Sum[k/(2*k+1),{k,0,n}],{n,0,18}]*Table[Product[2*j+1,{j,0,n}],{n,0,18}]
FullSimplify[Table[((n+1)/2 - HarmonicNumber[n + 1/2]/4 - Log[2]/2) * (2*n+1)!!, {n, 0, 20}]] (* Vaclav Kotesovec, Apr 14 2020 *)

Formula

a(n) = (2*n+1)!!*(Sum_{k=0..n} k/(2*k+1)).
Recurrence: a(n) = 4*n*a(n-1)-(2*n-1)^2*a(n-2)+(2*n-1)!!.

A334066 a(n) = (2n-1)!!*(Sum_{k=1..n}k/(2*k-1)).

Original entry on oeis.org

1, 5, 34, 298, 3207, 40947, 605076, 10157220, 190915965, 3971997585, 90613969110, 2249113016430, 60338869272675, 1739831420490975, 53656981894391400, 1762410972384203400, 61421841416041392825, 2263752327235180060125, 87970054921758957890250
Offset: 1

Views

Author

Greg Huber, Apr 13 2020

Keywords

Examples

			a(4)=298 since 1/1+2/3+3/5+4/7=298/105=298/(7!!).

Crossrefs

Cf. A004041.

Programs

  • Mathematica
    Table[Sum[k/(2*k-1), {k, 1, n}], {n, 1, 19}]*Table[Product[2*j-1, {j, 1, n}], {n, 1, 19}]
FullSimplify[Table[(n/2 + HarmonicNumber[n - 1/2]/4 + Log[2]/2) * (2*n-1)!!, {n, 1, 20}]] (* Vaclav Kotesovec, Apr 14 2020 *)

Formula

a(n) = (2n-1)!!*(Sum_{k=1..n}k/(2*k-1)).
Recurrence: a(n) = 2*a(n-1) + (2*n-3)^2*a(n-2) + (2*n-1)!!.
Previous Showing 11-18 of 18 results.

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

Content is available under The OEIS End-User License Agreement.