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A154647 Triangle, T(n, k) = [x^k]( p(x, n) ), where (1/2)*(1-x)^(n+1) * Sum_{j >= 0} ((4*j + 3)^n + (4*j+1)^n )*x^j, read by rows.

%I A154647 #8 May 27 2024 07:16:36
%S A154647 1,2,2,5,22,5,14,178,178,14,41,1308,3446,1308,41,122,9234,52084,52084,
%T A154647 9234,122,365,64082,692707,1434812,692707,64082,365,1094,442082,
%U A154647 8559030,32285474,32285474,8559030,442082,1094,3281,3048184,101121500,641507528,1151050534,641507528,101121500,3048184,3281
%N A154647 Triangle, T(n, k) = [x^k]( p(x, n) ), where (1/2)*(1-x)^(n+1) * Sum_{j >= 0} ((4*j + 3)^n + (4*j+1)^n )*x^j, read by rows.
%H A154647 G. C. Greubel, <a href="/A154647/b154647.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A154647 T(n, k) = [x^k]( p(x, n) ), where (1/2)*(1-x)^(n+1) * Sum_{j >= 0} ((4*j + 3)^n + (4*j+1)^n )*x^j.
%F A154647 T(n, n-k) = T(n, k).
%F A154647 Sum_{k=0..n} T(n, k) = A047053(n) (row sums).
%F A154647 T(n, 0) = T(n, n) = A007051(n). - _G. C. Greubel_, May 27 2024
%e A154647 Triangle begins as:
%e A154647      1;
%e A154647      2,      2;
%e A154647      5,     22,       5;
%e A154647     14,    178,     178,       14;
%e A154647     41,   1308,    3446,     1308,       41;
%e A154647    122,   9234,   52084,    52084,     9234,     122;
%e A154647    365,  64082,  692707,  1434812,   692707,   64082,    365;
%e A154647   1094, 442082, 8559030, 32285474, 32285474, 8559030, 442082, 1094;
%t A154647 m=12; p[x_, n_]= (1/2)*(1-x)^(n+1)*Sum[((4*j+3)^n + (4*j+1)^n)*x^j, {j,0,m +2}]; T[n_, k_]:= Coefficient[p[x, n], x, k];
%t A154647 Table[T[n,k], {n,0,m}, {k,0,n}]//Flatten
%o A154647 (Magma)
%o A154647 m:=12;
%o A154647 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A154647 p:= func< n,x | (1-x)^(n+1)*(&+[((4*j+3)^n+(4*j+1)^n)/2*x^j: j in [0..m+2]]) >;
%o A154647 T:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
%o A154647 [T(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, May 27 2024
%o A154647 (SageMath)
%o A154647 m=12
%o A154647 def p(x,n): return (1-x)^(n+1)*sum( ((4*j+3)^n +(4*j+1)^n)*x^j for j in range(m+2))/2
%o A154647 def T(n,k): return ( p(x,n) ).series(x, n+1).list()[k]
%o A154647 flatten([[T(n,k) for k in range(n+1)] for n in range(m+1)]) # _G. C. Greubel_, May 27 2024
%Y A154647 Cf. A007051, A047053 (row sums), A154646.
%K A154647 nonn,tabl
%O A154647 0,2
%A A154647 _Roger L. Bagula_, Jan 13 2009
%E A154647 Edited by _G. C. Greubel_, May 27 2024