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%I #21 Sep 08 2022 08:45:09
%S 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,
%T 449,41,1,1,49,721,3297,3297,721,49,1,1,57,1057,7161,14721,7161,1057,
%U 57,1,1,65,1457,13265,44961,44961,13265,1457,65,1,1,73,1921,22121,108353,192969,108353,22121,1921,73,1
%N Pascal-(1,7,1) array.
%C 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.
%H Vincenzo Librandi, <a href="/A081582/b081582.txt">Rows n = 0..100, flattened</a>
%F T(n,k) = Sum_{j = 0..n-k) binomial(n-k,j)*binomial*(k,j)*8^j.
%F Riordan array (1/(1 - x), x*(1 + 7*x)/(1 - x)).
%F 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).
%F Rows are the expansions of (1 + 7*x)^k/(1 - x)^(k+1).
%F T(n, k) = Hypergeometric2F1([-k, k-n], [1], 8). - _Jean-François Alcover_, May 24 2013
%F 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
%F Sum_{k=0..n} T(n, k) = A015519(n+1). - _G. C. Greubel_, May 26 2021
%e Rows begin
%e 1, 1, 1, 1, 1, ... A000012;
%e 1, 9, 17, 25, 33, ... A017077;
%e 1, 17, 97, 241, 449, ... A081593;
%e 1, 25, 241, 1161, 3297, ...
%e 1, 33, 449, 3297, 14721, ...
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 9, 1;
%e 1, 17, 17, 1;
%e 1, 25, 97, 25, 1;
%e 1, 33, 241, 241, 33, 1;
%e 1, 41, 449, 1161, 449, 41, 1;
%e 1, 49, 721, 3297, 3297, 721, 49, 1;
%e 1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1;
%t Table[ Hypergeometric2F1[-k, k-n, 1, 8], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, May 24 2013 *)
%o (Magma)
%o A081582:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
%o [A081582(n,k,7): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 26 2021
%o (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
%Y 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).
%Y Cf. A015519, A017077, A081593.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Mar 23 2003
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