A003724 -id:A003724 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Mar 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..576

Cf. A000557, A000807, A001861.

Cf. A003724, A136630, A352280.

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 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

E.g.f.: exp( 2 * sinh(x) ).

a(n) = Sum_{k=0..n} 2^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A050332 Number of factorizations of n into distinct numbers with an odd number of prime factors.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A026424, A045778, A050333. a(p^k)=A000700. a(A002110)=A003724.

Dirichlet g.f.: Product_{n is in A026424}(1+1/n^s).

a(n) = A050333(A101296(n)). - R. J. Mathar, May 26 2017

A124322 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 even size (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 3, 12, 25, 15, 37, 91, 60, 15, 128, 329, 315, 105, 457, 1415, 1533, 630, 105, 1872, 6297, 7623, 4410, 945, 8169, 29431, 42150, 27405, 7875, 945, 37600, 151085, 233475, 176715, 69300, 10395, 188685, 802099, 1365243, 1199220, 533610

Offset: 0

Emeric Deutsch, Oct 28 2006

Row n has 1+floor(n/2) terms. Sum of row n is the Bell number B(n)=A000110(n). Sum_{k=0..floor(n/2)} k*T(n,k) = A102287(n). T(n,0)=A003724(n).

			T(4,1) = 7 because we have 1234, 14|2|3, 1|24|3, 1|2|34, 13|2|4, 1|23|4 and 12|3|4.
Triangle starts:
   1;
   1;
   1,  1;
   2,  3;
   5,  7,  3;
  12, 25, 15;
  37, 91, 60, 15;
  ...

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

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000110, A102287, A003724, A124321.

G:=exp(sinh(z)+t*(cosh(z)-1)): Gser:=simplify(series(G,z=0,16)): for n from 0 to 13 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor(n/2)) 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)=0, 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

nn = 10; Range[0, nn]! CoefficientList[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 28 2012*)

E.g.f.: exp[sinh(z)+t(cosh(z)-1)].

A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

Original entry on oeis.org

1, 1, 3, 8, 28, 107, 459, 2151, 10931, 59700, 348146, 2155925, 14112377, 97266301, 703484851, 5323515156, 42040470092, 345670438963, 2953171501547, 26166317121747, 240047041176843, 2276607815242880, 22290187889601330, 225018607554567149, 2339331996135377345

Offset: 1

Geoffrey Critzer, Apr 02 2013

nonn

			a(4) = 8 because we have: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3},{4}}.

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A000110, A003724, A005046, A102286, A124322, A131719, A283424.

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)=0, x^j, 1), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*(i+1), i=0..degree(p)))(b(n-1$2)):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 08 2015
# second Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, t, (m-1)*
      b(n-1, t, m)+b(n-1, 1-t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n-1, 1$2):
seq(a(n), n=1..25);  # Alois P. Heinz, May 17 2023

nn=25;Drop[Range[0,nn]!CoefficientList[Series[Integrate[Exp[Cosh[x]-1]D[ Exp[Sinh[x]],x],x],{x,0,nn}],x],1]

E.g.f. A(x) satisfies: A'(x) = B'(x)*C(x) where B(x) is the e.g.f. for A003724 and C(x) is the e.g.f. for A005046.
a(n) = Sum_{k=0..floor((n-1)/2)} (k+1)*A124322(n-1,k). - Alois P. Heinz, Apr 02 2013
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022
From Alois P. Heinz, May 17 2023: (Start)
a(n) = Sum_{k=0..n-1} (-1)^k * A283424(n-1,k).
a(n) mod 2 = A131719(n+1). (End)

A293022 a(n) = n! * [x^n] exp(n*sinh(x)).

Original entry on oeis.org

1, 1, 4, 30, 320, 4380, 73152, 1443008, 32837632, 846829008, 24406425600, 777438110240, 27122380259328, 1028481186011968, 42119646804852736, 1852697321086064640, 87113569172046479360, 4360314205952755126528, 231474221309788621111296, 12990330117363343153415680

Offset: 0

Ilya Gutkovskiy, Sep 28 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..380

Cf. A003724, A134200.

N:= 30: # to get a(0)..a(N)
S:= series(exp(n*sinh(x)),x,N+1):
seq(n!*coeff(S,x,n),n=0..N); # Robert Israel, Sep 29 2017

Table[n! SeriesCoefficient[Exp[n Sinh[x]], {x, 0, n}], {n, 0, 19}]

A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3k + 1) / (3k + 1)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 37, 114, 478, 1907, 6777, 28414, 148579, 758916, 3580189, 18981485, 117883917, 720627553, 4193077474, 26795418840, 191751387094, 1352954503595, 9301704998742, 69285817230370, 559142785301527, 4453089770243547, 35182348161102172

Offset: 0

Ilya Gutkovskiy, Apr 08 2020

nonn

Number of partitions of n-set into blocks congruent to 1 mod 3.

