A077802 Sum of products of parts increased by 1 in hook partitions of n, where hook partitions are of the form h*1^(n-h).

1, 2, 7, 18, 41, 88, 183, 374, 757, 1524, 3059, 6130, 12273, 24560, 49135, 98286, 196589, 393196, 786411, 1572842, 3145705, 6291432, 12582887, 25165798, 50331621, 100663268, 201326563, 402653154, 805306337, 1610612704

Offset

0,2

Comments

It is not clear whether a(0) should be 1 or 0; this depends on whether the empty partition is a hook partition. By strict interpretation of the definition above, it is not; and except for n=0, there are exactly n hook partitions for each n. On the other hand, if defined as "a partition in whose Ferrers diagram every point is on the first row or column", the empty partition is a hook partition. - Franklin T. Adams-Watters, Jul 11 2009

Links

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Formula

From Vladeta Jovovic, Dec 05 2002: (Start)

a(n) = 3*2^n - n - 3, n > 0.

G.f.: x*(2-x)/(1-2*x)/(1-x)^2.

Recurrence: a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). (End)

Row sums of triangle A132048. Equals binomial transform of [1, 1, 4, 2, 4, 2, 4, 2, 4, ...]. - Gary W. Adamson, Aug 08 2007

a(n) = A125128(n) + A000225(n), n >= 1. - Miquel Cerda, Aug 07 2016

Example

The hook partitions of 4 are 4, 3+1, 2+1+1, 1+1+1+1; the corresponding products when parts are increased by 1 are 5, 8, 12, 16; and their sum is a(4) = 41.

Mathematica

s=0; lst={1}; Do[s+=(s-n); AppendTo[lst, Abs[s]], {n, 2, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)

Prog

(PARI) a(n)=if(n>0, 3*2^n - n - 3, 1) \\ Charles R Greathouse IV, Aug 08 2016

Crossrefs

Cf. A074141, A055010 (first differences), A042950 (second differences).

Cf. A132048.

Same as A095151 except for a(0). - Franklin T. Adams-Watters, Jul 11 2009

Sequence in context: A295054 A192955 A055503 * A095151 A147611 A007991

Adjacent sequences: A077799 A077800 A077801 * A077803 A077804 A077805

Keyword

easy,nonn

Author

Alford Arnold, Dec 02 2002

Extensions

More terms from John W. Layman, Dec 05 2002

Status

approved