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 A225076 #58 Mar 20 2022 02:16:59
%S A225076 1,1,22,1,1,236,1446,236,1,1,2178,58479,201244,58479,2178,1,1,19672,
%T A225076 1736668,19971304,49441990,19971304,1736668,19672,1,1,177134,46525293,
%U A225076 1356555432,9480267666,19107752148,9480267666,1356555432,46525293,177134,1
%N A225076 Triangle read by rows: absolute values of odd-numbered rows of A225356.
%C A225076 An equivalent definition: take the polynomials corresponding to rows 2, 4, 6, 8, ... of A060187, divide by x+1, and extract the coefficients. [Corrected by _Petros Hadjicostas_, Apr 17 2020]
%H A225076 G. C. Greubel, <a href="/A225076/b225076.txt">Rows n = 1..50 of the irregular triangle, flattened</a>
%F A225076 Triangle read by rows: row n gives coefficients in the expansion of the polynomial ((x - 1)^(2*n)/(x + 1)) * Sum_{k >= 0} (2*k + 1)^(2*n-1)*x^k. The infinite sum simplifies to a polynomial.
%F A225076 Sum_{m=0..2*n-2} T(n,m)*t^m = 2^(2*n-1) * (1-t)^(2*n) * LerchPhi(t, 1-2*n, 1/2)/(1 + t).
%F A225076 Sum_{k=1..n} T(n, k) = A002671(n-1).
%F A225076 T(n,m) = Sum_{k=0..m-1} (-1)^(m-k-1)*A060187(2*n,k+1) for n >= 1 and 1 <= m <= 2*n-1. - _Petros Hadjicostas_, Apr 17 2020
%e A225076 Triangle T(n,m) (for n >= 1 and 0 <= m <= 2*n - 2) begins as follows:
%e A225076 1;
%e A225076 1, 22, 1;
%e A225076 1, 236, 1446, 236, 1;
%e A225076 1, 2178, 58479, 201244, 58479, 2178, 1;
%e A225076 1, 19672, 1736668, 19971304, 49441990, 19971304, 1736668, 19672, 1;
%e A225076 ...
%t A225076 (* Power series via an infinite sum *)
%t A225076 p[x_,n_] = (x-1)^(2*n)*Sum[(2*k+1)^(2*n-1)*x^k,{k,0,Infinity}];
%t A225076 Table[CoefficientList[p[x,n]/(1+x),x],{n,1,10}]//Flatten
%t A225076 (* First alternative method: recurrence *)
%t A225076 t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k - (m-1))*t[n-1,k,m]];
%t A225076 T[n_, k_]:= T[n, k]= t[n+1,k+1,2]; (* t(n,k,2) = A060187 *)
%t A225076 Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(x+1), x], {n,14,2}]]
%t A225076 (* Second alternative method: polynomial expansion *)
%t A225076 p[t_] = Exp[t]*x/(-Exp[2*t] + x);
%t A225076 Flatten[Table[CoefficientList[(n!*(-1+x)^(n+1)/(x*(x+1)))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 1, 13, 2}]]
%o A225076 (Sage)
%o A225076 def A060187(n, k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )
%o A225076 def A225076(n,k): return sum( (-1)^(k-j-1)*A060187(2*n, j+1) for j in (0..k-1) )
%o A225076 flatten([[A225076(n, k) for k in (1..2*n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 19 2022
%Y A225076 Cf. A002671 (row sums), A034870, A060187, A171692, A225398.
%K A225076 nonn,tabf
%O A225076 1,3
%A A225076 _Roger L. Bagula_, Apr 26 2013
%E A225076 Edited by _N. J. A. Sloane_, May 06 2013, May 11 2013