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Search: id:A117142
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| A117142 |
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Number of partitions of n in which any two parts differ by at most 2. |
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+0
3
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| 1, 2, 3, 5, 6, 9, 10, 14, 15, 20, 21, 27, 28, 35, 36, 44, 45, 54, 55, 65, 66, 77, 78, 90, 91, 104, 105, 119, 120, 135, 136, 152, 153, 170, 171, 189, 190, 209, 210, 230, 231, 252, 253, 275, 276, 299, 300, 324, 325, 350, 351, 377, 378, 405, 406, 434, 435, 464, 465
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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G:=sum(x^k/[(1-x^k)(1-x^(k+1))(1-x^(k+2))], k=1..infinity). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is sum(x^k/product(1-x^j, j=k..k+b), k=1..infinity).
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EXAMPLE
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a(6)=9 because we have [6],[4,2],[3,3],[3,2,1],[3,1,1,1],[2,2,2],[2,2,1,1],[2,1,1,1,1], and [1,1,1,1,1,1] ([5,1] and [4,1,1] do not qualify).
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MAPLE
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g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(k+2)), k=1..75): gser:=series(g, x=0, 70): seq(coeff(gser, x^n), n=1..65); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]-P[j][1]<3 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions
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CROSSREFS
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Cf. A117143.
Sequence in context: A029451 A029455 A092213 this_sequence A076061 A025523 A128689
Adjacent sequences: A117139 A117140 A117141 this_sequence A117143 A117144 A117145
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
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