A218153 -id:A218153 - cp's OEIS frontend

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756

Offset: 0

N. J. A. Sloane, Dec 23 2003

nonn

Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008

Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016

			From _Gus Wiseman_, Oct 22 2018: (Start)
The a(6) = 22 set multipartitions of integer partitions of 6:
  (6)  (15)    (123)      (12)(12)      (1)(1)(1)(12)    (1)(1)(1)(1)(1)(1)
       (24)    (1)(14)    (1)(1)(13)    (1)(1)(1)(1)(2)
       (1)(5)  (1)(23)    (1)(2)(12)
       (2)(4)  (2)(13)    (1)(1)(1)(3)
       (3)(3)  (3)(12)    (1)(1)(2)(2)
               (1)(1)(4)
               (1)(2)(3)
               (2)(2)(2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row sums of A285229 and of A360763.

Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.

with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008
# second Maple program:
with(numtheory):
b:= proc(n, i)
      if n<0 or n>i*(i+1)/2 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n,i):= b(n-i, i-1) +b(n, i-1)
      fi
    end:
a:= proc(n) option remember; `if` (n=0, 1,
       add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 11 2011

max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)
b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018

A218552 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)A(x^k)^n) ).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 46, 107, 253, 604, 1463, 3573, 8812, 21901, 54837, 138145, 350068, 891529, 2281092, 5860471, 15113614, 39109461, 101521521, 264286160, 689820642, 1804890193, 4733051924, 12437565725, 32746931264, 86375236835, 228212881032, 603915863737, 1600500761487

Offset: 0

Paul D. Hanna, Nov 01 2012

nonn

Compare to the dual g.f. G(x) of A219232:

G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^n)^k) ).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 107*x^7 +...
where
log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
x^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
x^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
x^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 117*x^6/6 + 295*x^7/7 + 755*x^8/8 + 1933*x^9/9 + 5048*x^10/10 +...

Cf. A219232, A218551, A218153, A218576.

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m/m*prod(k=1,n\m+1,1+x^(m*k)*subst(A,x,x^k +x*O(x^n))^m))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)(1 + x^k)^n) ).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 25, 44, 79, 137, 237, 408, 689, 1162, 1946, 3231, 5342, 8776, 14340, 23326, 37758, 60847, 97670, 156145, 248697, 394719, 624343, 984360, 1547187, 2424581, 3788730, 5904230, 9176723, 14226914, 22002523, 33947526, 52258177, 80268131, 123028407

Offset: 0

Paul D. Hanna, Nov 02 2012

nonn

Compare to the dual g.f. of A219229:

exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 25*x^6 + 44*x^7 +...
where
log(A(x)) = x/1*((1+x*(1+x))*(1+x^2*(1+x^2))*(1+x^3*(1+x^3))*...) +
x^2/2*((1+x^2*(1+x)^2)*(1+x^4*(1+x^2)^2)*(1+x^6*(1+x^3)^2)*...) +
x^3/3*((1+x^3*(1+x)^3)*(1+x^6*(1+x^2)^3)*(1+x^9*(1+x^3)^3)*...) +
x^4/4*((1+x^4*(1+x)^4)*(1+x^8*(1+x^2)^4)*(1+x^12*(1+x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 11*x^4/4 + 26*x^5/5 + 39*x^6/6 + 57*x^7/7 + 99*x^8/8 + 142*x^9/9 + 208*x^10/10 +...

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from Paul D. Hanna)

Cf. A219229, A218575, A218552, A218153.

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,(1+x^(m*k)*(1+x^k+x*O(x^n))^m )))),n)}
for(n=0,50,print1(a(n),", "))

Conjecture: a(n) ~ c * d^n, where d = A060006 = 1.3247179572447... is the real root of the equation d*(d^2-1) = 1 and c = 43328430766.390... . - Vaclav Kotesovec, Apr 09 2016

A219231 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(nk)A(x^n)^k) ).

Original entry on oeis.org

1, 1, 2, 5, 15, 47, 160, 554, 1987, 7243, 26873, 100930, 383412, 1469673, 5679033, 22095308, 86489211, 340360513, 1345814572, 5344184197, 21303295069, 85216434084, 341960332173, 1376212103798, 5553269024152, 22463340663474, 91071265881382, 369996643180885, 1506118767637576

Offset: 0

Paul D. Hanna, Nov 16 2012

nonn

Compare to the dual g.f. G(x) of A218551:

G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*G(x^k)^n) ).

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 47*x^5 + 160*x^6 + 554*x^7 +...
where
log(A(x)) = x/(1*(1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)*...) +
x^2/(2*(1-x^2*A(x^2))*(1-x^4*A(x^2)^2)*(1-x^6*A(x^2)^3)*...) +
x^3/(3*(1-x^3*A(x^3))*(1-x^6*A(x^3)^2)*(1-x^9*A(x^3)^3)*...) +
x^4/(4*(1-x^4*A(x^4))*(1-x^8*A(x^4)^2)*(1-x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 39*x^4/4 + 146*x^5/5 + 594*x^6/6 + 2346*x^7/7 + 9543*x^8/8 + 38710*x^9/9 + 158448*x^10/10 +...

