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Search: id:A156289
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| A156289 |
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Triangle read by rows: w(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds |
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+0
2
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| 1, 1, 3, 1, 15, 15, 1, 63, 210, 105, 1, 255, 2205, 3150, 945, 1, 1023, 21120, 65835, 51975, 10395, 1, 4095, 195195, 1201200, 1891890, 945945, 135135, 1, 16383, 1777230, 20585565, 58108050, 54864810, 18918900, 2027025, 1, 65535, 16076985
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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w(n,k) is the number of partitions of 2n into k even sets S1,...,Sk where the first element of Sj is larger than the first element of Si when i<j.
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FORMULA
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recursion: w(n,1)=1 for 1<=n; w(n,k)=0 for 1<=n<k; w(n,k)=(2k-1)w(n-1,k-1)+k^2w(n-1,k) 1<k<=n.
generating function for the k-th column of the triangle w(i+k,k): G(k,x)=Sum(i=0,Infinity; w(i+k,k) *x^i) = Product(j=1,k; (2j-1)/(1-j^2*x).
Closed form expression for w(i+k,k): 2/(2k)!! * Sum(j=1,k; (-1)^(k+j) * Binomial(2k,k+j) * (j^2)^(k+i) where (2)!! = (2k)*(2k-2)*...*2
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EXAMPLE
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w(2,2)=15, w(4,4)=105, w(5,3)=2205, w(6,2)=1023
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MATHEMATICA
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w[n_, k_] := Which[n < k, 0, n == 1, 1, True, 2/Factorial2[2 k] Sum[(-1)^(k + j) Binomial[2 k, k + j] j^(2 n), {j, 1, k}]]
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CROSSREFS
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diagonal w(n, n) is A001147, sub-diagonal w(n+1, n) is A001880,
2-nd column variant w(n, 2)/3, for 2<=n, is A002450, 3-rd column variant w(n, 3)/15, for 3<=n, is A002451.
sum of the n-th row is A005046
Sequence in context: A014621 A144006 A113378 this_sequence A095922 A089278 A087071
Adjacent sequences: A156286 A156287 A156288 this_sequence A156290 A156291 A156292
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Hartmut F. W. Hoeft (hhoft(AT)emich.edu), Feb 07 2009
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