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.

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A011248 Twice A000364.

Original entry on oeis.org

2, 2, 10, 122, 2770, 101042, 5405530, 398721962, 38783024290, 4809759350882, 740742376475050, 138697748786275802, 31029068327114173810, 8174145018586247784722, 2504519282807259730936570, 883087786498046209107365642, 355038783159078578873329579330, 161446598471775796124336494906562
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000364, A009752.

Programs

  • Mathematica
    a[n_] := (-1)^n 4 Im[PolyLog[-2 n, I]];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Aug 18 2021 *)

Formula

E.g.f.: 2 - 2/Q(0), where Q(k)= 1 - (2*k+1)*(2*k+2)/x + 1/x*(2*k+1)*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
From Peter Luschny, Aug 18 2021: (Start)
a(n) = (-1)^n*4^(2*n+1)*(Bernoulli(2*n+1, 3/4) - Bernoulli(2*n+1, 1/4))/(2*n+1).
a(n) = (-1)^n*4*Im(PolyLog(-2*n, i)). (End)

A014547 n-th Euler (secant, zig) number A000364(n) is prime.

Original entry on oeis.org

2, 3, 19, 227, 255
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A000364.

Programs

  • Mathematica
    Select[Range[100], PrimeQ[EulerE[#]] &]/2 (* Harvey P. Dale, Aug 10 2011 *)

Extensions

Offset corrected by Arkadiusz Wesolowski, Oct 17 2011

A035163 Composite numbers k, not a power of 2, such that the E(k) == 1 (mod k), where E(k) is the k-th Euler number (A000364).

Original entry on oeis.org

15, 91, 289, 319, 435, 561, 692, 703, 1016, 1105, 1369, 1495, 1729, 1885, 1891, 2105, 2465, 2701, 2755, 2821, 3367, 4371, 5551, 6409, 6601, 7456, 8224, 8569, 8695, 8911, 9088, 10585, 10621, 11305, 11849, 12121, 12403, 13981, 14065, 15051, 15841, 16471, 17104
Offset: 1

Views

Author

Benoit Cloitre, Apr 06 2003

Keywords

Crossrefs

Cf. A000364, A081730.

Programs

  • Mathematica
    Select[Range[1000], CompositeQ[#] && #/2^IntegerExponent[#, 2] > 1 && Divisible[Abs[EulerE[2*#]] - 1, #] &] (* Amiram Eldar, Nov 26 2020 *)
  • PARI
    a000364(n)=subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1);
lista(nn) = {forcomposite(n=1, nn, if ( n != 2^valuation(n, 2), if (Mod(a000364(n), n) == 1, print1(n, ", "));););} \\ Michel Marcus, Apr 18 2015

Extensions

More terms from Hans Havermann, Apr 07 2003
a(23)-a(43) from Amiram Eldar, Nov 26 2020

A239322 Interleave (-1)^n*(A000182(n+1) - A000364(n)), -A028296(n+1).

Original entry on oeis.org

0, 1, 1, -5, -11, 61, 211, -1385, -6551, 50521, 303271, -2702765, -19665491, 199360981, 1704396331, -19391512145, -190473830831, 2404879675441, 26684005437391, -370371188237525
Offset: 0

Views

Author

Paul Curtz, Mar 28 2014

Keywords

Comments

Difference table of a(n):
0, 1, 1, -5, -11, 61, 211, -1385,...
1, 0, -6, -6, 72, 150, -1596,...
-1, -6, 0, 78, 78, -1746,...
-5, -6, 78, 0, -1824,...
11, 72, 78, -1824,...
61, -150, -1746,...
-211, -1596,...
-1385,...
etc.
a(n) is an autosequence (its inverse binomial transform is the signed sequence) of the first kind (its main diagonal is A000004=0's and the first two upper diagonal are the same). Like A000045(n).
Note that e(n) = A000111(n+1) - A000111(n) = 0, 0, 1, 3, 11, 45, 211,... is not in the OEIS. a(2n) = (-1)*(n+1)*e(2n).
Antidiagonals upon A000004:
1,
1,
-6, -5,
-6, -11,
78, 72, 61,
78, 150, 211,
Row sum: 1, 1, -11, -17, 211, 439,... .
b(n) = a(n) mod 9 = 0 followed by period 6: repeat 1, 1, 4, 7, 7, 4 is also an autosequence of the first kind.

