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A120759 Eigensequence for subpartitions of a partition. 0
1, 2, 5, 24, 527, 271156, 73452582161, 5395271857717411958088, 29108958418479344853405820427519529324955406, 847331460208759521535495911124086692972161538057881358236684093384849875943910959287454 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

Table of n, a(n) for n=0..9.

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).

EXAMPLE

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,...]

PROG

(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)

CROSSREFS

Cf. A115728.

Sequence in context: A025134 A076534 A095708 * A000895 A109306 A009560

Adjacent sequences:  A120756 A120757 A120758 * A120760 A120761 A120762

KEYWORD

eigen,nonn

AUTHOR

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

STATUS

approved

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Last modified May 26 15:29 EDT 2019. Contains 323597 sequences. (Running on oeis4.)