This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143683 #19 Oct 02 2024 04:17:37
%S A143683 1,1,1,1,10,1,1,19,19,1,1,28,118,28,1,1,37,298,298,37,1,1,46,559,1540,
%T A143683 559,46,1,1,55,901,4483,4483,901,55,1,1,64,1324,9856,21286,9856,1324,
%U A143683 64,1,1,73,1828,18388,67006,67006,18388,1828,73,1,1,82,2413,30808,164242,304300,164242,30808,2413,82,1
%N A143683 Pascal-(1,8,1) array.
%H A143683 Reinhard Zumkeller, <a href="/A143683/b143683.txt">Rows n = 0..125 of table, flattened</a>
%F A143683 Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
%F A143683 Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
%F A143683 Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
%F A143683 Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
%F A143683 T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - _Jean-François Alcover_, May 24 2013
%F A143683 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)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - _Peter Bala_, Mar 05 2017
%F A143683 Sum_{k=0..n} T(n,k) = A003683(n+1). - _G. C. Greubel_, May 27 2021
%e A143683 Square array begins as:
%e A143683 1, 1, 1, 1, 1, 1, 1, ... A000012;
%e A143683 1, 10, 19, 28, 37, 46, 55, ... A017173;
%e A143683 1, 19, 118, 298, 559, 901, 1324, ...
%e A143683 1, 28, 298, 1540, 4483, 9856, 18388, ...
%e A143683 1, 37, 559, 4483, 21286, 67006, 164242, ...
%e A143683 1, 46, 901, 9856, 67006, 304300, 1004590, ...
%e A143683 1, 55, 1324, 18388, 164242, 1004590, 4443580, ...
%e A143683 Antidiagonal triangle begins as:
%e A143683 1;
%e A143683 1, 1;
%e A143683 1, 10, 1;
%e A143683 1, 19, 19, 1;
%e A143683 1, 28, 118, 28, 1;
%e A143683 1, 37, 298, 298, 37, 1;
%e A143683 1, 46, 559, 1540, 559, 46, 1;
%e A143683 1, 55, 901, 4483, 4483, 901, 55, 1;
%t A143683 Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, May 24 2013 *)
%o A143683 (Haskell)
%o A143683 a143683 n k = a143683_tabl !! n !! k
%o A143683 a143683_row n = a143683_tabl !! n
%o A143683 a143683_tabl = map fst $ iterate
%o A143683 (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $
%o A143683 zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
%o A143683 -- _Reinhard Zumkeller_, Mar 16 2014
%o A143683 (Magma)
%o A143683 A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
%o A143683 [A143683(n,k,8): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 27 2021
%o A143683 (Sage) flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 27 2021
%Y A143683 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), A081582 (m = 7).
%Y A143683 Cf. A003683, A017173, A143680.
%K A143683 easy,nonn,tabl
%O A143683 0,5
%A A143683 _Paul Barry_, Aug 28 2008