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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A081558 Exponential generating function: exp(cosh(x)+2*x-1).

Original entry on oeis.org

1, 2, 5, 14, 44, 152, 575, 2354, 10379, 48902, 245240, 1301984, 7294589, 42959282, 265263185, 1712168654, 11528506124, 80783015192, 588097479635, 4439382164114, 34699233959759, 280381494540182, 2339287666524440, 20125268756209664, 178348602246900569
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Old definition was "Second binomial transform of expansion of exp(cosh(x))".
Binomial transform of A081557.

Links

Crossrefs

Cf. A005046, A081557.

Programs

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

Extensions

Definition edited by N. J. A. Sloane, Dec 12 2021

A124321 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} (or of any n-set) having k blocks of odd size (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 4, 0, 10, 0, 1, 0, 31, 0, 20, 0, 1, 31, 0, 136, 0, 35, 0, 1, 0, 379, 0, 441, 0, 56, 0, 1, 379, 0, 2500, 0, 1176, 0, 84, 0, 1, 0, 6556, 0, 11740, 0, 2730, 0, 120, 0, 1, 6556, 0, 59671, 0, 43870, 0, 5712, 0, 165, 0, 1, 0, 150349, 0, 378356, 0, 138622, 0
Offset: 0

Views

Author

Emeric Deutsch, Oct 28 2006

Keywords

Comments

Row sums are the Bell numbers (A000110).
Sum_{k=0..n} k*T(n,k) = A102286(n).
T(2*n,0) = A005046(n); T(2*n+1,0) = 0.

Examples

			T(3,1) = 4 because we have 123, 1|23, 12|3 and 13|2.
Triangle starts:
  1;
  0,  1;
  1,  0,  1;
  0,  4,  0,  1;
  4,  0, 10,  0,  1;
  0, 31,  0, 20,  0,  1;

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.

Links

Crossrefs

Cf. A000110, A102286, A005046, A124322.

Programs

  • Maple
    G:=exp(t*sinh(z)+cosh(z)-1): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    nn = 10; Range[0, nn]! CoefficientList[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 28 2012 *)

Formula

E.g.f.: G(t,z) = exp(t*sinh(z)+cosh(z)-1).

A307979 Expansion of e.g.f. exp((cosh(x) - cos(x))/2) (even powers only).

Original entry on oeis.org

1, 1, 3, 16, 133, 1576, 24783, 495496, 12245353, 364768576, 12838252443, 526095538816, 24781014246253, 1326767681420416, 80013978835916583, 5392682199766283776, 403287063337529642833, 33261775377836063850496, 3009257393136250807614003, 297176659119237977183973376
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Number of partitions of a 2n-set into blocks congruent to 2 mod 4.

Crossrefs

Cf. A003724, A005046, A013469, A016825, A035457, A291975, A306347, A307978.

Programs

  • Mathematica
    nmax = 19; Table[(CoefficientList[Series[Exp[(Cosh[x] - Cos[x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[n_] := a[n] = Sum[Boole[MemberQ[{2}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[2 n], {n, 0, 19}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp((cosh(x) - cos(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).

A352466 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2*n,2*k)^3 * k * a(n-k).

Original entry on oeis.org

1, 1, 109, 124876, 704029453, 13294133177626, 665514245564815384, 75462508236267111825685, 17305487139219914670764064013, 7368678746697280907127091048286734, 5449131877967324738667220718996986592734, 6632563741264033978048120096103173533343094035
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A005046, A061684, A352465, A352469.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[2 n, 2 k]^3 k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 11}]
nmax = 22; Take[CoefficientList[Series[Exp[Sum[x^(2 k)/(2 k)!^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3, {1, -1, 2}]

Formula

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^3 = exp( Sum_{n>=1} x^(2*n) / (2*n)!^3 ).

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).

A361804 Number of partitions of [n] with an equal number of even and odd block sizes.

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 45, 63, 1260, 1515, 25515, 104973, 510345, 5679765, 17252235, 263214318, 1207222380, 11863296915, 101718989235, 630468648873, 8281982665215, 48583038314415, 656006633919945, 5122900223419938, 54304561161840825, 605082149235374265
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2023

Keywords

Comments

Half the number of block sizes are even and the other half are odd.

