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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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5
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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; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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.
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REFERENCES
| 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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FORMULA
| 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).
From Peter Bala
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 wrt 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
| 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.
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MATHEMATICA
| 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}]]
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CROSSREFS
| Diagonal T(n, n) is A001147, sub-diagonal T(n+1, n) is A001880,
2-nd column variant T(n, 2)/3, for 2<=n, is A002450, 3-rd column variant
T(n, 3)/15, for 3<=n, is A002451.
Sum of the n-th row is A005046
Sequence in context: A193966 A113378 A178657 * A095922 A089278 A087071
Adjacent sequences: A156286 A156287 A156288 * A156290 A156291 A156292
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Hartmut F. W. Hoeft (hhoft(AT)emich.edu), Feb 07 2009
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