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

A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 8, 0, 1, 0, 4, 9, 18, 25, 0, 1, 0, 5, 12, 30, 70, 97, 0, 1, 0, 6, 15, 44, 135, 330, 434, 0, 1, 0, 7, 18, 60, 220, 705, 1694, 2095, 0, 1, 0, 8, 21, 78, 325, 1228, 3906, 9202, 10707, 0, 1, 0, 9, 24, 98, 450, 1905, 7196, 22953, 53334, 58194, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2022

Keywords

Examples

			Square array begins:
  1,  1,  1,   1,   1,   1, ...
  0,  0,  0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5, ...
  0,  3,  6,   9,  12,  15, ...
  0,  8, 18,  30,  44,  60, ...
  0, 25, 70, 135, 220, 325, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.
Column k=9 gives A357667.
Main diagonal gives A357682.
Cf. A292860.

Programs

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

Formula

T(n,k) = Sum_{j=0..floor(n/2)} k^j * Stirling2(n,2*j).
T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.

A357615 Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 3, 9, 30, 135, 705, 3906, 22953, 145053, 985800, 7136613, 54544485, 437961888, 3685605735, 32441696325, 297977767662, 2848636972971, 28278241848309, 290931124989546, 3097051613077269, 34064462020306473, 386600759467746528, 4521440483724439521
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2022

Keywords

Examples

			G.f. = 1 + 3*x^2 + 9*x^3 + 30*x^4 + 135*x^5 + 705*x^6 + ... - _Michael Somos_, Oct 06 2022

Links

Crossrefs

Column k=3 of A357681.
Cf. A065143, A264036.
Cf. A357572.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Cosh[Sqrt[3] * (Exp@x - 1)], {x, 0, n}]]; (* Michael Somos, Oct 06 2022 *)
  • PARI
    a(n) = sum(k=0, n\2, 3^k*stirling(n, 2*k, 2));
  • PARI
    my(x='x+O('x^30)); apply(round, Vec(serlaplace(cosh(sqrt(3) * (exp(x) - 1))))) \\ Michel Marcus, Oct 06 2022
  • PARI
    {a(n) = if(n<0, 0, n!*simplify(polcoeff( cosh(quadgen(12) * (exp(x + x*O(x^n)) - 1)), n)))}; /* Michael Somos, Oct 06 2022 */

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^k * Stirling2(n,2*k).
a(n) = ( Bell_n(sqrt(3)) + Bell_n(-sqrt(3)) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 3 * Sum_{k=0..n-1} binomial(n-1, k) * A357572(k).

A357725 Expansion of e.g.f. cos( sqrt(2) * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -2, -6, -10, 10, 190, 1106, 4438, 9978, -35250, -666622, -5657370, -35308182, -155215970, -128513870, 7051468022, 105057922906, 1042016038254, 8053738122466, 44608555196294, 48639210067658, -3200193654245442, -60669816166988654, -769281697485061994
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=2 of A357728.
Cf. A121867, A264036, A357736.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*(exp(x)-1)))))
  • PARI
    a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(2)*I)+Bell_poly(n, -sqrt(2)*I)))/2;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -2 * Sum_{k=0..n-1} binomial(n-1, k) * A357736(k).
a(n) = ( Bell_n(sqrt(2) * i) + Bell_n(-sqrt(2) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

A357661 Expansion of e.g.f. cosh( (exp(2*x) - 1)/sqrt(2) ).

Original entry on oeis.org

1, 0, 2, 12, 60, 320, 2040, 15568, 133648, 1230336, 11962400, 123144384, 1349008320, 15731096576, 194349866880, 2527082917120, 34392647418112, 488243791183872, 7216792525799936, 110936087161801728, 1771199461131500544, 29324602146652307456
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A024430, A357662, A357663.
Cf. A009153, A264036, A357664.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Cosh[(Exp[2x]-1)/Sqrt[2]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 23 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh((exp(2*x)-1)/sqrt(2)))))
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-k)*stirling(n, 2*k, 2));

Formula

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

A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 9, 27, 144, 945, 6273, 44226, 339399, 2796795, 24387786, 223853355, 2159078445, 21827316888, 230536050165, 2536213188519, 28994911890048, 343806474384045, 4220933769308205, 53566838971016418, 701650841036287275, 9473067208871584407
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2022

Keywords

Links

Crossrefs

Column k=9 of A357681.
Cf. A024430, A065143, A264036, A357615.
Cf. A027710, A357649, A357668.

Programs

  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1))))
  • PARI
    a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 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)))/2;

Formula

E.g.f.: cosh( 3 * (exp(x) - 1) ).
a(n) = Sum_{k=0..floor(n/2)} 9^k * Stirling2(n,2*k).
a(n) = ( Bell_n(3) + Bell_n(-3) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 9 * Sum_{k=0..n-1} binomial(n-1, k) * A357668(k).
Showing 1-7 of 7 results.

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