Cf. A218551, A219230, A219229, A219232, A218153.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*prod(k=1, n\m+1, 1/(1-x^(m*k)*subst(A, x, x^m +x*O(x^n))^k))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A219232 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)A(x^n)^k) ).

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 92, 284, 918, 3005, 10043, 33943, 116138, 400862, 1395228, 4889389, 17240482, 61117789, 217709832, 778841527, 2797066886, 10080379573, 36444817306, 132147553180, 480444008087, 1751033068088, 6396352141777, 23414462628460, 85878613308907, 315556155264918

Offset: 0

Paul D. Hanna, Nov 16 2012

nonn

Compare to the dual g.f. G(x) of A218552:

G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^k)^n) ).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 30*x^5 + 92*x^6 + 284*x^7 +...
where
log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)^3)*...) +
x^2/2*((1+x^2*A(x^2))*(1+x^4*A(x^2)^2)*(1+x^6*A(x^2)^3)*...) +
x^3/3*((1+x^3*A(x^3))*(1+x^6*A(x^3)^2)*(1+x^9*A(x^3)^3)*...) +
x^4/4*((1+x^4*A(x^4))*(1+x^8*A(x^4)^2)*(1+x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 86*x^5/5 + 321*x^6/6 + 1128*x^7/7 + 4163*x^8/8 + 15172*x^9/9 + 56078*x^10/10 +...

Cf. A218552, A219229, A219231, A219230, A218153, A218153.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A, x, x^m +x*O(x^n))^k)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A219260 G.f. satisfies: A(x) = exp(Sum_{n>=1} x^nA(x)^n/n Product_{k>=1} (1 + x^(nk)A(x^k)^n)).

Original entry on oeis.org

1, 1, 3, 10, 37, 143, 576, 2393, 10178, 44133, 194341, 866867, 3908454, 17784385, 81562890, 376637216, 1749704080, 8171760933, 38346326963, 180707395127, 854850922373, 4057990958069, 19324260613400, 92288612451684, 441919933724974, 2121281845071105, 10205443975074195

Offset: 0

Paul D. Hanna, Nov 16 2012

nonn

Compare to the dual g.f. G(x) of A219261:

G(x) = exp(Sum_{n>=1} x^n*G(x^n)/n * Product_{k>=1} (1+x^(n*k)*G(x^n)^k)).

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 576*x^6 + 2393*x^7 +...
where
log(A(x)) = x*A(x)/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
x^2*A(x)^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
x^3*A(x)^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
x^4*A(x)^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 461*x^5/5 + 2144*x^6/6 + 10109*x^7/7 + 48117*x^8/8 + 230998*x^9/9 + 1115875*x^10/10 +...

Cf. A218552, A219262, A219261, A218153.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A219261 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^nA(x^n)/n Product_{k>=1} (1 + x^(nk)A(x^n)^k) ).

Original entry on oeis.org

1, 1, 3, 9, 33, 124, 503, 2089, 8960, 39142, 173978, 783347, 3567123, 16395199, 75966835, 354447193, 1663921966, 7853325055, 37244059607, 177388171005, 848148206917, 4069483589180, 19588001935380, 94559416543623, 457697632011720, 2220852281129195, 10800560004895426

Offset: 0

Paul D. Hanna, Nov 16 2012

nonn

Compare to the dual g.f. G(x) of A219260:

G(x) = exp( Sum_{n>=1} x^n*G(x)^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^k)^n) ).

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 124*x^5 + 503*x^6 + 2089*x^7 +...
where
log(A(x)) = x*A(x)/1*((1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)^3)*...) +
x^2*A(x^2)/2*((1+x^2*A(x^2))*(1+x^4*A(x^2)^2)*(1+x^6*A(x^2)^3)*...) +
x^3*A(x^3)/3*((1+x^3*A(x^3))*(1+x^6*A(x^3)^2)*(1+x^9*A(x^3)^3)*...) +
x^4*A(x^4)/4*((1+x^4*A(x^4))*(1+x^8*A(x^4)^2)*(1+x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 396*x^5/5 + 1895*x^6/6 + 8989*x^7/7 + 43545*x^8/8 + 211645*x^9/9 + 1036560*x^10/10 +...

Cf. A219232, A219263, A219260, A218153.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*subst(A, x, x^m +x*O(x^n))/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A^k, x, x^m +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

cp's OEIS Frontend

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

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A218552 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n) ).

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A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).

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A219231 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^n)^k) ).

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A219232 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).

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A219260 G.f. satisfies: A(x) = exp(Sum_{n>=1} x^n*A(x)^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n)).

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A219261 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x^n)/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).

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A218552 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)A(x^k)^n) ).

A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)(1 + x^k)^n) ).

A219231 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(nk)A(x^n)^k) ).

A219232 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)A(x^n)^k) ).

A219260 G.f. satisfies: A(x) = exp(Sum_{n>=1} x^nA(x)^n/n Product_{k>=1} (1 + x^(nk)A(x^k)^n)).

A219261 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^nA(x^n)/n Product_{k>=1} (1 + x^(nk)A(x^n)^k) ).