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A142961
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Irregular triangle read by rows: coefficients of polynomials related to a family of convolutions of certain central binomial sequences.
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4
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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
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OFFSET
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0,4
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COMMENTS
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The row length sequence {r(k)} of this irregular triangle a(k, p) is given by r(0) = 1 = r(1) and r(k) = 2*(floor(k/2)) = A052928(k), k >= 2. This is {1,1,2,2,4,4,6,6,8,8,10,10,...}.
The array of the k-family of convolutions Sigma(k, n) := Sum_{p=0..n} p^k * binomial(2*p, p) * binomial(2*(n-p), n-p) can be written as Sigma(k, n) = ((4^n)*c(k, n) / A046161(k)) * Sum_{p=0..r(k)-1} a(k, p)*n^p, where c(k, n) = n for even k >= 2 and c(k, n) = n^2 for odd k >= 3, with c(0, n) = 1, c(1, n).
The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.
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LINKS
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FORMULA
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a(k, p)= [n^p] P(k, n) where c(k, n)*P(k, n) = A046161(k)*Sigma(k, n)/(4^n), with c(k, n) and the array Sigma(k, n) given above. A046161(k) are the denominators of binomial(2*k, k)/4^k: [1, 2, 8, 16, 128, 256, 1024, 2048, 32768, 65536, 262144,...].
Sigma(k, n)/4^n = Sum_{p=0..min(n, k)} binomial(n, p)*(2*p -1)!!*S2(k, p)/2^p, 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_{p=0..k} (S2(k, p)*x^p*d_x^p), with the derivative operator d_x.)
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EXAMPLE
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The irregular triangle a(k, p) begins:
k\p 0 1 2 3 4 5 ...
0: 1
1: 1
2: 1 3
3: 3 5
4: 2 -1 30 35
5: -10 5 70 63
6: 8 -2 -75 35 315 231
...
k=3: Sigma(3, n) = Sum_{p=0..n} p^3 * binomial(2*p, p) * binomial(2*(n-p), n-p) = (4*n/16)*n^2*(3 + 5*n), for n >= 0. This is the sequence {0, 2, 52, 648, 5888, 44800, 304128, 1906688, 11272192, 63700992, ...}.
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CROSSREFS
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KEYWORD
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sign,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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