OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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
EXAMPLE
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;
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
PROG
(Magma)
m:=12;
R<x>:=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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 13 2009
EXTENSIONS
Edited by G. C. Greubel, May 27 2024
STATUS
approved