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 1-8 of 8 results.

A081562 Binomial transform of expansion of exp(2cosh(x)), A000807.

Original entry on oeis.org

1, 1, 3, 7, 27, 91, 423, 1807, 9747, 49651, 303183, 1777447, 12072987, 79587691, 593485623, 4327497727, 35069154147, 279393234211, 2440577314143, 21043100301847, 196825339400427, 1822706292362011, 18153886768953543
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Links

Crossrefs

Cf. A000807, A081563.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(2*Cosh(x)+x-2) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 13 2019
  • Maple
    seq(coeff(series(exp(2*cosh(x)+x-2), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019
  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[2 Cosh[x] + x - 2], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(2*cosh(x)+x-2) )) \\ G. C. Greubel, Aug 13 2019
  • Sage
    [factorial(n)*( exp(2*cosh(x)+x-2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 13 2019

Formula

E.g.f.: exp(x)+exp(2*cosh(x))/e^2 = exp(2*cosh(x)+x-2).

A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)*Bell(k)*Bell(n-k).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 6, 6, 5, 15, 20, 24, 20, 15, 52, 75, 100, 100, 75, 52, 203, 312, 450, 500, 450, 312, 203, 877, 1421, 2184, 2625, 2625, 2184, 1421, 877, 4140, 7016, 11368, 14560, 15750, 14560, 11368, 7016, 4140, 21147, 37260, 63144, 85260, 98280, 98280, 85260, 63144, 37260, 21147
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			   1;
   1,  1;
   2,  2,   2;
   5,  6,   6,   5;
  15, 20,  24,  20, 15;
  52, 75, 100, 100, 75, 52;
  ...

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 80.

Crossrefs

Cf. A000110, row sums give A001861.
Columns include A000110 and A052889.
Cf. A000807.

Programs

  • Maple
    A033306 := proc(n,k)
    if k < 0 or k > n then
        0;
    else
        binomial(n,k)*combinat[bell](k)*combinat[bell](n-k) ;
    end if;
end proc: # R. J. Mathar, Mar 21 2013
# second Maple program:
b:= proc(n) option remember; expand(`if`(n>0, add(
     (x^j+1)*b(n-j)*binomial(n-1, j-1), j=1..n), 1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 30 2019
  • Mathematica
    t[n_, k_] := Binomial[n, k] * BellB[k] * BellB[n-k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Formula

E.g.f.: exp(exp(x*y)+exp(x)-2).
Sum_{k=0..2n} (-1)^k * T(2n,k) = A000807(n). - Alois P. Heinz, Feb 13 2024

Extensions

Edited by Vladeta Jovovic, Sep 17 2003

A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

Original entry on oeis.org

1, 2, 4, 10, 32, 114, 448, 1978, 9472, 48738, 270336, 1595114, 9965568, 65852882, 457326592, 3329243546, 25356271616, 201326396098, 1663597019136, 14279558011850, 127044810702848, 1170023757062450, 11136610150121472, 109395885009537402, 1107781178494025728, 11549900930966957346
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2022

Keywords

Links

Crossrefs

Cf. A000557, A000807, A001861.
Cf. A003724, A136630, A352280.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[2 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k * Binomial[n, k] * BellB[k, -1] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 27 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

Formula

E.g.f.: exp( 2 * sinh(x) ).
a(n) = Sum_{k=0..n} 2^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A080527 Expansion of e.g.f. exp(3*cosh(x))/e^3 (even powers only).

Original entry on oeis.org

1, 3, 30, 543, 14745, 546618, 26119965, 1547467743, 110507211390, 9310580616243, 910032696123645, 101790848712790218, 12883623878563854105, 1827803943114479006043, 288318381606931126782270, 50215020818975432279332743, 9597691024295026236008687265
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Periodic zeros suppressed.

Examples

			exp(3*cosh(x))/exp(3) = 1 + 3*x^2/2! + 30*x^4/4! + ...

Crossrefs

Cf. A000807, A005046, A081565.

Programs

  • Mathematica
    With[{nn=30},Take[CoefficientList[Series[Exp[3Cosh[x]]/E^3,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Dec 15 2013 *)

Formula

E.g.f.: exp(3*cosh(x)-3).
a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * a(n-k). - Ilya Gutkovskiy, Mar 10 2022

A081563 Second binomial transform of expansion of exp(2*cosh(x)).

Original entry on oeis.org

1, 2, 6, 20, 78, 332, 1566, 7940, 43518, 253532, 1573566, 10295540, 71069598, 513897932, 3893187486, 30741656420, 252979075518, 2161184079932, 19161309456126, 175782239098580, 1667967153565278, 16331180476591532
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Binomial transform of A081562.

Links

Crossrefs

Cf. A000807, A081562.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(2*Cosh(x)+2*x-2) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 13 2019
  • Maple
    seq(coeff(series(exp(2*cosh(x)+2*x-2), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019
  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[2 Cosh[x] + 2 x - 2], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(2*cosh(x)+2*x-2) )) \\ G. C. Greubel, Aug 13 2019
  • Sage
    [factorial(n)*( exp(2*cosh(x)+2*x-2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 13 2019

Formula

E.g.f.: exp(2*x) * exp(2*cosh(x))/e^2 = exp(2*cosh(x)+2*x-2).

A352254 Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).

Original entry on oeis.org

1, 1, 5, 48, 753, 16880, 507579, 19509042, 927229553, 53126200872, 3597373129635, 283321938437318, 25614466939850169, 2629191169850594388, 303549146372282854883, 39103024746814973908890, 5581172267077778765676129, 877211696663645448333041072, 151002471269513108372760683523
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A000248, A000807, A003724, A003727, A005046, A009233, A351937, A352253.

Programs

  • Mathematica
    nmax = 36; Take[CoefficientList[Series[Exp[x Sinh[x]/2], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * k * a(n-k).

A352624 Expansion of e.g.f. exp(exp(x) + cosh(x) - 2).

Original entry on oeis.org

1, 1, 3, 8, 31, 122, 579, 2886, 16139, 95358, 611111, 4128830, 29709695, 224400022, 1785322699, 14841968646, 129015458195, 1167021383902, 10979895178511, 107113768171950, 1082508179141031, 11308614423992102, 121995294474174963, 1356835055606851286, 15542964081299602811
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2022

Keywords

Links

Crossrefs

Cf. A000110, A000807, A001861, A003724, A005046, A352327, A352617.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(2-(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 24 2022
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[Exp[x] + Cosh[x] - 2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (3 + (-1)^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 + (-1)^k) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A000110(n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A000807(k) * A003724(n-2*k).

A352277 a(0) = 1; a(n) = -2 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * a(n-k).

Original entry on oeis.org

1, -2, 10, -62, 250, 3538, -109430, 376738, 64406170, -1496149262, -66387156950, 4120939699138, 114360544465210, -16447057086702062, -315993884108535350, 99921676927889325538, 1478937314465295441370, -907773678752741550637262, -14225447208333541085396630
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2022

Keywords

Crossrefs

Cf. A000807, A210657, A260884, A352278.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = -2 Sum[Binomial[2 n - 1, 2 k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 36; Take[CoefficientList[Series[Exp[2 (1 - Cosh[x])], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]

Formula

E.g.f.: exp( 2 * (1 - cosh(x)) ) (even powers only).
Showing 1-8 of 8 results.

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

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