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A156289
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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.
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18
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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
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OFFSET
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1,3
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COMMENTS
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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.
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
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
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REFERENCES
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L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.
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LINKS
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FORMULA
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Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=n<k;
T(n,k) = (2k-1)*T(n-1,k-1) + k^2*T(n-1,k) 1<k<=n.
Generating function for the k-th column of the triangle T(i+k,k):
G(k,x) = Sum(i=0,Infinity; T(i+k,k)*x^i) = Product(j=1,k; (2j-1)/(1-j^2*x).
Closed form expression for T(n,k):
T(n,k) = 2/(k!*2^k)*sum {j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).
GENERATING FUNCTION
E.g.f. (including a constant 1):
(1)... F(x,z) = exp(x*(cosh(z)-1)
= sum{n>=0} R(n,x)*z^(2*n)/(2*n)!
= 1 + x*z^2/2! + (x + 3*x^2)*z^4/4! + (x + 15*x^2 + 15*x^3)*z^6/6 + ....
ROW POLYNOMIALS
The row polynomials R(n,x) begin
... R(1,x) = x
... R(2,x) = x + 3*x^2
... R(3,x) = x + 15*x^2 + 15*x^3.
The egf F(x,z) satisfies the partial differential equation
(2)... d^2/dz^2(F) = x*F + x*(2*x+1)*F' + x^2*F'',
where ' denotes differentiation with respect to x. Hence the row polynomials satisfy the recurrence relation
(3)... R(n+1,x) = x*{R(n,x) + (2*x+1)*R'(n,x) + x*R''(n,x)}
with R(0,x) = 1. The recurrence relation for T(n,k) given above follows from this.
(4)... T(n,k) = (2*k-1)!!*A036969(n,k).
(End)
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EXAMPLE
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The triangle begins
n\k|..1.....2......3......4......5......6
=========================================
.1.|..1
.2.|..1.....3
.3.|..1....15.....15
.4.|..1....63....210....105
.5.|..1...255...2205...3150....945
.6.|..1..1023..21120..65835..51975..10395
..
T(3,3) = 15. The 15 partitions of the set [6] into three even blocks are:
(12)(34)(56), (12)(35)(46), (12)(36)(45),
(13)(24)(56), (13)(25)(46), (13)(26)(45),
(14)(23)(56), (14)(25)(36), (14)(26)(35),
(15)(23)(46), (15)(24)(36), (15)(26)(34),
(16)(23)(45), (16)(24)(35), (16)(25)(34).
Examples of recurrence relation
T(4,3) = 5*T(3,2) + 9*T(3,3) = 5*15 + 9*15 = 210;
T(6,5) = 9*T(5,4) + 25*T(5,5) = 9*3150 + 25*945 = 51975.
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
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MAPLE
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T := proc(n, k) option remember; `if`(k = 0 and n = 0, 1, `if`(n < 0, 0,
(2*k-1)*T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 1 to 8 do seq(T(n, k), k=1..n) od; # Peter Luschny, Sep 04 2017
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MATHEMATICA
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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}]]
(* alternate computation with function triangle[] defined in A257490 *)
a[n_]:=Map[Apply[Plus, #]&, triangle[n], {2}]
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CROSSREFS
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2nd column variant T(n, 2)/3, for 2<=n, is A002450.
3rd column variant T(n, 3)/15, for 3<=n, is A002451.
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KEYWORD
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AUTHOR
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STATUS
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approved
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