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A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

%I A225415 #14 Jun 13 2025 07:41:49
%S A225415 1,1,58,1,1,1556,12006,1556,1,1,39054,1461615,5647300,1461615,39054,1,
%T A225415 1,976552,135028828,1838120344,4873361350,1838120344,135028828,976552,
%U A225415 1,1,24414050,11462721645,414730580760,3221733789330,6783391017228,3221733789330,414730580760,11462721645,24414050,1
%N A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.
%H A225415 G. C. Greubel, <a href="/A225415/b225415.txt">Rows n = 1..50 of the irregular triangle, flattened</a>
%F A225415 T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142459(2*n, j+1). - _G. C. Greubel_, Mar 19 2022
%e A225415 Triangle begins:
%e A225415   1;
%e A225415   1,     58,         1;
%e A225415   1,   1556,     12006,       1556,          1;
%e A225415   1,  39054,   1461615,    5647300,    1461615,      39054,         1;
%e A225415   1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1;
%t A225415 (* First program *)
%t A225415 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 A225415 T[n_, k_]:= T[n, k] = t[n+1, k+1,4]; (* t(n,k,4) = A142459 *)
%t A225415 Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n,1,14,2}]]
%t A225415 (* Second program *)
%t A225415 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,4) = A142459 *)
%t A225415 T[n_, k_]:= T[n, k]= Sum[ (-1)^(k-j-1)*t[2*n,j+1,4], {j,0,k-1}];
%t A225415 Table[T[n, k], {n,12}, {k,2*n-1}]//Flatten (* _G. C. Greubel_, Mar 19 2022 *)
%o A225415 (Sage)
%o A225415 @CachedFunction
%o A225415 def T(n, k, m):
%o A225415     if (k==1 or k==n): return 1
%o A225415     else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
%o A225415 def A142459(n, k): return T(n, k, 4)
%o A225415 def A225415(n,k): return sum( (-1)^(k-j-1)*A142459(2*n, j+1) for j in (0..k-1) )
%o A225415 flatten([[A225415(n, k) for k in (1..2*n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 19 2022
%Y A225415 The m=4 triangle in the sequence A034870 (m=0), A171692 (m=1), A225076 (m=2), A225398 (m=3).
%K A225415 nonn,tabf,easy
%O A225415 1,3
%A A225415 _Roger L. Bagula_, May 07 2013
%E A225415 Edited by _N. J. A. Sloane_, May 11 2013