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A077802
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Sum of products of parts increased by 1 in hook partitions of n; where hook partitions are of the form h*1^(n-h).
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 11 2009]
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FORMULA
| 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). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 05 2002
Row sums of triangle A132048. Equals binomial transform of [1, 1, 4, 2, 4, 2, 4, 2, 4,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 08 2007
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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.
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MATHEMATICA
| s=0; lst={1}; Do[s+=(s-n); AppendTo[lst, Abs[s]], {n, 2, 4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]
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CROSSREFS
| Cf. A074141, A055010(first differences), A042950(second differences).
Cf. A132048.
Same as A095151 except for a(0). [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 11 2009]
Sequence in context: A054111 A192955 A055503 * A095151 A147611 A007991
Adjacent sequences: A077799 A077800 A077801 * A077803 A077804 A077805
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KEYWORD
| easy,nonn
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), Dec 02 2002
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EXTENSIONS
| More terms from John W. Layman (layman(AT)math.vt.edu), Dec 05 2002
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