login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A136630 Triangular array: T(n,k) counts the partitions of the set [n] into k odd sized blocks. 11
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 0, 10, 0, 1, 0, 0, 16, 0, 20, 0, 1, 0, 1, 0, 91, 0, 35, 0, 1, 0, 0, 64, 0, 336, 0, 56, 0, 1, 0, 1, 0, 820, 0, 966, 0, 84, 0, 1, 0, 0, 256, 0, 5440, 0, 2352, 0, 120, 0, 1, 0, 1, 0, 7381, 0, 24970, 0, 5082, 0, 165, 0, 1, 0, 0, 1024, 0, 87296, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

For partitions into blocks of even size see A156289.

Essentially the unsigned matrix inverse of triangle A121408.

From Peter Bala, Jul 28 2014: (Start)

Define a polynomial sequence x_(n) by setting x_(0) = 1 and for n = 1,2,... setting x_(n) = x*(x + n - 2)*(x + n - 4)*...*(x + n - 2*(n - 1)). Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k) for n = 0,1,2,.... An example is given below.

Let M denote the lower unit triangular array A119467 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/ having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle, omitting the first row and column, equals the infinite matrix product M(0)*M(1)*M(2)*.... (End)

REFERENCES

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 225-226.

LINKS

Table of n, a(n) for n=0..83.

Ch. A. Charalambides, Central factorial numbers and related expansions, Fib. Quarterly, Vol. 19, No 5, Dec 1981, pp. 451-456.

FORMULA

G.f. for column k: x^k/Product_{j=0..[k/2]} (1 - (2*j + k-2*[k/2])^2 * x^2).

G.f. for column 2*k: x^(2*k)/Product_{j=0..k} (1 - (2*j)^2*x^2).

G.f. for column 2*k+1: x^(2*k+1)/Product_{j=0..k} (1 - (2*j+1)^2*x^2).

From Peter Bala, Feb 21 2011 (Start)

T(n,k) = 1/(2^k*k!)*sum {j = 0..k}(-1)^(k-j)*binomial(k,j)*(2*j-k)^n,

Recurrence relation T(n+2,k) = T(n,k-2) + k^2*T(n,k).

E.g.f.: F(x,z) = exp(x*sinh(z)) = sum {n = 0..inf} R(n,x)*z^n/n! = 1 + x*z + x^2*z^2/2! + (x+x^3)*z^3/3! + ....

The row polynomials R(n,x) begin

R(1,x) = x

R(2,x) = x^2

R(3,x) = x+x^3.

The e.g.f. F(x,z) satisfies the partial differential equation d^2/dz^2(F) = x^2*F + x*F' + x^2*F'' where ' denotes differentiation w.r.t. x.

Hence the row polynomials satisfy the recurrence relation R(n+2,x) = x^2*R(n,x) +x*R'(n,x) +x^2*R''(n,x) with R(0,x) = 1.

The recurrence relation for T(n,k) given above follows from this.

(End)

For the corresponding triangle of ordered partitions into odd-sized blocks see A196776. Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of exp(t*M) lists the row polynomials for the present triangle. - Peter Bala, Oct 06 2011

Row generating polynomials equal D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A196776. - Peter Bala, Dec 06 2011

From Peter Bala, Jul 28 2014: (Start)

E.g.f.: exp(t*sinh(x)) = 1 + t*x + t^2*x^2/2! + (t + t^3)*x^3/3! + ....

Hockey-stick recurrence: T(n+1,k+1) = sum {i = 0..floor((n-k)/2)} binomial(n,2*i)*T(n-2*i,k).

Recurrence equation for the row polynomials R(n,t):

R(n+1,t) = t*sum {k = 0..floor(n/2)} binomial(n,2*k)*R(n-2*k,t) with R(0,t) = 1. (End)

EXAMPLE

Triangle begins:

1;

0, 1;

0, 0, 1;

0, 1, 0, 1;

0, 0, 4, 0, 1;

0, 1, 0, 10, 0, 1;

0, 0, 16, 0, 20, 0, 1;

0, 1, 0, 91, 0, 35, 0, 1;

0, 0, 64, 0, 336, 0, 56, 0, 1;

0, 1, 0, 820, 0, 966, 0, 84, 0, 1;

0, 0, 256, 0, 5440, 0, 2352, 0, 120, 0, 1;

0, 1, 0, 7381, 0, 24970, 0, 5082, 0, 165, 0, 1;

T(5,3) = 10. The ten partitions of the set [5] into 3 odd-sized blocks are

(1)(2)(345), (1)(3)(245), (1)(4)(235), (1)(5)(234), (2)(3)(145),

(2)(4)(135), (2)(5)(134), (3)(4)(125), (3)(5)(124), (4)(5)(123).

Connection constants: Row 5 = [0,1,0,10,0,1]. Hence, with the polynomial sequence x_(n) as defined in the Comments section we have x^5 = x_(1) + 10*x_(3) + x_(5) = x + 10*x*(x+1)*(x-1) + x*(x+3)*(x+1)*(x-1)*(x-3).

MAPLE

A136630 := proc (n, k) option remember; if k < 0 or n < k then 0 elif k = n then 1 else procname(n-2, k-2) + k^2*procname(n-2, k) end if end proc: seq(seq(A136630(n, k), k = 1 .. n), n = 1 .. 12);

# Peter Bala, Jul 27 2014

MATHEMATICA

t[n_, k_] := Coefficient[ x^k/Product[ 1 - (2*j + k - 2*Quotient[k, 2])^2*x^2, {j, 0, k/2}] + x*O[x]^n, x, n]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Nov 22 2013, after Pari *)

PROG

(PARI) {T(n, k)=polcoeff(x^k/prod(j=0, k\2, 1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)), n)}

CROSSREFS

Cf. A121408; A136631 (antidiagonal sums), A003724 (row sums), A136632; A002452 (column 3), A002453 (column 5); A008958 (central factorial triangle), A156289. A185690, A196776.

Sequence in context: A036859 A036861 A120324 * A111728 A143784 A147311

Adjacent sequences:  A136627 A136628 A136629 * A136631 A136632 A136633

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 14 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 20 21:20 EST 2014. Contains 249754 sequences.