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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.
18

%I #38 Feb 18 2021 00:28:18

%S 1,1,3,1,15,15,1,63,210,105,1,255,2205,3150,945,1,1023,21120,65835,

%T 51975,10395,1,4095,195195,1201200,1891890,945945,135135,1,16383,

%U 1777230,20585565,58108050,54864810,18918900,2027025,1,65535,16076985

%N 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 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 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 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 L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.

%H Michael De Vlieger, <a href="/A156289/b156289.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)

%H 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 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 J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>

%F Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=n<k;

%F T(n,k) = (2k-1)*T(n-1,k-1) + k^2*T(n-1,k) 1<k<=n.

%F Generating function for the k-th column of the triangle T(i+k,k):

%F G(k,x) = Sum(i=0,Infinity; T(i+k,k)*x^i) = Product(j=1,k; (2j-1)/(1-j^2*x).

%F Closed form expression for T(n,k):

%F T(n,k) = 2/(k!*2^k)*sum {j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).

%F From _Peter Bala_, Feb 21 2011: (Start)

%F GENERATING FUNCTION

%F E.g.f. (including a constant 1):

%F (1)... F(x,z) = exp(x*(cosh(z)-1)

%F = sum{n>=0} R(n,x)*z^(2*n)/(2*n)!

%F = 1 + x*z^2/2! + (x + 3*x^2)*z^4/4! + (x + 15*x^2 + 15*x^3)*z^6/6 + ....

%F ROW POLYNOMIALS

%F The row polynomials R(n,x) begin

%F ... R(1,x) = x

%F ... R(2,x) = x + 3*x^2

%F ... R(3,x) = x + 15*x^2 + 15*x^3.

%F The egf F(x,z) satisfies the partial differential equation

%F (2)... d^2/dz^2(F) = x*F + x*(2*x+1)*F' + x^2*F'',

%F where ' denotes differentiation with respect to x. Hence the row polynomials satisfy the recurrence relation

%F (3)... R(n+1,x) = x*{R(n,x) + (2*x+1)*R'(n,x) + x*R''(n,x)}

%F with R(0,x) = 1. The recurrence relation for T(n,k) given above follows from this.

%F (4)... T(n,k) = (2*k-1)!!*A036969(n,k).

%F (End)

%e The triangle begins

%e n\k|..1.....2......3......4......5......6

%e =========================================

%e .1.|..1

%e .2.|..1.....3

%e .3.|..1....15.....15

%e .4.|..1....63....210....105

%e .5.|..1...255...2205...3150....945

%e .6.|..1..1023..21120..65835..51975..10395

%e ..

%e T(3,3) = 15. The 15 partitions of the set [6] into three even blocks are:

%e (12)(34)(56), (12)(35)(46), (12)(36)(45),

%e (13)(24)(56), (13)(25)(46), (13)(26)(45),

%e (14)(23)(56), (14)(25)(36), (14)(26)(35),

%e (15)(23)(46), (15)(24)(36), (15)(26)(34),

%e (16)(23)(45), (16)(24)(35), (16)(25)(34).

%e Examples of recurrence relation

%e T(4,3) = 5*T(3,2) + 9*T(3,3) = 5*15 + 9*15 = 210;

%e T(6,5) = 9*T(5,4) + 25*T(5,5) = 9*3150 + 25*945 = 51975.

%e 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 T := proc(n,k) option remember; `if`(k = 0 and n = 0, 1, `if`(n < 0, 0,

%p (2*k-1)*T(n-1, k-1) + k^2*T(n-1, k))) end:

%p for n from 1 to 8 do seq(T(n,k), k=1..n) od; # _Peter Luschny_, Sep 04 2017

%t 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 (* alternate computation with function triangle[] defined in A257490 *)

%t a[n_]:=Map[Apply[Plus,#]&,triangle[n],{2}]

%t (* _Hartmut F. W. Hoft_, Apr 26 2015 *)

%Y Diagonal T(n, n) is A001147, subdiagonal T(n+1, n) is A001880.

%Y 2nd column variant T(n, 2)/3, for 2<=n, is A002450.

%Y 3rd column variant T(n, 3)/15, for 3<=n, is A002451.

%Y Sum of the n-th row is A005046.

%Y Cf. A241171, A257468, A257490, A096162.

%K easy,nonn,tabl

%O 1,3

%A _Hartmut F. W. Hoft_, Feb 07 2009