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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. 15
 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; text; internal format)
 OFFSET 1,3 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. 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 REFERENCES L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62. L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened) Richard Olu Awonusika, 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), The Journal of Analysis (2020). Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020. J. Riordan, Letter, Jul 06 1978 FORMULA Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=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) 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  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 MAPLE 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 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}]] (* alternate computation with function triangle[] defined in A257490 *) a[n_]:=Map[Apply[Plus, #]&, triangle[n], {2}] (* Hartmut F. W. Hoft, Apr 26 2015 *) CROSSREFS Diagonal T(n, n) is A001147, subdiagonal T(n+1, n) is A001880. 2nd column variant T(n, 2)/3, for 2<=n, is A002450. 3rd column variant T(n, 3)/15, for 3<=n, is A002451. Sum of the n-th row is A005046. Cf. A241171, A257468, A257490, A096162. Sequence in context: A113378 A178657 A257490 * A095922 A263632 A284861 Adjacent sequences:  A156286 A156287 A156288 * A156290 A156291 A156292 KEYWORD easy,nonn,tabl AUTHOR Hartmut F. W. Hoft, Feb 07 2009 STATUS approved

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)