A185895 -id:A185895 - cp's OEIS frontend

A007837 Number of partitions of n-set with distinct block sizes.

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847

Offset: 0

Arnold Knopfmacher

nonn

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..700
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011

nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)

{my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Christian G. Bower

a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A345762 E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).

Original entry on oeis.org

1, -1, -1, 2, 0, 29, -135, 727, -1967, -6074, 94510, 1548051, -41361089, 408842095, 213929807, -41951737904, 130060640466, 10569226878107, -229371598130229, 3327344803563111, -31418096993670379, -383829978086171112, 17799865170898698140, 220582224147105677385

Offset: 0

Seiichi Manyama, Jun 26 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A028343, A087906, A185895, A209902, A330199, A345758, A346039, A346057.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-x^k)^(1/k!))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, (exp(x^k)-1)/k))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d-1)!)*x^k/k))))

a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, sumdiv(k, d, 1/(d-1)!)*a(n-k)/(n-k)!));

E.g.f.: exp( -Sum_{k>=1} (exp(x^k) - 1)/k ).
E.g.f.: exp( -Sum_{k>=1} A087906(k)*x^k/k! ).
a(n) = -(n-1)! * Sum_{k=1..n} (Sum_{d|k} 1/(d-1)!) * a(n-k)/(n-k)! for n > 0.

A294495 E.g.f.: Product_{k>0} (1-x^k/k!)^k.

Original entry on oeis.org

1, -1, -2, 3, 14, 45, -156, -1225, -3396, -105, 226760, 1175229, 4084200, -35683219, -585896962, -3512021955, -14398868176, 198247498911, 3131185307832, 29821940715413, 122481857683680, -1187008881417051, -31616420134183522, -419944298964036771

Offset: 0

Seiichi Manyama, Nov 01 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..470

Cf. A185895.

Cf. A032315, A174661, A294494.

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1-x^k/k!)^k)))

A319218 Expansion of e.g.f. Product_{k>=1} (1 - x^k/(k - 1)!).

Original entry on oeis.org

1, -1, -2, 3, 8, 75, -216, -175, -3816, -36225, 189800, 325149, 2375460, 25547951, 386162126, -3290670825, -6316583056, -59290501809, -310987223208, -4836373835707, -86500419684420, 1119358992256239, 3043733432729198, 26408738842522959, 169835931388147464

Offset: 0

Ilya Gutkovskiy, Sep 13 2018

sign

Cf. A032299, A076900, A185895, A319219.

seq(n!*coeff(series(mul((1 - x^k/(k - 1)!),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019

nmax = 24; CoefficientList[Series[Product[(1 - x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[-Sum[Sum[x^(j k)/(k (j - 1)!^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[-d (d - 1)!^(-k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*((j - 1)!)^k)).

A346314 Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / (n!)^2).

Original entry on oeis.org

1, -1, -1, 8, 15, 124, -3340, -9311, -102641, -1880812, 150047424, 692058289, 8916106452, 167039809897, 7435628931289, -1381243302601067, -9407162843960561, -165954439670564988, -3103870029424074136, -123659189880256295879, -10671656695397289496160

Offset: 0

Ilya Gutkovskiy, Jul 13 2021

sign

Cf. A183229, A183240, A185895, A346315.

nmax = 20; CoefficientList[Series[Product[(1 - x^k/(k!)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[(Binomial[n, k] k!)^2 k Sum[1/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

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

A371551 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^2.

Original entry on oeis.org

1, -2, 0, 10, -4, -42, -258, 306, 5980, 3142, 61730, -794334, -3299074, -8459830, 40220390, 1550926110, 1631691740, 43693916390, -125593997262, -4079362135854, -32054212967294, -33715330874838, -600410923342450, 9383532800084966, 329821022627776798

Offset: 0

Ilya Gutkovskiy, Mar 27 2024

sign

Cf. A002107, A032312, A185895, A371552.

nmax = 24; CoefficientList[Series[Product[(1 - x^k/k!)^2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A371552 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^3.

Original entry on oeis.org

1, -3, 3, 18, -57, -138, 246, 4281, 13383, -156906, -450822, -957729, 23375886, 289894875, -179027895, -3403581357, -174968380137, -419588974650, 4439383168602, 50400469832883, 1027067921064738, 428364930324489, -18456487538087145, -1019962180000311267

Offset: 0

Ilya Gutkovskiy, Mar 27 2024

sign

Cf. A010816, A032314, A185895, A371551.

nmax = 23; CoefficientList[Series[Product[(1 - x^k/k!)^3, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A345759 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k / k!).

Original entry on oeis.org

1, -1, -2, -2, 7, 78, 513, 2665, 9406, -13902, -789143, -11806456, -140040408, -1463842226, -13377115923, -95264642343, -198034245627, 11021440199748, 322964047973519, 6617250866231379, 118668721540190350, 1965786734149801960, 30348547043773563767

Offset: 0

Seiichi Manyama, Jun 26 2021

sign

Stirling transform of A185895.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A048993, A140585, A185895, A305547.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k/k!)))

a(n) = Sum_{k=0..n} Stirling2(n,k) * A185895(k).

cp's OEIS Frontend

A007837 Number of partitions of n-set with distinct block sizes.

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A345762 E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).

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A294495 E.g.f.: Product_{k>0} (1-x^k/k!)^k.

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A319218 Expansion of e.g.f. Product_{k>=1} (1 - x^k/(k - 1)!).

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A346314 Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / (n!)^2).

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A371551 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^2.

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Mathematica

A371552 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^3.

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A345759 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k / k!).

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