A005046 -id:A005046 - cp's OEIS frontend

A115278 Number of partitions of {1,...,2*n} into even sized blocks such that no block size is repeated.

1, 1, 1, 16, 29, 256, 14422, 49141, 490429, 10758400, 1797335306, 9458619391, 133756636598, 2528529510391, 137864810180749, 53441183229799381, 410251032050409469, 7615997734377068128, 167055180095977694194, 6741819165851219788075, 738863335901972011745434

Offset: 0

Christian G. Bower, Jan 18 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..370

Cf. A005046, A007837, A032310, A115277.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!* b[n - i*j, i - 2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[2 n, 2 n]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

E.g.f.: B(x) of b(n) where b(2*n)=a(n), b(2*n+1)=0. B(x)=Product {m >= 1} (1+x^(2*m)/(2*m)!).

A275679 Number of set partitions of [n] with alternating block size parities.

Original entry on oeis.org

1, 1, 1, 4, 3, 20, 43, 136, 711, 1606, 12653, 36852, 250673, 1212498, 6016715, 45081688, 196537387, 1789229594, 8963510621, 76863454428, 512264745473, 3744799424978, 32870550965259, 219159966518160, 2257073412153459, 15778075163815474, 165231652982941085

Offset: 0

Alois P. Heinz, Aug 05 2016

nonn

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 3: 1234, 1|23|4, 1|24|3.
a(5) = 20: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 14|2|35, 15|2|34.

Alois P. Heinz, Table of n, a(n) for n = 0..592

Cf. A003724, A005046, A007837, A038041, A275309, A275310, A275311, A275312, A275313, A286076, A361804.

b:= proc(n, t) option remember; `if`(n=0, 1, add(
      `if`((i+t)::odd, b(n-i, 1-t)*binomial(n-1, i-1), 0), i=1..n))
    end:
a:= n-> `if`(n=0, 1, b(n, 0)+b(n, 1)):
seq(a(n), n=0..35);

b[n_, t_] := b[n, t] = If[n==0, 1, Sum[If[OddQ[i+t], b[n-i, 1-t] * Binomial[n-1, i-1], 0], {i, 1, n}]]; a[n_] := If[n==0, 1, b[n, 0] + b[n, 1]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

A327000 A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 6, 3, 9, 26, 10, 117, 68, 100, 35, 2574, 4500, 517, 365, 126, 70005, 748616, 199155, 4163, 1302, 462, 2082759, 192426260, 282846568, 10499643, 36180, 4606, 1716, 65061234, 59688349943, 799156187475, 141482705378, 663488532, 341733, 16284

Offset: 0

Peter Luschny, Aug 12 2019

			Array starts:
n\k [  3    4        5            6                 7 ]
[0]    1,   6,       26,          100,              365, ...            [A125107]
[1]    1,   9,       68,          517,              4163, ...           [A048742]
[2]    3,   117,     4500,        199155,           10499643, ...       [A326995]
[3]    10,  2574,    748616,      282846568,        141482705378, ...   [A327002]
[4]    35,  70005,   192426260,   799156187475,     4961959681629275, ...
[5]    126, 2082759, 59688349943, 3097220486457142, 278271624962638244163, ...
   A001700,

Cf. A327001, A309522, A125107, A048742, A001700, A291973, A326999, A326995, A327002.

ListTools:-Flatten([seq(seq(A309522(n-k, k) - A327001(n-k, k), k=3..n), n=3..10)]);

The columns for k = 0, 1, 2 are suppressed as they are identical 0.
A(0, k) = A000108(k) - A011782(k).
A(1, k) = A000142(k) - A000110(k).
A(2, k) = A002105(k) - A005046(k-1) for k >= 1.
A(3, k) = A018893(k) - A291973(k).
A(4, k) = A326999(k) - A291975(k).

A331611 E.g.f.: exp(1 / (2 - cosh(x)) - 1) (even powers only).

