A120759 Eigensequence for subpartitions of a partition.

1, 2, 5, 24, 527, 271156, 73452582161, 5395271857717411958088, 29108958418479344853405820427519529324955406, 847331460208759521535495911124086692972161538057881358236684093384849875943910959287454

Offset: 0

Author

Franklin T. Adams-Watters and Paul D. Hanna, Jul 03 2006

Keywords

• eigen
• nonn

Comments

Let this sequence, A, be a partition P=A, then the total number of subpartitions of the partition P is equal to A. See A115728 for the definition of subpartitions of a partition.

Examples

			At n=4, the recurrence gives:
a(4) = a(3)^2 + 1 - Sum_{k=0..2} (-1)^(4-k)*a(k)*C(a(k),4-k)
= a(3)^2 + 1 - [a(0)*C(a(0),4) - a(1)*C(a(1),3) + a(2)*C(a(2),2)]
= 24^2 + 1 - [1*0 - 2*0 + 5*C(5,2)] = 24^2 + 1 - 5*10 = 527.
The recurrence extracts a(n) from the g.f.:
1/(1-x) = 1*(1-x) + 2*x*(1-x)^2 + 5*x^2*(1-x)^5 + 24*x^3*(1-x)^24 +...
+ a(n)*x^n*(1-x)^a(n) +...
The number of digits of a(n) base 10 begins:
[1,1,1,2,3,6,11,22,44,87,174,348,696,1391,...]
		

Crossrefs

Cf. A115728.

Programs

PARI

a(n)=if(n==0,1,a(n-1)^2+1-sum(k=0,n-2,(-1)^(n-k)*a(k)*binomial(a(k),n-k)))

PARI

a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^a(k)), n)

Formula

a(n) = a(n-1)^2 + 1 - Sum_{k=0..n-2} (-1)^(n-k)*a(k)*C(a(k),n-k) for n>=1, with a(0)=1.

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^a(n).