%I #28 Sep 08 2022 08:45:09
%S 1,1,1,1,7,1,1,13,13,1,1,19,61,19,1,1,25,145,145,25,1,1,31,265,595,
%T 265,31,1,1,37,421,1585,1585,421,37,1,1,43,613,3331,6145,3331,613,43,
%U 1,1,49,841,6049,17401,17401,6049,841,49,1,1,55,1105,9955,40105,65527,40105,9955,1105,55,1
%N Pascal-(1,5,1) 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 A016921, A081589, A081590. Coefficients of the row polynomials in the Newton basis are given by A013613.
%H Vincenzo Librandi, <a href="/A081580/b081580.txt">Rows n = 0..100, flattened</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%F Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 5*T(n-1, k-1) + T(n-1, k).
%F Rows are the expansions of (1+5*x)^k/(1-x)^(k+1).
%F From _Paul Barry_, Aug 28 2008: (Start)
%F Number triangle T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*5^j.
%F Riordan array (1/(1-x), x*(1+5*x)/(1-x)). (End)
%F T(n, k) = Hypergeometric2F1([-k, k-n], [1], 6). - _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)*(6*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 12*x + 36*x^2/2) = 1 + 13*x + 61*x^2/2! + 145*x^3/3! + 265*x^4/4! + 421*x^5/5! + .... - _Peter Bala_, Mar 05 2017
%F Sum_{k=0..n} T(n, k, 3) = A002532(n+1). - _G. C. Greubel_, May 26 2021
%e Square array begins as:
%e 1, 1, 1, 1, 1, ... A000012;
%e 1, 7, 13, 19, 25, ... A016921;
%e 1, 13, 61, 145, 265, ... A081589;
%e 1, 19, 145, 595, 1585, ... A081590;
%e 1, 25, 265, 1585, 6145, ...
%e The triangle begins as:
%e 1;
%e 1, 1;
%e 1, 7, 1;
%e 1, 13, 13, 1;
%e 1, 19, 61, 19, 1;
%e 1, 25, 145, 145, 25, 1;
%e 1, 31, 265, 595, 265, 31, 1;
%e 1, 37, 421, 1585, 1585, 421, 37, 1;
%e 1, 43, 613, 3331, 6145, 3331, 613, 43, 1;
%e 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1;
%e 1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1; - _Philippe Deléham_, Mar 15 2014
%t Table[Hypergeometric2F1[-k, k-n, 1, 6], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, May 24 2013 *)
%o (Magma)
%o A081580:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
%o [A081580(n,k,5): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 26 2021
%o (Sage) flatten([[hypergeometric([-k, k-n], [1], 6).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), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).
%Y Cf. A002532, A016921, A081589, A081590.
%K nonn,tabl,easy
%O 0,5
%A _Paul Barry_, Mar 23 2003