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A156289 Triangle read by rows: T(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.

%I A156289 #45 Feb 03 2025 22:53:09
%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: T(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 T(n,k) is the number of partitions of a set of size 2*n into k blocks of even size [Comtet]. For partitions into odd sized blocks see A136630.
%C A156289 See A241171 for the triangle of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. - _Peter Bala_, Aug 20 2014
%C A156289 This triangle T(n,k) gives the sum over the M_3 multinomials A036040 for the partitions of 2*n with k even parts, for 1 <= k <= n. See the triangle A257490 with sums over the entries with k parts, and the _Hartmut F. W. Hoft_ program. - _Wolfdieter Lang_, May 13 2015
%D A156289 L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
%D A156289 L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.
%H A156289 Michael De Vlieger, <a href="/A156289/b156289.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)
%H A156289 Richard Olu Awonusika, <a href="https://doi.org/10.1007/s41478-020-00272-8">On Jacobi polynomials P_k(alpha, beta) and coefficients c_j^L(alpha, beta) (k >= 0, L = 5,6; 1 <= j <= L; alpha, beta > -1)</a>, The Journal of Analysis (2020).
%H A156289 Thomas Browning, <a href="https://arxiv.org/abs/2010.13256">Counting Parabolic Double Cosets in Symmetric Groups</a>, arXiv:2010.13256 [math.CO], 2020.
%H A156289 J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>
%F A156289 Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=n<k; T(n,k) = (2*k-1)*T(n-1,k-1) + k^2*T(n-1,k) 1<k<=n.
%F A156289 Generating function for the k-th column of the triangle T(i+k,k):
%F A156289 G(k,x) = Sum_{i>=0} T(i+k,k)*x^i = Product_{j=1..k} (2*j-1)/(1-j^2*x).
%F A156289 Closed form expression: T(n,k) = (2/(k!*2^k))*Sum_{j=1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).
%F A156289 From _Peter Bala_, Feb 21 2011: (Start)
%F A156289 GENERATING FUNCTION
%F A156289 E.g.f. (including a constant 1):
%F A156289 (1)... F(x,z) = exp(x*(cosh(z)-1))
%F A156289 = Sum_{n>=0} R(n,x)*z^(2*n)/(2*n)!
%F A156289 = 1 + x*z^2/2! + (x + 3*x^2)*z^4/4! + (x + 15*x^2 + 15*x^3)*z^6/6! + ....
%F A156289 ROW POLYNOMIALS
%F A156289 The row polynomials R(n,x) begin
%F A156289 ... R(1,x) = x
%F A156289 ... R(2,x) = x + 3*x^2
%F A156289 ... R(3,x) = x + 15*x^2 + 15*x^3.
%F A156289 The egf F(x,z) satisfies the partial differential equation
%F A156289 (2)... d^2/dz^2(F) = x*F + x*(2*x+1)*F' + x^2*F'',
%F A156289 where ' denotes differentiation with respect to x. Hence the row polynomials satisfy the recurrence relation
%F A156289 (3)... R(n+1,x) = x*{R(n,x) + (2*x+1)*R'(n,x) + x*R''(n,x)}
%F A156289 with R(0,x) = 1. The recurrence relation for T(n,k) given above follows from this.
%F A156289 (4)... T(n,k) = (2*k-1)!!*A036969(n,k).
%F A156289 (End)
%e A156289 The triangle begins
%e A156289   n\k|..1.....2......3......4......5......6
%e A156289   =========================================
%e A156289   .1.|..1
%e A156289   .2.|..1.....3
%e A156289   .3.|..1....15.....15
%e A156289   .4.|..1....63....210....105
%e A156289   .5.|..1...255...2205...3150....945
%e A156289   .6.|..1..1023..21120..65835..51975..10395
%e A156289   ..
%e A156289 T(3,3) = 15. The 15 partitions of the set [6] into three even blocks are:
%e A156289   (12)(34)(56), (12)(35)(46), (12)(36)(45),
%e A156289   (13)(24)(56), (13)(25)(46), (13)(26)(45),
%e A156289   (14)(23)(56), (14)(25)(36), (14)(26)(35),
%e A156289   (15)(23)(46), (15)(24)(36), (15)(26)(34),
%e A156289   (16)(23)(45), (16)(24)(35), (16)(25)(34).
%e A156289 Examples of recurrence relation
%e A156289  T(4,3) = 5*T(3,2) + 9*T(3,3) = 5*15 + 9*15 = 210;
%e A156289  T(6,5) = 9*T(5,4) + 25*T(5,5) = 9*3150 + 25*945 = 51975.
%e A156289  T(4,2) = 28 + 35 = 63 (M_3 multinomials A036040 for partitions of 8 with 3 even parts, namely (2,6) and (4^2)). - _Wolfdieter Lang_, May 13 2015
%p A156289 T := proc(n,k) option remember; `if`(k = 0 and n = 0, 1, `if`(n < 0, 0,
%p A156289 (2*k-1)*T(n-1, k-1) + k^2*T(n-1, k))) end:
%p A156289 for n from 1 to 8 do seq(T(n,k), k=1..n) od; # _Peter Luschny_, Sep 04 2017
%t A156289 T[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}]]
%t A156289 (* alternate computation with function triangle[] defined in A257490 *)
%t A156289 a[n_]:=Map[Apply[Plus,#]&,triangle[n],{2}]
%t A156289 (* _Hartmut F. W. Hoft_, Apr 26 2015 *)
%Y A156289 Diagonal T(n, n) is A001147, subdiagonal T(n+1, n) is A001880.
%Y A156289 2nd column variant T(n, 2)/3, for 2<=n, is A002450.
%Y A156289 3rd column variant T(n, 3)/15, for 3<=n, is A002451.
%Y A156289 Sum of the n-th row is A005046.
%Y A156289 Cf. A241171, A257468, A257490, A096162.
%K A156289 easy,nonn,tabl
%O A156289 1,3
%A A156289 _Hartmut F. W. Hoft_, Feb 07 2009