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A142963
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Table of coefficients of row polynomials of certain o.g.f.s.
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14
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1, 1, 2, 1, 10, 4, 1, 30, 72, 8, 1, 74, 516, 464, 16, 1, 166, 2584, 7016, 2864, 32, 1, 354, 10740, 64240, 84480, 17376, 64, 1, 734, 40008, 450280, 1321760, 949056, 104704, 128, 1, 1498, 139108, 2681296, 14713840, 24198976, 10223488, 629248, 256, 1, 3030, 462264, 14341992
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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 M. Greiter, June 27, 2008.
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LINKS
| W. Lang, First 10 rows and more.
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FORMULA
| 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, Feb 20 2009: (Start)
For the triangle coefficients the following relation holds: T(n,m) = (m+1)*T(n-1,m) + (4*n-4*m-2)*T(n-1,m-1) with T(n,m=0) = 1 and T(n,m=n-1) = 2^(n-1), n >= 1 and 0 <= m <= n-1. (End)
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EXAMPLE
| [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.
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MAPLE
| Contribution from Johannes W. Meijer, Sep 28 2011: (Start)
A142963 := proc(n, m): if n=m+1 then 2^(n-1); elif m=0 then 1 ; elif m<0 or m>n-1 then 0; else (m+1)*procname(n-1, m)+(4*n-4*m-2)*procname(n-1, m-1); end if; end proc: seq(seq(A142963(n, m), m=0..n-1), n=1..9); # (End)
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CROSSREFS
| Left hand column sequences 2*A142964, 4*A142965, 8*A142966, 16*A142968.
Row sums A142967.
Contribution from Johannes W. Meijer, Feb 20 2009: (Start)
A156919 and this sequence can be mapped onto A156920.
Cf. A156921, A156925, A156927, A156933
Right hand column sequences 2^n*A000340, 2^n*A156922, 2^n*A156923, 2^n*A156924 (End)
Sequence in context: A144275 A011268 A163235 * A099755 A202483 A110682
Adjacent sequences: A142960 A142961 A142962 * A142964 A142965 A142966
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 15 2008
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EXTENSIONS
| Minor edits by Johannes W. Meijer, Sep 28 2011
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