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 0.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 0.
a(5) = 15: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 45: 12|34|5|6, 12|35|4|6, 12|3|45|6, 12|36|4|5, 12|3|46|5, 12|3|4|56, 13|24|5|6, 13|25|4|6, 13|2|45|6, 13|26|4|5, 13|2|46|5, 13|2|4|56, 14|23|5|6, 15|23|4|6, 1|23|45|6, 16|23|4|5, 1|23|46|5, 1|23|4|56, 14|25|3|6, 14|2|35|6, 14|26|3|5, 14|2|36|5, 14|2|3|56, 15|24|3|6, 1|24|35|6, 16|24|3|5, 1|24|36|5, 1|24|3|56, 15|2|34|6, 1|25|34|6, 16|2|34|5, 1|26|34|5, 1|2|34|56, 15|26|3|4, 15|2|36|4, 15|2|3|46, 16|25|3|4, 1|25|36|4, 1|25|3|46, 16|2|35|4, 1|26|35|4, 1|2|35|46, 16|2|3|45, 1|26|3|45, 1|2|36|45.

Links

Crossrefs

Cf. A000110, A003724, A004767, A004773, A005046, A275679.

Programs

  • Maple
    b:= proc(n, x, y) option remember; `if`(abs(x-y)>2*n, 0,
     `if`(n=0, 1, b(n-1, x+1, y)+`if`(x>0, b(n-1, x-1, y+1)*x, 0)+
     `if`(y>0, b(n-1, x+1, y-1)*y, 0)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..33);

Formula

a(n) mod 5 = 3 for n in { A004767 }, a(n) mod 5 = 1 for n = 0 and a(n) mod 5 = 0 for all other n (n in { A004773 } \ { 0 }).
a(n) mod 3 = 0 for n >= 1.

A089004 Number of partitions of an n-element set that have at least one odd block.

Original entry on oeis.org

1, 1, 5, 11, 52, 172, 877, 3761, 21147, 109419, 678570, 4063248, 27644437, 186525861, 1382958545, 10323844183, 82864869804, 675378319788, 5832742205057, 51386368744773, 474869816156751, 4486977535640087
Offset: 1

Views

Author

Vladeta Jovovic, Nov 02 2003

Keywords

Links

Crossrefs

Cf. A000110, A005046, A003724, A086543, A087137.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
       max(t, `if`(j=0, 0, irem(i, 2)))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Cosh[x]-1](Exp[Sinh[x]]-1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 04 2018 *)

Formula

E.g.f.: exp(cosh(x)-1)*(exp(sinh(x))-1).

A089005 Number of partitions of n-set with at least one even block.

Original entry on oeis.org

0, 1, 3, 10, 40, 166, 749, 3683, 19275, 107806, 640970, 4024912, 26653653, 185401581, 1350624721, 10282222002, 81592209580, 673535269054, 5773214891137, 51291776763863, 471617190143567, 4481375500319334, 43947651280912186, 444258975094335440
Offset: 1

Views

Author

Vladeta Jovovic, Nov 02 2003

Keywords

Links

Crossrefs

Cf. A000110, A005046, A003724, A059838, A047967.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
       max(t, `if`(j=0, 0, 1-irem(i, 2)))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i<1, 0, Sum[multinomial[n, {n - i j} ~Join~ Table[i, {j}]]/j! b[n - i j, i - 1, Max[t, If[j == 0, 0, 1 - Mod[i, 2]]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Nov 18 2020, after Maple *)
  • PARI
    my(x='x+O('x^30)); concat(0, Vec(serlaplace(exp(sinh(x))*(exp(cosh(x)-1)-1))))

Formula

E.g.f.: exp(sinh(x))*(exp(cosh(x)-1)-1).

A096619 Number of partitions of a 2*n-element set with exactly two odd blocks.

Original entry on oeis.org

1, 10, 136, 2500, 59671, 1786060, 65222431, 2843052040, 145349748316, 8590361117290, 579887365929301, 44257224641241160, 3785653479578940061, 360188281690273321750, 37868568207290527576096, 4373779619483505303462160, 552095790104596359907313731
Offset: 1

Views

Author

Vladeta Jovovic, Aug 14 2004

Keywords

Links

Crossrefs

Cf. A005046.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<0, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1, t-`if`(irem(i, 2)=1, j, 0) ), j=0..n/i)))
    end:
a:= n-> b(2*n$2, 2):
seq(a(n), n=1..20);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<0, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, t-If[Mod[i, 2] == 1, j, 0]], {j, 0, n/i}]]]; a[n_] := b[2*n, 2*n, 2]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)

Formula

E.g.f.: 1/2*exp(cosh(x)-1)*(sinh(x))^2. More generally, number of partitions of an n-element set with exactly k odd blocks is 1/k!*exp(cosh(x)-1)*(sinh(x))^k.

Extensions

More terms from Emeric Deutsch, Nov 16 2004
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