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 A156732 #12 Mar 01 2021 02:03:56
%S A156732 0,1,1,4,0,4,9,2,2,9,16,10,0,10,16,25,27,5,5,27,25,36,56,28,0,28,56,
%T A156732 36,49,100,84,14,14,84,100,49,64,162,192,84,0,84,192,162,64,81,245,
%U A156732 375,270,42,42,270,375,245,81
%N A156732 Triangle T(n, k) = ((n-2*k)^2/(n-k+1))*binomial(n+1, k+1), read by rows.
%C A156732 The general formula for integrals of the form int (x^(2*k))/((arcsin(x))^2) dx involves the triangular sequence t(2*k, n). For example, the solution to the integral Integral x^6/((arcsin(x))^2) dx involves the following sequence: -5*Si(arcsin(x)) + 27*Si(3*arcsin(x)) - 25*Si(5*arcsin(x)), where Si represents the sine integral. The sequence of integers 5, 27, 25 corresponds to the sixth row of this triangular sequence. The general formula for the integral Integral x^(2*k)/((arcsin(x))^2) dx is: (1/(2^(2*k)))*( -((2^(2*k)*sqrt(1-x^2)*(x^(2*k)))/arcsin(x) )+ (-1)^(k+1)*(1+(2*k))*Si((1+2*k)*arcsin(x)) + Sum_{n=1..k} (-1)^n*((1-2*n)^2/(k-n+1))*binomial(2*k, k+n)*Si((2*n-1)*arcsin(x)) ). - _John M. Campbell_, Sep 22 2010
%D A156732 J. Riordan, Combinatorial Identities, Wiley, 1968.
%H A156732 G. C. Greubel, <a href="/A156732/b156732.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A156732 John. M. Campbell, <a href="https://arxiv.org/abs/1009.0236">Double Series Involving Binomial Coefficients and the Sine Integral</a>, arXiv:1009.0236 [math.NT], 2010, p. 3-4. [From _John M. Campbell_, Sep 22 2010]
%F A156732 T(n, k) = ((n-2*k)^2/(n-k+1))*binomial(n+1, k+1).
%F A156732 Sum_{k=0..floor(n/2)} T(n-1, k-1) = 2^n.
%F A156732 From _G. C. Greubel_, Feb 28 2021: (Start)
%F A156732 T(n, k) = T(n, n-k).
%F A156732 T(n, k) = ((n-2*k)^2/(n+2))*binomial(n+2, k+1).
%F A156732 Sum_{k=0..n} T(n, k) = 2*(2^(n+1) -n-2) = 4*A002662(n) + 2*n^2. (End)
%e A156732 Triangle begins as:
%e A156732 0;
%e A156732 1, 1;
%e A156732 4, 0, 4;
%e A156732 9, 2, 2, 9;
%e A156732 16, 10, 0, 10, 16;
%e A156732 25, 27, 5, 5, 27, 25;
%e A156732 36, 56, 28, 0, 28, 56, 36;
%e A156732 49, 100, 84, 14, 14, 84, 100, 49;
%e A156732 64, 162, 192, 84, 0, 84, 192, 162, 64;
%e A156732 81, 245, 375, 270, 42, 42, 270, 375, 245, 81;
%t A156732 T[n_, k_]:= ((n-2*k)^2/(n-k+1))*Binomial[n+1, k+1];
%t A156732 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 28 2021 *)
%o A156732 (Sage)
%o A156732 def A156732(n, k): return ((n-2*k)^2/(n+2))*binomial(n+2, k+1)
%o A156732 flatten([[A156732(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 28 2021
%o A156732 (Magma)
%o A156732 A156732:= func< n,k | ((n-2*k)^2/(n+2))*Binomial(n+2, k+1) >;
%o A156732 [A156732(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 28 2021
%Y A156732 Cf. A000290, A002662.
%K A156732 nonn,tabl
%O A156732 0,4
%A A156732 _Roger L. Bagula_, Feb 14 2009
%E A156732 Edited by _G. C. Greubel_, Feb 28 2021