A225398 - OEIS

%I #15 Mar 20 2022 02:17:04

%S 1,1,38,1,1,676,4806,676,1,1,10914,362895,1346780,362895,10914,1,1,

%T 174752,20554588,263879264,683233990,263879264,20554588,174752,1,1,

%U 2796190,1063096365,35677598760,267248150610,554291429748,267248150610,35677598760,1063096365,2796190,1

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

%H G. C. Greubel, <a href="/A225398/b225398.txt">Rows n = 1..50 of the irregular triangle, flattened</a>

%F From _G. C. Greubel_, Mar 19 2022: (Start)

%F T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142458(2*n, j+1).

%F T(n, n-k) = T(n, k). (End)

%e Triangle begins:

%e   1;

%e   1,     38,        1;

%e   1,    676,     4806,       676,         1;

%e   1,  10914,   362895,   1346780,    362895,     10914,        1;

%e   1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1;

%t (* First program *)

%t 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 T[n_, k_]:= T[n, k]= t[n+1, k+1, 3]; (* t(n,k,3) = A142458 *)

%t Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n, 1, 14, 2}]]

%t (* Second program *)

%t 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[n_, k_]:= A225398[n, k]= Sum[(-1)^(k-j-1)*t[2*n,j+1,3], {j,0,k-1}];

%t Table[A225398[n, k], {n,12}, {k,2*n-1}] //Flatten (* _G. C. Greubel_, Mar 19 2022 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k, m):

%o     if (k==1 or k==n): return 1

%o     else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)

%o def A142458(n, k): return T(n, k, 3)

%o def A225398(n,k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )

%o flatten([[A225398(n, k) for k in (1..2*n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 19 2022

%Y Cf. A034870, A171692, A225076.

%K nonn,tabf

%O 1,3

%A _Roger L. Bagula_, Apr 26 2013 (Entered by _N. J. A. Sloane_, May 06 2013)

%E Edited by _N. J. A. Sloane_, May 11 2013