Examples

			a(0)=1-1=0, a(1)=-(-1)=1, a(2)=2-1=1, a(3)=-5, a(4)=-(16-5)=-11.

Crossrefs

Cf. Zig (A000364) and Zag (A000182) give Euler A000111(n).

A007313 Reversion of g.f. for Euler (secant) numbers A000364.

Original entry on oeis.org

1, -1, -3, -41, -1035, -40721, -2291331, -174783865, -17394878523, -2192620580129, -341767803858867, -64587124941406473, -14555427555355014123, -3857289820949315456817
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000364.

Extensions

Extended with signs by Christian G. Bower, Oct 15 1999

A069042 Numbers k such that A000364(k) == 1 (mod k^2).

Original entry on oeis.org

1, 2, 17, 37, 1153, 1303
Offset: 1

Views

Author

Benoit Cloitre, Apr 06 2003

Keywords

Crossrefs

Cf. A000364, A081730.

Programs

  • Mathematica
    Select[Range[1, 10000], Divisible[Abs[EulerE[2*#]] - 1, #^2] &] (* Amiram Eldar, Jun 03 2017 *)
  • PARI
    is(k) = Mod(abs(eulerfrac(2*k)), k^2) == 1; \\ Jinyuan Wang, Mar 13 2025

Extensions

More terms from Hans Havermann, Apr 07 2003
a(1) prepended by Jinyuan Wang, Mar 13 2025

A106051 Number of divisors of the Euler number E(2n) (A000364).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 16, 4, 8, 8, 12, 8, 16, 4, 128, 4, 32, 8, 64, 2, 48, 8, 64, 8, 16, 16, 128, 32, 128, 4, 192, 32, 64, 32, 64, 8, 512, 32, 32, 4, 96, 16, 64, 16, 64, 8, 64, 16, 2048, 32, 64, 8, 32, 32, 512, 32, 1024, 64, 32, 16, 96, 16, 512, 256, 2048, 8, 32
Offset: 0

Views

Author

John W. Layman, May 06 2005

Keywords

Comments

Notice that all listed terms are powers of 2 except for the 10th, 20th and 30th. It would be interesting to know whether this pattern continues. Note: Various sources give differing values for the Euler numbers. A000364 gives {1,1,5,61,1385,50521,2702765,199360981,19391512145,...}, whereas Mathematica gives {1,0,-1,0,5,0,-61,0,1385,0,-50521,...}.

Examples

			E(4) = 1385 has divisors {1,5,277,1385}, so a(4) = 4.

Links

Crossrefs

Cf. A000005, A000364.

Programs

  • Maple
    a:= n-> numtheory[tau](abs(euler(2*n))):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 03 2023
  • Mathematica
    a ={};For[n=0, n<=33, n++, {Eu=EulerE[2*n];L=Length[Divisors[Eu]];a=Append[a, L]}];a

Formula

a(n) = A000005(A000364(n)).

Extensions

a(34)-a(49) from Robert G. Wilson v, May 09 2005
a(46) corrected and more terms from Sean A. Irvine, Mar 03 2023

A135594 a(n) = (1/2^n) * Sum_{i=0..n} (-1)^(n-i) * binomial(n,i) * A000364(i).

Original entry on oeis.org

1, 0, 1, 6, 73, 1380, 37801, 1417626, 69802993, 4369750440, 339034806001, 31935510092046, 3590398569115513, 474937566660074700, 73024143791301120601, 12914495107705743175266, 2603190607000627341985633, 593297406341867021292734160
Offset: 0

Views

Author

Vladeta Jovovic, Feb 25 2008

Keywords

Comments

Let k be a positive integer. It appears that reducing the sequence {a(n): n >= 1} modulo k produces a periodic sequence with period a divisor of phi(k) unless k is of the form 2^j, when the period equals k. For example, modulo 7 the sequence becomes [0, 1, 6, 3, 1, 1, 0, 1, 6, 3, 1, 1, 0, 1, 6, 3, 1, 1, ...], with an apparent period of 6 = phi(7), while modulo 8 the sequence becomes [0, 1, 6, 1, 4, 1, 2, 1, 0, 1, 6, 1, 4, 1, 2, 1, 0, 1, 6, 1, 4, 1, 2, 1, ...] with an apparent period of 8. - Peter Bala, May 07 2023

References

  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 4.2.2.(b).

