1,3

The o.g.f.s G(k,x) for the k-family of sequences S(k,n):= sum(p^k*binomial(2*p,p)*binomial(2*(n-p),n-p),p=0..n), k=0,1,... (convolution of two sequences involving the central binomial coefficients) are 1/(1-4*x) for k=0 and 2*x*P(k,x)/(1-4*x)^(k+1) for k=1,2,..., with the row polynomials P(k,x):= sum(a(n,m),x^m,m=0..k-1).

The author was led to compute the sums S(k,n) by a question asked by Dr. M. Greiter, June 27, 2008.

W. Lang, First 10 rows and more.

G(k,x)= sum(S2(k,p)*((2*p)!/p!)*x^p/(1-4*x)^(p+1),p=0..k), k>=0 (here k>=1), with the Stirling2 triangle S2(k,p):=A048993(k,p). (Proof from the product of the o.g.f.s of the two convoluted sequences and the normal ordering (x^d_x)^k = sum(S2(k,p)*x^p*d_x^p,p=0..k), with the derivative operator d_x.)

a(k,m)= [x^m]P(k,x) = [x^m] ((1-4*x)^(k+1))*G(k,x)/(2*x), k>=1, m=0,1,..k-1.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: (Start)

For the triangle coefficients the following relation holds: WL(n,m) = (m+1)*WL(n-1,m) + (4*n-4*m-2)*WL(n-1,m-1) with WL(n,m=0)=1 and WL(n,m=n-1)=2^(n-1). In view of the offset n=1,2,3, ... and m=0,1,..,n-1.

(End)

[1];[1,2];[1,10,4];[1,30,72,8];[1,74,516,464,16];...

k=3: P(3,x)= 1+10*x+4*x^2. G(3,x)=2*x*(1+10*x+4*x^2)/(1-4*x)^4.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: (Start)

nmax:=7; mmax:=nmax-1: for n from 1 to nmax do WL[n, 0]:=1 end do: for m from 1 to mmax do WL[1, m]:=0 end do: for n from 2 to nmax do for m from 1 to mmax do WL[n, m]:=(m+1)*WL[n-1, m] + (4*n-4*m-2)*WL[n-1, m-1] end do end do: for n from 1 to nmax do for m from 0 to n-1 do WL[(n-1)*(n)/2+m+1]:=WL[n, m] end do end do: a:=n-> WL[n]: seq(a(n), n=1..(1/2)*(nmax^2+nmax));

(End)

Column sequences are 2*A142964, 4*A142965, 8*A142966.

Row sums A142967.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: (Start)

A156919 and this sequence can be mapped onto A156920.

Cf. A156921, A156925, A156927, A156933

Another (left hand) column sequence is 16*A142968.

Right hand column sequences are 2^n*A000340, 2^n*A156922, 2^n*A156923, 2^n*A156924

(End)

Sequence in context: A144275 A011268 A163235 this_sequence A099755 A110682 A110327

Adjacent sequences: A142960 A142961 A142962 this_sequence A142964 A142965 A142966

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 15 2008