A257468 -id:A257468 - cp's OEIS frontend

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.

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

Offset: 1

Hartmut F. W. Hoft, Feb 07 2009

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

			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

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.

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

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.

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

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 *)

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

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

G(k,x) = Sum_{i>=0} T(i+k,k)*x^i = Product_{j=1..k} (2*j-1)/(1-j^2*x).

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).

From Peter Bala, Feb 21 2011: (Start)

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)

A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 28, 35, 210, 105, 1, 45, 210, 630, 1575, 3150, 945, 1, 66, 495, 462, 1485, 13860, 5775, 13860, 51975, 51975, 10395, 1, 91, 1001, 3003, 3003, 45045, 42042, 105105, 45045, 630630, 525525, 315315, 1576575, 945945, 135135

Offset: 1

Hartmut F. W. Hoft, Apr 26 2015

The length of row n is given by A000041(n).
Each entry in this irregular triangle is the quotient of the respective entries in A257468 and A096162, which is the multinomial called M_3 in Abramowitz-Stegun.
Has the same structure as the triangles in A036036, A096162, A115621 and A257468.

			Brackets group all partitions of the same length when there is more than one partition.
n/m  1    2          3           4    5
1:   1
2:   1    3
3:   1   15         15
4:   1  [28  35]   210         105
5:   1  [45 210]  [630 1575]  3150  945
...
n = 6:  1 [66 495 462] [1485 13860 5775] [13860 51975] 51975  0395
Replacing the bracketed numbers by their sums yields the triangle of A156289.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], pp. 831-832.

Cf. A000041, A036040, A036036, A096162, A115621, A156289, A257468.

(* triangle2574868[] and triangle096162[] are defined as functions triangle[] in the respective sequences A257468 and A096162 *)
triangle[n_] := triangle257468[n]/triangle096162[n]
a[n_] := Flatten[triangle[n]]
a[7] (* data *)

Edited by Wolfdieter Lang, May 11 2015

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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.

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