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A081582
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Pascal-(1,7,1) array.
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13
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1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 97, 25, 1, 1, 33, 241, 241, 33, 1, 1, 41, 449, 1161, 449, 41, 1, 1, 49, 721, 3297, 3297, 721, 49, 1, 1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1, 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1, 1, 73, 1921, 22121, 108353, 192969, 108353, 22121, 1921, 73, 1
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OFFSET
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0,5
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COMMENTS
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One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A017077, A081593, A081594. Coefficients of the row polynomials in the Newton basis are given by A013614.
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LINKS
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FORMULA
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T(n,k) = Sum_{j = 0..n-k) binomial(n-k,j)*binomial*(k,j)*8^j.
Riordan array (1/(1 - x), x*(1 + 7*x)/(1 - x)).
Square array T(n, k) defined by T(n, 0) = T(0, k)=1, T(n, k) = T(n, k-1) + 7*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1 + 7*x)^k/(1 - x)^(k+1).
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(8*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 16*x + 64*x^2/2) = 1 + 17*x + 97*x^2/2! + 241*x^3/3! + 449*x^4/4! + 721*x^5/5! + .... - Peter Bala, Mar 05 2017
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EXAMPLE
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Rows begin
1, 25, 241, 1161, 3297, ...
1, 33, 449, 3297, 14721, ...
Triangle begins:
1;
1, 1;
1, 9, 1;
1, 17, 17, 1;
1, 25, 97, 25, 1;
1, 33, 241, 241, 33, 1;
1, 41, 449, 1161, 449, 41, 1;
1, 49, 721, 3297, 3297, 721, 49, 1;
1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1;
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MATHEMATICA
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Table[ Hypergeometric2F1[-k, k-n, 1, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, May 24 2013 *)
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PROG
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(Magma)
A081582:= func< n, k, q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
(Sage) flatten([[hypergeometric([-k, k-n], [1], 8).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021
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CROSSREFS
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Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A143683 (m = 8).
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KEYWORD
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AUTHOR
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STATUS
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approved
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