cp's OEIS Frontend

A112554 Riordan array (c(x^2)^2, x*c(x^2)), c(x) the g.f. of A000108.

%I A112554 #30 Feb 11 2023 02:44:22
%S A112554 1,0,1,2,0,1,0,3,0,1,5,0,4,0,1,0,9,0,5,0,1,14,0,14,0,6,0,1,0,28,0,20,
%T A112554 0,7,0,1,42,0,48,0,27,0,8,0,1,0,90,0,75,0,35,0,9,0,1,132,0,165,0,110,
%U A112554 0,44,0,10,0,1,0,297,0,275,0,154,0,54,0,11,0,1,429,0,572,0,429,0,208,0,65,0,12,0,1
%N A112554 Riordan array (c(x^2)^2, x*c(x^2)), c(x) the g.f. of A000108.
%C A112554 Inverse of A112552.
%C A112554 The n-th row polynomial (in descending powers of x) is equal to the n-th degree Taylor polynomial of the polynomial function (1 - x^4)*(1 + x^2)^n about 0. For example, when n = 6, (1 - x^4)*(1 + x^2)^6 = 1 + 6*x^2 + 14*x^4 + 14*x^6 + O(x^8). - _Peter Bala_, Feb 19 2018
%H A112554 G. C. Greubel, <a href="/A112554/b112554.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A112554 Peter Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>
%F A112554 Sum_{k=0..n} T(n, k) = binomial(n+1, floor(n/2)) = A037952(n+1).
%F A112554 T(n, k) = ((1 + (-1)^(n-k))/2)*binomial(n, floor((n-k)/2)) - binomial(n, floor((n-k-4)/2 )). - _Peter Bala_, Feb 19 2018
%e A112554 Triangle begins
%e A112554    1;
%e A112554    0, 1;
%e A112554    2, 0,  1;
%e A112554    0, 3,  0, 1;
%e A112554    5, 0,  4, 0, 1;
%e A112554    0, 9,  0, 5, 0, 1;
%e A112554   14, 0, 14, 0, 6, 0, 1;
%p A112554 seq(seq((1 + (-1)^(n-k))/2*( binomial(n, floor((n - k)/2)) - binomial(n, floor((n - k - 4)/2 )) ), k = 0..n), n = 0..10); # _Peter Bala_, Feb 19 2018
%t A112554 T[n_, k_] := (1 + (-1)^(n-k))/2 (Binomial[n, Floor[(n-k)/2]] - Binomial[n, Floor[(n-k-4)/2]]);
%t A112554 Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%o A112554 (Sage) # Algorithm of L. Seidel (1877)
%o A112554 # Prints the first n rows of a signed version of the triangle.
%o A112554 def Signed_A112554_triangle(n) :
%o A112554     D = [0]*(n+4); D[1] = 1
%o A112554     b = False; h = 2
%o A112554     for i in range(2*n+2) :
%o A112554         if b :
%o A112554             for k in range(h,0,-1) : D[k] += D[k-1]
%o A112554             h += 1
%o A112554         else :
%o A112554             for k in range(1,h, 1) : D[k] -= D[k+1]
%o A112554         b = not b
%o A112554         if b and i > 0 : print([D[z] for z in (2..h-1)])
%o A112554 Signed_A112554_triangle(13) # _Peter Luschny_, May 01 2012
%o A112554 (Magma)
%o A112554 A112554:= func< n,k | ((1+(-1)^(n-k))/2)*(Binomial(n, Floor((n-k)/2)) - Binomial(n, Floor((n-k-4)/2))) >;
%o A112554 [A112554(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jan 13 2022
%Y A112554 Row sums A037952, matrix inverse A112552.
%Y A112554 Cf. A000108, A037952 (row sums), A112552, A112553.
%K A112554 easy,nonn,tabl
%O A112554 0,4
%A A112554 _Paul Barry_, Sep 13 2005