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

Author

Alford Arnold, Dec 02 2002

Keywords

• easy
• nonn

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

Examples

			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.
		

Links

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

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

Programs

Mathematica

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

PARI

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

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

Extensions

More terms from John W. Layman, Dec 05 2002