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%I #24 Dec 13 2022 22:57:13
%S 1,1,3,8,21,54,137,344,856,2113,5179,12614,30548,73595,176455,421215,
%T 1001388,2371678,5597245,13166069,30873728,72185937,168313391,
%U 391428622,908058205,2101629502,4853215947,11183551059,25718677187,59030344851,135237134812,309274516740
%N First differences of the binomial transform of the partition numbers (A000041).
%C a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.
%C From _Gus Wiseman_, Dec 12 2022: (Start)
%C 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:
%C () ((1)) ((2)) ((3))
%C ((11)) ((12))
%C ((1)(1)) ((21))
%C ((111))
%C ((1)(2))
%C ((2)(1))
%C ((11)(1))
%C ((1)(1)(1))
%C The case of constant lengths is A101509.
%C The case of strictly decreasing lengths is A129519.
%C The case of sequences of partitions is A141199.
%C The case of twice-partitions is A358831.
%C (End)
%H Alois P. Heinz, <a href="/A218482/b218482.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
%F G.f.: Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
%F G.f.: Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
%F G.f.: exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
%F 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).
%F 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.
%F a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - _Vaclav Kotesovec_, Jun 25 2015
%e G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
%e The g.f. equals the product:
%e 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) *...
%e and also equals the series:
%e 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 +...
%p b:= proc(n) option remember;
%p add(combinat[numbpart](k)*binomial(n,k), k=0..n)
%p end:
%p a:= n-> b(n)-b(n-1):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 19 2014
%t Flatten[{1, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* _Vaclav Kotesovec_, Jun 25 2015 *)
%o (PARI) {a(n)=sum(k=0,n,(binomial(n,k)-if(n>0,binomial(n-1,k)))*numbpart(k))}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) {a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1-x)^k/((1-x)^k-X^k)),n)}
%o (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)}
%o (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)}
%o (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)}
%o (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)}
%o (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)}
%Y Cf. A000041, A000219, A011782, A055887, A063834, A075900, A098407, A101509, A103446, A129519, A141199, A218481.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 29 2012
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