Original entry on oeis.org

1, 1, 10, 241, 10585, 732826, 73233205, 9955632961, 1764233731270, 394629336427021, 108652463882802505, 36084903957564392206, 14217903951354603567385, 6554505383225768210009041, 3493988190176442653240091010, 2131975894217009666242489287001

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A005046, A050351, A075729, A094088, A331608, A331612.

nmax = 15; Table[(CoefficientList[Series[Exp[1/(2 - Cosh[x]) - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A094088[0] = 1; A094088[n_] := A094088[n] = Sum[Binomial[2 n, 2 k] A094088[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] A094088[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * A094088(k) * a(n-k).

a(n) ~ 2^(2*n + 1/4) * exp(1/(2*sqrt(3)*log(2 + sqrt(3))) - 2/3 + sqrt(8*n/log(2 + sqrt(3)))/3^(1/4) - 2*n) * n^(2*n - 1/4) / (3^(1/8) * log(2 + sqrt(3))^(2*n + 1/4)). - Vaclav Kotesovec, Jan 26 2020

A081559 Expansion of e.g.f.: exp(cosh(2*x)-1), even powers only.

Original entry on oeis.org

1, 4, 64, 1984, 97024, 6713344, 615829504, 71654785024, 10243143368704, 1755968011239424, 354197952894337024, 82788022987201183744, 22140953727834378993664, 6703959915806302859689984, 2277487386474356139699994624, 861378969099073547571187154944

Offset: 0

Paul Barry, Mar 22 2003

Periodic zeros suppressed.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A005046, A081560.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Cosh(2*x)-1) )); [Factorial(2*n-2)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Aug 13 2019

seq(coeff(series(exp(cosh(2*x)-1), x, 2*n+1)*factorial(2*n), x, 2*n), n = 0 .. 15); # G. C. Greubel, Aug 13 2019

With[{nn = 30}, CoefficientList[Series[Exp[Cosh[2*x]-1], {x, 0, nn}], x] Range[0, nn]!][[1 ;; ;; 2]] (* G. C. Greubel, Aug 13 2019 *)

my(x='x+O('x^30)); v=Vec(serlaplace( exp(cosh(2*x)-1) )); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, Aug 13 2019

[factorial(2*n)*( exp(cosh(2*x)-1) ).series(x, 2*n+1).list()[2*n] for n in (0..15)] # G. C. Greubel, Aug 13 2019

E.g.f.: exp(cosh(2*x))/e = exp(cosh(2*x)-1).

Definition amended by Georg Fischer, Dec 03 2021

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)

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).

A047689 Numerators of coefficients in Taylor series for exp(cos(x)-1).

Original entry on oeis.org

1, -1, 1, -31, 379, -1639, 150349, -4373461, 39074491, -6698486371, 30708135509, -4934550701809, 1319703681935929, -99896787342523081, 2121075416425574701, -72929243565543262421, 83877585692383961052499, -2410048463569672917849859

Offset: 0

N. J. A. Sloane

			1 - (1/2)*x^2 + (1/6)*x^4 - (31/720)*x^6 + (379/40320)*x^8 - (1639/907200)*x^10 + ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

Cf. A047690, A005046.

A047690 Denominators of coefficients in Taylor series for exp(cos(x)-1).

Original entry on oeis.org

1, 2, 6, 720, 40320, 907200, 479001600, 87178291200, 5230697472000, 6402373705728000, 221172909834240000, 281000181944401920000, 620448401733239439360000, 403291461126605635584000000, 76222086152928465125376000000, 24113896346562823512391680000000

Offset: 0

N. J. A. Sloane

			1 - (1/2)*x^2 + (1/6)*x^4 - (31/720)*x^6 + (379/40320)*x^8 - (1639/907200)*x^10 + ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

Cf. A047689, A005046.

Cf. A097322.

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

Paul Barry, Mar 22 2003

Periodic zeros suppressed.

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

Cf. A000807, A005046, A081565.

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 *)

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

cp's OEIS Frontend

A115278 Number of partitions of {1,...,2*n} into even sized blocks such that no block size is repeated.

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A275679 Number of set partitions of [n] with alternating block size parities.

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A327000 A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.

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A331611 E.g.f.: exp(1 / (2 - cosh(x)) - 1) (even powers only).

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Mathematica

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A081559 Expansion of e.g.f.: exp(cosh(2*x)-1), even powers only.

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

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Maple

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A047689 Numerators of coefficients in Taylor series for exp(cos(x)-1).

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