A218482 First differences of the binomial transform of the partition numbers (A000041).

1, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812, 309274516740

Offset: 0

Author

Paul D. Hanna, Oct 29 2012

Keywords

• nonn

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)

Examples

			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 +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A000219, A011782, A055887, A063834, A075900, A098407, A101509, A103446, A129519, A141199, A218481.

Programs

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 *)

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)}

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