%I A156289
%S A156289 1,1,3,1,15,15,1,63,210,105,1,255,2205,3150,945,1,1023,21120,65835,
%T A156289 51975,10395,1,4095,195195,1201200,1891890,945945,135135,1,16383,
%U A156289 1777230,20585565,58108050,54864810,18918900,2027025,1,65535,16076985
%N A156289 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
%C A156289 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.
%F A156289 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.
%F A156289 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).
%F A156289 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
%e A156289 w(2,2)=15, w(4,4)=105, w(5,3)=2205, w(6,2)=1023
%t A156289 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}]]
%Y A156289 diagonal w(n, n) is A001147, sub-diagonal w(n+1, n) is A001880,
%Y A156289 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.
%Y A156289 sum of the n-th row is A005046
%Y A156289 Sequence in context: A014621 A144006 A113378 this_sequence A095922 A089278 
               A087071
%Y A156289 Adjacent sequences: A156286 A156287 A156288 this_sequence A156290 A156291 
               A156292
%K A156289 easy,nonn,tabl
%O A156289 1,3
%A A156289 Hartmut F. W. Hoeft (hhoft(AT)emich.edu), Feb 07 2009