A024429 -id:A024429 - cp's OEIS frontend

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A096648 Number of partitions of an n-set with odd number of even blocks.

0, 1, 3, 7, 25, 106, 434, 2045, 10707, 57781, 338195, 2115664, 13796952, 95394573, 692462671, 5235101739, 41436754261, 341177640610, 2915100624274, 25866987547865, 237448494222575, 2252995117706961, 22078799199129799, 222971522853648704, 2319210969809731600

Offset: 1

Vladeta Jovovic, Aug 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A000701.

Cf. A024429, A024430, A096647.

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,
      irem(t+`if`(irem(i, 2)=0, j, 0), 2)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015

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, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, Mod[t+If[Mod[i, 2] == 0, j, 0], 2]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[ a[n], {n, 1, 30}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)
With[{nn=30},Rest[CoefficientList[Series[Exp[Sinh[x]]Sinh[Cosh[x]-1], {x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Sep 03 2016 *)

E.g.f.: exp(sinh(x))*sinh(cosh(x)-1).
a(2*n) = A024429(2*n) and a(2*n+1) = A024430(2*n+1). - Jonathan Vos Post, Oct 19 2005
a(n) = sum{k=0..n, if(mod(n-k,2)=1, A048993(n,k), 0)}. - Paul Barry, May 19 2006

More terms from Emeric Deutsch, Nov 16 2004

A121870 Monthly Problem 10791, second expression.

Original entry on oeis.org

1, 1, 2, 9, 61, 554, 6565, 96677, 1716730, 36072181, 881242577, 24674241834, 783024550969, 27896201305769, 1106485798248706, 48517267642373105, 2337333266369553253, 123040664089658462650, 7043260281573138384701, 436533086101058798529933

Offset: 0

N. J. A. Sloane, Sep 05 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..325
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A024429, A024430, A121867, A121868, A121869.

List([0..25], n-> (Sum([0..Int(n/2)], k-> Stirling2(n,2*k)*(-1)^(k)) )^2 + (Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k+1)))^2 ); # G. C. Greubel, Oct 08 2019

C:= ComplexField(); a:= func< n | Round(Abs( (&+[I^k*StirlingSecond(n,k): k in [0..n]])^2 )) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

Maple

A121870a:= proc(a) local i, t: i:=1: t:=0: for i to 100 do t:=t+evalf((i^(a-1))*(I)^i/(i)!): od: RETURN(round(abs(t^2))): end: a:= A121870a(n); # Russell Walsmith, Apr 18 2008 # Alternate: seq(abs(BellB(n,I))^2, n=0..30); # Robert Israel, Oct 15 2017

Mathematica

Table[Abs[BellB[n, I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)

PARI

a(n) = abs( (sum(k=0,n, I^k*stirling(n,k,2)))^2 ); vector(25, n, a(n-1)) \\ G. C. Greubel, Oct 08 2019

Sage

[abs( sum(I^k*stirling_number2(n,k) for k in (0..n))^2 ) for n in (0..25)] # G. C. Greubel, Oct 08 2019

Formula

a(n) = A121867(n)^2 + A121868(n)^2.

From Gary W. Adamson, Jul 22 2011: (Start)

sqrt(a(n)) = upper left term in M^n as to the modulus of a polar term; M = an infinite square production matrix in which a column of (i, i, i, ...) is appended to the right of Pascal's triangle, as follows (with i = sqrt(-1)):

1, i, 0, 0, 0, ...

1, 1, i, 0, 0, ...

1, 2, 1, i, 0, ...

1, 3, 3, 1, i, ...

... (End)

a(n) = |B_n(i)|^2, where B_n(x) is the n-th Bell polynomial, i = sqrt(-1) is the imaginary unit. - Vladimir Reshetnikov, Oct 15 2017

a(n) ~ (n*exp(-1 + Re(LambertW(i*n)) / Abs(LambertW(i*n))^2) / Abs(LambertW(i*n)))^(2*n) / Abs(1 + LambertW(i*n)), where i is the imaginary unit. - Vaclav Kotesovec, Jul 28 2021

A330021 Expansion of e.g.f. exp(sinh(exp(x) - 1)).

Original entry on oeis.org

1, 1, 2, 6, 25, 128, 754, 5001, 37048, 303930, 2732395, 26657106, 280039786, 3149224991, 37729906686, 479570263690, 6442902231289, 91186621152460, 1355582225366134, 21112253012491481, 343672026658191836, 5834977672879651390, 103130592695715620419

Offset: 0

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Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

nonn

Comments

Stirling transform of A003724.

