OFFSET
0,3
COMMENTS
a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.
From Gus Wiseman, Dec 12 2022: (Start)
Also the number of sequences of compositions (A133494) with weakly decreasing lengths and total sum n. For example, the a(0) = 1 through a(3) = 8 sequences are:
() ((1)) ((2)) ((3))
((11)) ((12))
((1)(1)) ((21))
((111))
((1)(2))
((2)(1))
((11)(1))
((1)(1)(1))
The case of constant lengths is A101509.
The case of strictly decreasing lengths is A129519.
The case of sequences of partitions is A141199.
The case of twice-partitions is A358831.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
G.f.: Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
G.f.: Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
G.f.: exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
G.f.: exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).
G.f.: Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
The g.f. equals the product:
A(x) = (1-x)/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1 + x*(1-x)/((1-x)-x)^2 + x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2 + x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2 + x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...
MAPLE
b:= proc(n) option remember;
add(combinat[numbpart](k)*binomial(n, k), k=0..n)
end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..50); # Alois P. Heinz, Aug 19 2014
MATHEMATICA
Flatten[{1, Table[Sum[Binomial[n-1, k]*PartitionsP[k+1], {k, 0, n-1}], {n, 1, 30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)
PROG
(PARI) {a(n)=sum(k=0, n, (binomial(n, k)-if(n>0, binomial(n-1, k)))*numbpart(k))}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(prod(k=1, n, (1-x)^k/((1-x)^k-X^k)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(sum(m=0, n, x^m*(1-x)^(m*(m-1)/2)/prod(k=1, m, ((1-x)^k - X^k))), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(sum(m=0, n, x^(m^2)*(1-X)^m/prod(k=1, m, ((1-x)^k - x^k)^2)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(exp(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(exp(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m)), n)}
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(prod(k=1, n, (1 + x^k/(1-X)^k)^valuation(2*k, 2)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 29 2012
STATUS
approved