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A112465 Riordan array (1/(1+x), x/(1-x)).

%I A112465 #39 Apr 19 2025 18:08:56
%S A112465 1,-1,1,1,0,1,-1,1,1,1,1,0,2,2,1,-1,1,2,4,3,1,1,0,3,6,7,4,1,-1,1,3,9,
%T A112465 13,11,5,1,1,0,4,12,22,24,16,6,1,-1,1,4,16,34,46,40,22,7,1,1,0,5,20,
%U A112465 50,80,86,62,29,8,1,-1,1,5,25,70,130,166,148,91,37,9,1,1,0,6,30,95,200,296,314,239,128,46,10,1
%N A112465 Riordan array (1/(1+x), x/(1-x)).
%C A112465 Inverse is A112466. Note that C(n,k) = Sum_{j = 0..n-k} C(j+k-1, j).
%H A112465 Reinhard Zumkeller, <a href="/A112465/b112465.txt">Rows n = 0..125 of triangle, flattened</a>
%H A112465 Roland Bacher, <a href="http://arxiv.org/abs/1509.09054">Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle</a>, arXiv:1509.09054 [math.CO], 2015.
%H A112465 E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Mathematics, 34 (2005) pp. 101-122.
%H A112465 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A112465 Number triangle T(n, k) = Sum_{j=0..n-k} (-1)^(n-k-j)*C(j+k-1, j).
%F A112465 T(2*n, n) = A072547(n) (main diagonal). - _Paul Barry_, Apr 08 2011
%F A112465 From _Reinhard Zumkeller_, Jan 03 2014: (Start)
%F A112465 T(n, k) = T(n-1, k-1) + T(n-1, k), 0 < k < n, with T(n, 0) = (-1)^n and T(n, n) = 1.
%F A112465 T(n, k) = A108561(n, n-k). (End)
%F A112465 T(n, k) = T(n-1, k-1) + T(n-2, k) + T(n-2, k-1), T(0, 0) = 1, T(1, 0) = -1, T(1, 1) = 1, T(n, k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 11 2014
%F A112465 exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 + x + x^2/2! + x^3/3!) = -1 + 2*x^2/2! + 6*x^3/3! + 13*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 21 2014
%e A112465 Triangle starts
%e A112465    1;
%e A112465   -1, 1;
%e A112465    1, 0, 1;
%e A112465   -1, 1, 1,  1;
%e A112465    1, 0, 2,  2,  1;
%e A112465   -1, 1, 2,  4,  3,  1;
%e A112465    1, 0, 3,  6,  7,  4,  1;
%e A112465   -1, 1, 3,  9, 13, 11,  5, 1;
%e A112465    1, 0, 4, 12, 22, 24, 16, 6, 1;
%e A112465 Production matrix begins
%e A112465   -1, 1;
%e A112465    0, 1, 1;
%e A112465    0, 0, 1, 1;
%e A112465    0, 0, 0, 1, 1;
%e A112465    0, 0, 0, 0, 1, 1;
%e A112465    0, 0, 0, 0, 0, 1, 1;
%e A112465    0, 0, 0, 0, 0, 0, 1, 1;
%e A112465    0, 0, 0, 0, 0, 0, 0, 1, 1; - _Paul Barry_, Apr 08 2011
%t A112465 T[n_, k_]:= Sum[Binomial[j+k-1, j]*(-1)^(n-k-j), {j, 0, n-k}];
%t A112465 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Jul 23 2018 *)
%o A112465 (Haskell)
%o A112465 a112465 n k = a112465_tabl !! n !! k
%o A112465 a112465_row n = a112465_tabl !! n
%o A112465 a112465_tabl = iterate f [1] where
%o A112465    f xs'@(x:xs) = zipWith (+) ([-x] ++ xs ++ [0]) ([0] ++ xs')
%o A112465 -- _Reinhard Zumkeller_, Jan 03 2014
%o A112465 (Magma)
%o A112465 A112465:= func< n,k | (-1)^(n+k)*(&+[(-1)^j*Binomial(j+k-1,j): j in [0..n-k]]) >;
%o A112465 [A112465(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Apr 18 2025
%o A112465 (SageMath)
%o A112465 def A112465(n,k): return (-1)^(n+k)*sum((-1)^j*binomial(j+k-1,j) for j in range(n-k+1))
%o A112465 print(flatten([[A112465(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 18 2025
%Y A112465 Cf. A059260, A108561, A112466, A112468.
%Y A112465 Columns: A033999(n) (k=0), A000035(n) (k=1), A004526(n) (k=2), A002620(n-1) (k=3), A002623(n-4) (k=4), A001752(n-5) (k=5), A001753(n-6) (k=6), A001769(n-7) (k=7), A001779(n-8) (k=8), A001780(n-9) (k=9), A001781(n-10) (k=10), A001786(n-11) (k=11), A001808(n-12) (k=12).
%Y A112465 Diagonals: A000012(n) (k=n), A023443(n) (k=n-1), A152947(n-1) (k=n-2), A283551(n-3) (k=n-3).
%Y A112465 Main diagonal: A072547.
%Y A112465 Sums: A078008 (row), A078024 (diagonal), A092220 (signed diagonal), A280560 (signed row).
%K A112465 easy,sign,tabl
%O A112465 0,13
%A A112465 _Paul Barry_, Sep 06 2005