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 1-7 of 7 results.

A065143 a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.

Original entry on oeis.org

1, 0, 4, 12, 44, 220, 1228, 7196, 45004, 303900, 2201676, 16920860, 136966860, 1163989788, 10364408140, 96463232284, 935872773068, 9440653262620, 98809201693260, 1071131795708188, 12007932126074060
Offset: 0

Views

Author

Karol A. Penson, Oct 17 2001

Keywords

Comments

Stirling transform of A199572 (aerated powers of 4).

Links

Crossrefs

Column k=4 of A357681.
Cf. A199572, A264036, A357598.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n,k]*(1+(-1)^k)*2^k/2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
Table[(BellB[n, 2] + BellB[n, -2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
  • PARI
    a(n) = sum(k=0, n, stirling(n,k,2)*(1+(-1)^k)*2^k/2); \\ Michel Marcus, Nov 02 2015
  • PARI
    x='x+O('x^50); Vec(serlaplace(cosh(2*exp(x)-2))) \\ G. C. Greubel, Nov 16 2017

Formula

Representation as a sum of an infinite series: a(n) = exp(2)*Sum_{k = 0..infinity} ((2*k)^n*2^(2*k)/(2*k)!) - sinh(2)*sum_{k = 0..infinity}(k^n*2^k/k!), for n >= 0.
E.g.f.: cosh(2*exp(x)-2). - Vladeta Jovovic, Sep 14 2003
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n * cosh(2*exp(r)-2) / (r^n * (exp(n) * sqrt(4*exp(2*r)*r^2/n + 1-n+r))), where r is the root of the equation -2*exp(r)*r*tanh(2-2*exp(r)) = n.
(a(n)/n!)^(1/n) ~ exp(1/LambertW(n/2)) / LambertW(n/2).
(End)
a(n) = (Bell_n(2) + Bell_n(-2))/2, where Bell_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Nov 01 2015
a(n) = 1; a(n) = 4 * Sum_{k=0..n-1} binomial(n-1, k) * A357598(k). - Seiichi Manyama, Oct 12 2022

A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].

Original entry on oeis.org

0, 1, 1, 3, 13, 55, 241, 1171, 6357, 37567, 236521, 1574331, 11068333, 82110535, 640794337, 5239439011, 44723250501, 397481121295, 3671081354137, 35176098791115, 349120380267421, 3583273413146647, 37975511840454673, 415004245048757299, 4670891190907818165
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 01 2015

Keywords

Comments

a(n) without the leading zero [1, 1, 3, 13, 55, ...] is the binomial transform of A264036.

Examples

			G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...

Links

Crossrefs

Cf. A077957, A264036.
Cf. A024429, A357572, A357598.

Programs

  • Mathematica
    Table[(BellB[n, Sqrt[2]] - BellB[n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 24}]
  • PARI
    vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k+1).
a(n) = (Bell_n(sqrt(2)) - Bell_n(-sqrt(2)))/(2*sqrt(2)), where Bell_n(x) is n-th Bell polynomial.
Bell_n(sqrt(2)) = A264036(n) + a(n)*sqrt(2).
E.g.f.: sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A264036(k). - Seiichi Manyama, Oct 12 2022

A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

Original entry on oeis.org

0, 1, 1, 4, 19, 85, 406, 2191, 13105, 84190, 573121, 4127521, 31434184, 252388957, 2126998693, 18740283556, 172134162631, 1644920020417, 16324076578870, 167938152551491, 1787952325142341, 19667748794844550, 223217829954224029, 2610546296216999197
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2022

Keywords

Links

Crossrefs

Cf. A024429, A264037, A357598.
Cf. A027710, A357615, A357737.

Programs

  • PARI
    a(n) = sum(k=0, (n-1)\2, 3^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3))-Bell_poly(n, -sqrt(3)))/(2*sqrt(3)));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(sqrt(3)) - Bell_n(-sqrt(3)) )/(2 * sqrt(3)), where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357615(k).

A357666 Expansion of e.g.f. sinh( (exp(4*x) - 1)/2 )/2.

Original entry on oeis.org

0, 1, 4, 20, 160, 1872, 25024, 348224, 5055488, 78571776, 1332573184, 24695206912, 493816963072, 10492449771520, 234399640633344, 5480635606908928, 134015043318054912, 3427700843478056960, 91642829715498336256, 2556218693498006929408
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A024429, A357664, A357665.
Cf. A357598, A357617, A357663.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(4*x)-1)/2)/2)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 4^(n-1-k)*stirling(n, 2*k+1, 2));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 4^(n-1-k) * Stirling2(n,2*k+1).

A357668 Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.

Original entry on oeis.org

0, 1, 1, 10, 55, 307, 2026, 14779, 114157, 933616, 8110261, 74525167, 719925328, 7279859485, 76855303981, 845280487018, 9663800287483, 114601481983855, 1407040763488354, 17856103120048783, 233883061849700137, 3157648445216335528, 43887908697233605489
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2022

Keywords

Links

Crossrefs

Cf. A024429, A264037, A357572, A357598.
Cf. A027710, A356572, A357667.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(3*(exp(x)-1))/3)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 9^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 3)-Bell_poly(n, -3)))/6;

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(3) - Bell_n(-3) )/6, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357667(k).

A357599 Expansion of e.g.f. sinh(2 * log(1+x)) / 2.

Original entry on oeis.org

0, 1, -1, 6, -30, 180, -1260, 10080, -90720, 907200, -9979200, 119750400, -1556755200, 21794572800, -326918592000, 5230697472000, -88921857024000, 1600593426432000, -30411275102208000, 608225502044160000, -12772735542927360000, 281000181944401920000
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2022

Keywords

Crossrefs

Cf. A001710, A133799, A357598.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*log(1+x))/2)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 4^k*stirling(n, 2*k+1, 1));
  • PARI
    a(n) = if(n<3, 0^n-(-1)^n, (-1)^(n+1)*(n+1)!/4);

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling1(n,2*k+1).
a(n) = (-1)^(n+1) * (n+1)!/4 for n > 2.

A357738 Expansion of e.g.f. sin( 2 * (exp(x) - 1) )/2.

Original entry on oeis.org

0, 1, 1, -3, -23, -83, -119, 973, 11145, 69805, 278281, 33165, -12794231, -157150355, -1271714807, -7108146611, -11364216951, 380051588653, 6923479542025, 78935931180813, 669998027706505, 3602978599128301, -8825050911646199, -598024924863875123
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2022

Keywords

Links

Crossrefs

Cf. A357598, A357727.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Sin[2(Exp[x]-1)]/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 19 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sin(2*(exp(x)-1))/2)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, (-4)^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2*I)-Bell_poly(n, -2*I)))/(4*I);

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^(k) * Stirling2(n,2*k+1).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357727(k).
a(n) = ( Bell_n(2 * i) - Bell_n(-2 * i) )/(4 * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
Showing 1-7 of 7 results.

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