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.

1, 2, 2, 5, 22, 5, 14, 178, 178, 14, 41, 1308, 3446, 1308, 41, 122, 9234, 52084, 52084, 9234, 122, 365, 64082, 692707, 1434812, 692707, 64082, 365, 1094, 442082, 8559030, 32285474, 32285474, 8559030, 442082, 1094, 3281, 3048184, 101121500, 641507528, 1151050534, 641507528, 101121500, 3048184, 3281

Offset: 0

Author

Roger L. Bagula, Jan 13 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
     1;
     2,      2;
     5,     22,       5;
    14,    178,     178,       14;
    41,   1308,    3446,     1308,       41;
   122,   9234,   52084,    52084,     9234,     122;
   365,  64082,  692707,  1434812,   692707,   64082,    365;
  1094, 442082, 8559030, 32285474, 32285474, 8559030, 442082, 1094;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A007051, A047053 (row sums), A154646.

Programs

Magma

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
p:= func< n,x | (1-x)^(n+1)*(&+[((4*j+3)^n+(4*j+1)^n)/2*x^j: j in [0..m+2]]) >;
T:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
[T(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, May 27 2024

Mathematica

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];
Table[T[n,k], {n,0,m}, {k,0,n}]//Flatten

SageMath

m=12
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
def T(n,k): return ( p(x,n) ).series(x, n+1).list()[k]
flatten([[T(n,k) for k in range(n+1)] for n in range(m+1)]) # G. C. Greubel, May 27 2024

Formula

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.

T(n, n-k) = T(n, k).

Sum_{k=0..n} T(n, k) = A047053(n) (row sums).

T(n, 0) = T(n, n) = A007051(n). - G. C. Greubel, May 27 2024

Extensions

Edited by G. C. Greubel, May 27 2024