Exponential transform of A024429.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..485

Crossrefs

Cf. A003724, A009218, A011800, A024429, A080831.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add( binomial(n-1, j-1)*irem(j, 2)*g(n-j), j=1..n)) end: b:= proc(n, m) option remember; `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=0..22); # Alois P. Heinz, Jun 23 2023

Mathematica

nmax = 22; CoefficientList[Series[Exp[Sinh[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A003724(k).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A024429(k) * a(n-k).

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset: 0

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Author

Gus Wiseman, Aug 01 2021

Keywords

nonn

Examples

The a(0) = 1 through a(7) = 14 partitions: (1) (3) (5) (7) (9) (11) (13) (15) (4,2,1) (4,3,2) (5,4,2) (6,4,3) (6,5,4) (5,3,1) (6,3,2) (6,5,2) (7,5,3) (6,2,1) (6,4,1) (7,4,2) (7,6,2) (7,3,1) (7,5,1) (8,4,3) (8,2,1) (8,3,2) (8,5,2) (8,4,1) (8,6,1) (9,3,1) (9,4,2) (10,2,1) (9,5,1) (10,3,2) (10,4,1) (11,3,1) (12,2,1) (5,4,3,2,1)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Odd bisection of A067659, which is ranked by A030059.

The even version is the even bisection of A067661.

The case of all odd parts is counted by A069911 (non-strict: A078408).

The non-strict version is A160786, ranked by A340931.

The non-strict even version is A236913, ranked by A340784.

The even-length version is A343942 (non-strict: A236914).

The even-sum version is A344650 (non-strict: A236559 or A344611).

A000009 counts partitions with all odd parts, ranked by A066208.

A000009 counts strict partitions, ranked by A005117.

A027193 counts odd-length partitions, ranked by A026424.

A027193 counts odd-maximum partitions, ranked by A244991.

A058695 counts partitions of odd numbers, ranked by A300063.

A340385 counts partitions with odd length and maximum, ranked by A340386.

Other cases of odd length:

- A024429 set partitions

- A089677 ordered set partitions

- A166444 compositions

- A174726 ordered factorizations

- A332304 strict compositions

- A339890 factorizations

Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1)))) end: a:= n-> b(2*n+1$2, 0): seq(a(n), n=0..80); # Alois P. Heinz, Aug 05 2021

Mathematica

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]

Extensions

More terms from Alois P. Heinz, Aug 05 2021

A357668 Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.

Original entry on oeis.org

0, 1, 1, 10, 55, 307, 2026, 14779, 114157, 933616, 8110261, 74525167, 719925328, 7279859485, 76855303981, 845280487018, 9663800287483, 114601481983855, 1407040763488354, 17856103120048783, 233883061849700137, 3157648445216335528, 43887908697233605489

Offset: 0

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Author

Seiichi Manyama, Oct 08 2022

Keywords

nonn

Links

Seiichi Manyama, Table of n, a(n) for n = 0..547

Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Cf. A024429, A264037, A357572, A357598.

Cf. A027710, A356572, A357667.

Programs

PARI

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(3*(exp(x)-1))/3)))

PARI

a(n) = sum(k=0, (n-1)\2, 9^k*stirling(n, 2*k+1, 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)))/6;

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * Stirling2(n,2*k+1).

a(n) = ( Bell_n(3) - Bell_n(-3) )/6, where Bell_n(x) is n-th Bell polynomial.

a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357667(k).

A296543 Expansion of e.g.f. tanh(exp(x)-1).

Original entry on oeis.org

0, 1, 1, -1, -11, -33, 61, 1367, 7253, -12561, -580499, -4701497, 4669765, 580325215, 6636339165, 1365901495, -1122870368715, -17289945450289, -31110588453299, 3713822629274023, 74717183313957413, 280555705771423039, -19253195126787261507, -496715617694137066089, -3008746115751273626347

Offset: 0

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Author

Ilya Gutkovskiy, Dec 15 2017

Keywords

sign

Examples

tanh(exp(x)-1) = x/1! + x^2/2! - x^3/3! - 11*x^4/4! - 33*x^5/5! + 61*x^6/6! + 1367*x^7/7! + ...

Crossrefs

Cf. A009752, A024429, A024430, A080832, A296544.

Programs

Maple

a:=series(tanh(exp(x)-1),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019

Mathematica

nmax = 24; CoefficientList[Series[Tanh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]! nmax = 24; CoefficientList[Series[Sinh[Exp[x] - 1]/Cosh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]! nmax = 24; CoefficientList[Series[(Exp[x] - 1)/(1 + ContinuedFractionK[(Exp[x] - 1)^2, 2 k - 1, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

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

E.g.f.: (exp(x)-1)/(1 + (exp(x)-1)^2/(3 + (exp(x)-1)^2/(5 + (exp(x)-1)^2/(7 + (exp(x)-1)^2/(9 + ...))))), a continued fraction.