Cf. A000110, A003724, A193437, A306347, A333882, A333883.

nmax = 27; CoefficientList[Series[Exp[Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 3]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]
nmax = 30; CoefficientList[Series[Exp[Exp[x]/3 - 2*Sin[Pi/6 - Sqrt[3]*x/2] / (3*Exp[x/2])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

E.g.f.: exp(exp(x)/3 - 2*sin(Pi/6 - sqrt(3)*x/2) / (3*exp(x/2))). - Vaclav Kotesovec, Apr 15 2020

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(n-3*k-1). - Seiichi Manyama, Sep 22 2023

A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2sqrt(x)) - BesselJ(0,2sqrt(x))) / 2 ).

Original entry on oeis.org

1, 1, 2, 7, 40, 321, 3356, 45123, 752256, 15018433, 355378732, 9823042923, 311510611072, 11242338245009, 458052976883672, 20851748359005567, 1054108827258438656, 58860837547461314049, 3606677286494115444812, 241397002229305033296603, 17579096976247770110062080

Offset: 0

Ilya Gutkovskiy, Jul 11 2021

nonn

Cf. A003724, A023998.

nmax = 20; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[x]] - BesselJ[0, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, 2 k + 1]^2 (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 20}]

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

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

A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).

Original entry on oeis.org

1, 3, 9, 30, 117, 516, 2493, 13152, 75177, 460272, 3003921, 20806176, 152114013, 1169842368, 9435180357, 79553524224, 699531782481, 6400932102912, 60820145019801, 599036357936640, 6105903392066373, 64309189153428480, 698936466350352717, 7828833281592926208, 90270159223293364473

Offset: 0

Ilya Gutkovskiy, Mar 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..565

Cf. A027710, A080527, A107403.

Cf. A003724, A136630, A352279.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Exp[3 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

E.g.f.: exp( 3 * sinh(x) ).

a(n) = Sum_{k=0..n} 3^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A352617 Expansion of e.g.f. exp( exp(x) + sinh(x) - 1 ).

Original entry on oeis.org

1, 2, 5, 16, 60, 254, 1199, 6206, 34827, 210264, 1355992, 9288954, 67279309, 513149498, 4107383185, 34398823888, 300629113292, 2735356900806, 25857446103571, 253472859754918, 2572266378189583, 26981781750668760, 292136508070103208, 3260640536587635410, 37472102225288489529

Offset: 0

Ilya Gutkovskiy, Mar 24 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..565

Cf. A000110, A001861, A003724, A005046, A352279, A352326.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(1+(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 24 2022

nmax = 24; CoefficientList[Series[Exp[Exp[x] + Sinh[x] - 1], {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}]

my(x='x+O('x^30)); Vec(serlaplace(exp( exp(x) + sinh(x) - 1 ))) \\ Michel Marcus, Mar 24 2022

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..n} binomial(n,k) * A000110(k) * A003724(n-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A352279(n-2*k).

A381277 Expansion of e.g.f. exp(sinh(3*x) / 3).

Original entry on oeis.org

1, 1, 1, 10, 37, 172, 1477, 8416, 74377, 683344, 5836969, 67102048, 699721453, 8268521536, 107106298093, 1347611617792, 19462095444241, 279380302430464, 4247519795325649, 68946703997616640, 1122787065355425973, 19697500164381137920, 351304020205694058133

Offset: 0

Seiichi Manyama, Feb 18 2025

nonn

Cf. A003724, A009229.

Cf. A136630.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, 3^(n-k)*a136630(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * binomial(n-1,2*k) * a(n-2*k-1).

a(n) = Sum_{k=0..n} 3^(n-k) * A136630(n,k).

cp's OEIS Frontend

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

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A050332 Number of factorizations of n into distinct numbers with an odd number of prime factors.

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A124322 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 even size (0<=k<=floor(n/2)).

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A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

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Maple

Mathematica

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A293022 a(n) = n! * [x^n] exp(n*sinh(x)).

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Maple

Mathematica

A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3*k + 1) / (3*k + 1)!).

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A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))) / 2 ).

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A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

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A352617 Expansion of e.g.f. exp( exp(x) + sinh(x) - 1 ).

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

A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3k + 1) / (3k + 1)!).

A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2sqrt(x)) - BesselJ(0,2sqrt(x))) / 2 ).

A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).