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A142961 Coefficients of polynomials related to a convolution of certain central binomial sequences. 1
1, 1, 1, 3, 3, 5, -2, 1, 30, 35, -10, 5, 70, 63, 8, -2, -75, 35, 315, 231, 56, -14, -245, 105, 693, 429, -272, 36, 2268, -525, -5880, 2310, 12012, 6435, -2448, 324, 9660, -2037, -16632, 6006, 25740, 12155, 3968, -304, -31260, 3840, 73395, -14091, -90090, 30030, 109395, 46189, 43648, -3344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Sigma(k,n):=sum(p^k*binomial(2*p,p)*binomial(2*(n-p),n-p),p=0..n) = (4^n)*c(n)*sum(a(k,p)*n^p,p=0..r(k)-1)/A046161(n), with r(k), the row lengths, given below and c(n)=n for even n>=2 and c(n)=n^2 for odd n>=3. c(0)=1=c(1).

The author was led to compute such sums by a question asked by M. Greiter, June 27, 2008.

The row lengths of this tabf array are r(k) = [1,1,2,2,4,4,6,6,8,8,10,10,...]. r(0)=1=r(1) and r(k)= 2*(floor(k/2)), k=2,3,...

LINKS

Table of n, a(n) for n=0..53.

W. Lang, first 11 rows and more.

FORMULA

a(k,p)= [n^p](A046161(n)*Sigma(k,n)/((4^n)*c(n)) with the convolution Sigma(k,n) and c(n) given above. A046161(n) are the denominators of binomial(2*n,n)/4^n: [1, 2, 8, 16, 128, 256, 1024, 2048, 32768, 65536, 262144,...].

Sigma(k,n)/4^n = sum(binomial(n,p)*(2*p -1)!!*S2(k,p)/2^p, p=0..min(n,k)), with the double factorials (2*p -1)!!= A001147(p), with (-1)!!:=1 and the Stirling numbers of the second kind S2(k,p):=A048993(k,p).(Proof from the product of the o.g.f.s 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.)

EXAMPLE

[1];[1];[1,3];[3,5];[2,1,30,35];[10,5,70,63];[8,2,75,35,315,231];...

CROSSREFS

Sequence in context: A209389 A105104 A229087 * A101777 A204154 A267089

Adjacent sequences:  A142958 A142959 A142960 * A142962 A142963 A142964

KEYWORD

sign,easy,tabf

AUTHOR

Wolfdieter Lang Sep 15 2008

STATUS

approved

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Last modified December 4 23:10 EST 2016. Contains 278755 sequences.