A296545 Expansion of e.g.f. arcsinh(exp(x)-1).

Original entry on oeis.org

0, 1, 1, 0, -5, -15, 46, 735, 2185, -33390, -453479, -364155, 57806200, 681966285, -3289884779, -197798065920, -1815938249585, 33917006295885, 1155429901407646, 5691720408045315, -408736165211351795, -10271257189100959590, 23948813753053818421, 6626731340918542069425, 124356774945741129842320

Offset: 0

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Author

Ilya Gutkovskiy, Dec 15 2017

Keywords

sign

Examples

arcsinh(exp(x)-1) = x/1! + x^2/2! - 5*x^4/4! - 15*x^5/5! + 46*x^6/6! + 735*x^7/7! + ...

Links

Robert Israel, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A014307, A024429, A121868.

Programs

Maple

S:= series(arcsinh(exp(x)-1),x,41): seq(coeff(S,x,j)*j!,j=0..40); # Robert Israel, Dec 17 2017

Mathematica

nmax = 24; CoefficientList[Series[ArcSinh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]! nmax = 24; CoefficientList[Series[-Log[1 - Exp[x] + Sqrt[1 + (1 - Exp[x])^2]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: -log(1 - exp(x) + sqrt(1 + (1 - exp(x))^2)).

A341448 Heinz numbers of integer partitions of type OO.

Original entry on oeis.org

6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240

Offset: 1

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Author

Gus Wiseman, Feb 15 2021

Keywords

nonn

Comments

These partitions are defined to have an odd number of odd parts and an odd number of even parts. They also have even length and odd sum.

Examples

The sequence of partitions together with their Heinz numbers begins: 6: (2,1) 74: (12,1) 141: (15,2) 14: (4,1) 77: (5,4) 142: (20,1) 15: (3,2) 86: (14,1) 143: (6,5) 24: (2,1,1,1) 93: (11,2) 145: (10,3) 26: (6,1) 95: (8,3) 150: (3,3,2,1) 33: (5,2) 96: (2,1,1,1,1,1) 152: (8,1,1,1) 35: (4,3) 104: (6,1,1,1) 158: (22,1) 38: (8,1) 106: (16,1) 161: (9,4) 51: (7,2) 119: (7,4) 177: (17,2) 54: (2,2,2,1) 122: (18,1) 178: (24,1) 56: (4,1,1,1) 123: (13,2) 185: (12,3) 58: (10,1) 126: (4,2,2,1) 201: (19,2) 60: (3,2,1,1) 132: (5,2,1,1) 202: (26,1) 65: (6,3) 135: (3,2,2,2) 204: (7,2,1,1) 69: (9,2) 140: (4,3,1,1) 209: (8,5)

Crossrefs

Note: A-numbers of ranking sequences are in parentheses below.

The case of odd parts, length, and sum is counted by A078408 (A300272).

The type EE version is A236913 (A340784).

These partitions (for odd n) are counted by A236914.

A000009 counts partitions into odd parts (A066208).

A026804 counts partitions whose least part is odd (A340932).

A027193 counts partitions of odd length/maximum (A026424/A244991).

A058695 counts partitions of odd numbers (A300063).

A160786 counts odd-length partitions of odd numbers (A340931).

A340101 counts factorizations into odd factors.

A340385 counts partitions of odd length and maximum (A340386).

A340601 counts partitions of even rank (A340602).

A340692 counts partitions of odd rank (A340603).

Cf. A000700, A024429, A027187, A106529, A117409, A174725, A257541, A325134, A339890, A340102, A340604.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; Select[Range[100],OddQ[Count[primeMS[#],?EvenQ]]&&OddQ[Count[primeMS[#],?OddQ]]&]

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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A096648 Number of partitions of an n-set with odd number of even blocks.

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Maple

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A121870 Monthly Problem 10791, second expression.

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GAP

Magma

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Sage

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A330021 Expansion of e.g.f. exp(sinh(exp(x) - 1)).

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Maple

Mathematica

Formula

A346634 Number of strict odd-length integer partitions of 2n + 1.

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Examples

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Crossrefs

Programs

Maple

Mathematica

Extensions

A357668 Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.

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Crossrefs

Programs

PARI

PARI

PARI

Formula

A296543 Expansion of e.g.f. tanh(exp(x)-1).

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Examples

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Programs

Maple

Mathematica

Formula

A296545 Expansion of e.g.f. arcsinh(exp(x)-1).

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Maple

Mathematica

Formula