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%I A081581 #25 Sep 08 2022 08:45:09
%S A081581 1,1,1,1,8,1,1,15,15,1,1,22,78,22,1,1,29,190,190,29,1,1,36,351,848,
%T A081581 351,36,1,1,43,561,2339,2339,561,43,1,1,50,820,5006,9766,5006,820,50,
%U A081581 1,1,57,1128,9192,28806,28806,9192,1128,57,1,1,64,1485,15240,68034,116208,68034,15240,1485,64,1
%N A081581 Pascal-(1,6,1) array.
%C A081581 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 A016993, A081591, A081592. Coefficients of the row polynomials in the Newton basis are given by A013614.
%H A081581 Vincenzo Librandi, <a href="/A081581/b081581.txt">Rows n = 0..100, flattened</a>
%F A081581 Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 6*T(n-1, k-1) + T(n-1, k).
%F A081581 Rows are the expansions of (1+6*x)^k/(1-x)^(k+1).
%F A081581 T(n, k) = Hypergeometric2F1([-k, k-n], [1], 7). - _Jean-François Alcover_, May 24 2013
%F A081581 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)*(7*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 14*x + 49*x^2/2) = 1 + 15*x + 78*x^2/2! + 190*x^3/3! + 351*x^4/4! + 561*x^5/5! + .... - _Peter Bala_, Mar 05 2017
%F A081581 From _G. C. Greubel_, May 26 2021: (Start)
%F A081581 T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 6.
%F A081581 Sum_{k=0..n} T(n, k, 6) = A083099(n+1). (End)
%e A081581 Rows start as:
%e A081581   1,  1,   1,    1,    1, ... A000012;
%e A081581   1,  8,  15,   22,   29, ... A016993;
%e A081581   1, 15,  78,  190,  351, ... A081591;
%e A081581   1, 22, 190,  848, 2339, ...
%e A081581   1, 29, 351, 2339, 9766, ...
%e A081581 The triangle starts as:
%e A081581   1;
%e A081581   1,  1;
%e A081581   1,  8,   1;
%e A081581   1, 15,  15,    1;
%e A081581   1, 22,  78,   22,    1;
%e A081581   1, 29, 190,  190,   29,   1;
%e A081581   1, 36, 351,  848,  351,  36,  1;
%e A081581   1, 43, 561, 2339, 2339, 561, 43, 1;
%t A081581 Table[Hypergeometric2F1[-k, k-n, 1, 7], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, May 24 2013 *)
%o A081581 (PARI) t(n, k) = sum(j=0, n-k, binomial(n-k, j)*binomial(k, j)*7^j) \\ _Michel Marcus_, May 24 2013
%o A081581 (Magma)
%o A081581 A081581:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
%o A081581 [A081581(n,k,6): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 26 2021
%o A081581 (Sage) flatten([[hypergeometric([-k, k-n], [1], 7).simplify() for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 26 2021
%Y A081581 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), A081582 (m = 7), A143683 (m = 8).
%Y A081581 Cf. A016993, A081591, A083099.
%K A081581 easy,nonn,tabl
%O A081581 0,5
%A A081581 _Paul Barry_, Mar 23 2003

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