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A246118 T(n,k), for n,k >= 1, is the number of partitions of the set [n] into k blocks, where, if the blocks are arranged in order of their minimal element, the odd indexed blocks are all singletons. 2
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 11, 6, 1, 0, 1, 5, 26, 23, 9, 1, 0, 1, 6, 57, 72, 50, 12, 1, 0, 1, 7, 120, 201, 222, 86, 16, 1, 0, 1, 8, 247, 522, 867, 480, 150, 20, 1, 0, 1, 9, 502, 1291, 3123, 2307, 1080, 230, 25, 1, 0, 1, 10, 1013, 3084, 10660, 10044, 6627, 2000, 355, 30, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Unsigned matrix inverse of A246117. Analog of the Stirling numbers of the second kind, A048993.

This is the triangle of connection constants between the monomial polynomials x^n and the polynomial sequence [x, x^2, x^2*(x - 1), x^2*(x - 1)^2, x^2*(x - 1)^2*(x - 2), x^2*(x - 1)^2*(x - 2)^2, ...]. An example is given below.

Except for differences in offset, this triangle is the Galton array G(floor(k/2),1) in the notation of Neuwirth with inverse array G(-floor(n/2),1).

Essentially the same as A256161. - Peter Bala, Apr 14 2018

LINKS

Table of n, a(n) for n=1..78.

Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.

Emrah Kiliç and Helmut Prodinger, Identities with Squares of Binomial Coefficients: an Elementary and Explicit Approach, Publications de l'Institut Mathématique (Beograd) (N.S.), Vol.99(113) (2016), 243-248. See p. 248.

E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.

FORMULA

T(n,k) = Sum_{i = 0..n-1} Stirling2(i, floor(k/2))*Stirling2(n-i-1, floor((k - 1)/2)) for n,k >= 1.

Recurrence equation: T(1,1) = 1, T(n,1) = 0 for n >= 2; T(n,k) = 0 for k > n; otherwise T(n,k) = floor(k/2)*T(n-1,k) + T(n-1,k-1).

O.g.f. (with an extra 1): A(z) = 1 + Sum_{k >= 1} (x*z)^k/( ( Product_{i = 1..floor((k-1)/2)} (1 - i*z) ) * ( Product_{i = 1..floor(k/2)} (1 - i*z) ) ) = 1 + x*z + x^2*z^2 + (x^2 + x^3)*z^3 + (x^2 + 2*x^3 + x^4)*z^4 + .... satisfies A(z) = 1 + x*z + x^2*z^2/(1 - z)*A(z/(1 - z)).

k-th column generating function z^k/( ( Product_{i = 1..floor((k-1)/2)} (1 - i*z) ) * ( Product_{i = 1..floor(k/2)} (1 - i*z) ) ).

Recurrence for row polynomials: R(n,x) = x^2*Sum_{k = 0..n-2} binomial(n-2,k)*R(k,x) with initial conditions R(0,x) = 1 and R(1,x) = x. Compare with the recurrence satisfied by the Bell polynomials: Bell(n,x) = x*Sum_{k = 0..n-1} binomial(n-1,k) * Bell(k,x).

Row sums are A007476.

EXAMPLE

Triangle begins

n\k| 1    2    3    4    5    6    7    8

1  | 1

2  | 0    1

3  | 0    1    1

4  | 0    1    2    1

5  | 0    1    3    4    1

6  | 0    1    4   11    6    1

7  | 0    1    5   26   23    9    1

8  | 0    1    6   57   72   50   12    1

...

Connection constants: Row 6 = (0, 1, 4, 11, 6, 1) so

x^6 = x^2 + 4*x^2*(x - 1) + 11*x^2*(x - 1)^2 + 6*x^2*(x - 1)^2*(x - 2) + x^2*(x - 1)^2*(x - 2)^2.

Row 5 = [0, 1, 3, 4, 1]. There are 9 set partitions of {1,2,3,4,5} of the type described in the Name section:

= = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Number of      Set partitions                Count

blocks

= = = = = = = = = = = = = = = = = = = = = = = = = = = = =

2                {1}{2,3,4,5}                   1

3           {1}{2,4,5}{3}, {1}{2,3,5}{4},

            {1}{2,3,4}{5}                       3

4          {1}{2,3}{4}{5}, {1}{2,4}{3}{5},

           {1}{2,5}{3}{4}, {1}{2}{3}{4,5}       4

5          {1}{2}{3}{4}{5}                      1

MATHEMATICA

Flatten[Table[Table[Sum[StirlingS2[j, Floor[k/2]] * StirlingS2[n-j-1, Floor[(k-1)/2]], {j, 0, n-1}], {k, 1, n}], {n, 1, 12}]] (* Vaclav Kotesovec, Feb 09 2015 *)

CROSSREFS

Cf. A000295 (column 4), A007476 (row sums), A008277, A045618 (column 5), A048993, A246117 (unsigned matrix inverse), A256161.

Sequence in context: A210391 A071921 A003992 * A171882 A214075 A322267

Adjacent sequences:  A246115 A246116 A246117 * A246119 A246120 A246121

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Aug 14 2014

STATUS

approved

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Last modified December 15 14:21 EST 2018. Contains 318149 sequences. (Running on oeis4.)