Crossrefs

Cf. A000657, A005799.

Programs

  • Maple
    A000364 := proc(n) option remember ; (2*n)!*coeftayl(sec(x),x=0,2*n) ; end: A135594 := proc(n) add((-1)^(n-i)*binomial(n,i)*A000364(i),i=0..n)/2^n ; end: seq(A135594(n),n=0..20) ; # R. J. Mathar, Mar 14 2008
f:=sec(z): fser:=series(f,z=0,63): for n from 0 to 60 do b[n]:=factorial(n)*coeff(fser,z,n) end do: a:= proc(n) options operator, arrow: add((-1)^(n-k)*binomial(n,k)*b[2*k],k=0..n)/2^n end proc: seq(a(n),n=0..16); # Emeric Deutsch, Mar 17 2008
  • Mathematica
    Table[(-1)^n*Sum[Binomial[n, k]*EulerE[2*k], {k, 0, n}]/2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jun 08 2019 *)

Formula

G.f.: 1/Q(0), where Q(k)= 1 + x - x*(2*k+1)*(k+1)/(1 - x*(2*k+1)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ 2^(3*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019
Conjecture: e.g.f. as a continued fraction: 2*exp(-t)/(2 - (1-exp(-4*t))/(2 - (1-exp(-8*t))/(2 - (1-exp(-12*t))/(2 - ... )))) = 1 + t^2/2! + 6*t^3/3! + 73*t^4/4! + .... Cf. A000657 and A005799. - Peter Bala, Dec 21 2019

Extensions

More terms from R. J. Mathar and Emeric Deutsch, Mar 03 2008

A064670 Triangle T(n,k) (1 <= k <= n) where the first column (T(n,1)) is the sequence of secant numbers A000364.

Original entry on oeis.org

1, 1, 4, 5, 20, 36, 61, 244, 504, 576, 1385, 5540, 11916, 17280, 14400, 50521, 202084, 442224, 697536, 792000, 518400, 2702765, 10811060, 23870196, 39202560, 50198400, 47174400, 25401600, 199360981, 797443924, 1769923944
Offset: 1

Views

Author

Jose L. Arregui (arregui(AT)posta.unizar.es), Oct 09 2001

Keywords

Comments

Replacing m*m by m*(m+1) in the formula, the first column gives the tangent numbers A000182.

Examples

			T(4,3) = 9*(T(3,2) + T(3,3)) = 9*(20+36) = 504.

Crossrefs

Cf. A064190, A000182, A000354.

Programs

  • Mathematica
    T[1, 1] = 1; T[n_, k_] /; 1 <= k <= n := T[n, k] = k^2*Sum[T[n-1, j], {j, k-1, n-1}]; T[, ] = 0; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 25 2016 *)

Formula

T(n+1, m) = m*m*Sum_{k = m-1..n} T(n, k) (T(n, 0) = 0).

Extensions

More terms from Philippe Deléham, Sep 22 2005

A081731 a(n) = (A000364(n+2)-A000364(n))/60.

Original entry on oeis.org

1, 23, 841, 45023, 3321841, 323146823, 40078005241, 6172529945423, 1155774491891041, 258569396539480823, 68116719340312179241, 20870735447824438473023, 7358996769608563523829841, 2958635655331631430113322023, 1345380961533244150651061562841, 687031380335435376384294050107823
Offset: 1

Views

Author

Benoit Cloitre, Apr 06 2003

Keywords

Crossrefs

Cf. A000364, A060075.

Programs

  • Mathematica
    a[n_] := (Abs[EulerE[2*n+4]] - Abs[EulerE[2*n]]) / 60; Array[a, 16] (* Amiram Eldar, May 03 2025 *)
  • PARI
    a(n) = (abs(eulerfrac(2*n+4)) - abs(eulerfrac(2*n))) / 60; \\ Amiram Eldar, May 03 2025
Previous Showing 11-20 of 265 results. Next

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

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