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A225398 Triangle read by rows: absolute values of odd-numbered rows of A225433.

%I A225398 #20 Jun 13 2025 07:40:38
%S A225398 1,1,38,1,1,676,4806,676,1,1,10914,362895,1346780,362895,10914,1,1,
%T A225398 174752,20554588,263879264,683233990,263879264,20554588,174752,1,1,
%U A225398 2796190,1063096365,35677598760,267248150610,554291429748,267248150610,35677598760,1063096365,2796190,1
%N A225398 Triangle read by rows: absolute values of odd-numbered rows of A225433.
%H A225398 G. C. Greubel, <a href="/A225398/b225398.txt">Rows n = 1..50 of the irregular triangle, flattened</a>
%F A225398 From _G. C. Greubel_, Mar 19 2022: (Start)
%F A225398 T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142458(2*n, j+1).
%F A225398 T(n, n-k) = T(n, k). (End)
%e A225398 Triangle begins:
%e A225398   1;
%e A225398   1,     38,        1;
%e A225398   1,    676,     4806,       676,         1;
%e A225398   1,  10914,   362895,   1346780,    362895,     10914,        1;
%e A225398   1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1;
%t A225398 (* First program *)
%t A225398 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 A225398 T[n_, k_]:= T[n, k]= t[n+1, k+1, 3]; (* t(n,k,3) = A142458 *)
%t A225398 Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n, 1, 14, 2}]]
%t A225398 (* Second program *)
%t A225398 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(n,k,3) = A142458 *)
%t A225398 A225398[n_, k_]:= A225398[n, k]= Sum[(-1)^(k-j-1)*t[2*n,j+1,3], {j,0,k-1}];
%t A225398 Table[A225398[n, k], {n,12}, {k,2*n-1}] //Flatten (* _G. C. Greubel_, Mar 19 2022 *)
%o A225398 (Sage)
%o A225398 @CachedFunction
%o A225398 def T(n, k, m):
%o A225398     if (k==1 or k==n): return 1
%o A225398     else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
%o A225398 def A142458(n, k): return T(n, k, 3)
%o A225398 def A225398(n,k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )
%o A225398 flatten([[A225398(n, k) for k in (1..2*n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 19 2022
%Y A225398 Cf. A034870, A171692, A225076.
%K A225398 nonn,tabf,easy
%O A225398 1,3
%A A225398 _Roger L. Bagula_, Apr 26 2013 (Entered by _N. J. A. Sloane_, May 06 2013)
%E A225398 Edited by _N. J. A. Sloane_, May 11 2013