OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = (1-x)^(n+1) * Sum_{k >= 0} ( (q*k + 1)^n + (q*(k+1) - 1)^n )*x^k, and q = 4.
T(n, k) f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = q^n * (1-x)^(n+1) * ( LerchPhi(x, -n, 1/q) + LerchPhi(x, -n, (q-1)/q) ), and q = 4.
T(n, n-k) = T(n, k).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 35, 1;
1, 329, 329, 1;
1, 2535, 6811, 2535, 1;
1, 18225, 103925, 103925, 18225, 1;
1, 127435, 1384685, 2868895, 1384685, 127435, 1;
1, 881977, 17115873, 64568761, 64568761, 17115873, 881977, 1;
MATHEMATICA
m:=13;
p[x_, n_, q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j, 0, m+ 2}];
f[n_, k_, q_]:= Coefficient[Series[p[x, n, q], {x, 0, m+2}], x, k];
T[n_, k_, q_]:= f[n, k, q] - f[n, 0, q] + 1;
Table[T[n, k, 4], {n, 0, m}, {k, 0, n}]//Flatten
PROG
(Magma)
m:=13;
R<x>:=PowerSeriesRing(Integers(), m+2);
p:= func< x, n, q | (1-x)^(n+1)*(&+[((q*j+1)^n + (q*(j+1)-1)^n)*x^j: j in [0..m+2]]) >;
f:= func< n, k, q | Coefficient(R!( p(x, n, q) ), k) >;
T:= func< n, k, q | f(n, k, q) - f(n, 0, q) + 1 >; // T = A176199
[T(n, k, 4): k in [0..n], n in [0..m]]; // G. C. Greubel, Jun 18 2024
(SageMath)
m=13
def p(x, n, q): return (1-x)^(n+1)*sum(((q*j+1)^n + (q*(j+1)-1)^n)*x^j for j in range(m+3))
def f(n, k, q): return ( p(x, n, q) ).series(x, n+1).list()[k]
def T(n, k, q): return f(n, k, q) - f(n, 0, q) + 1 # T = A176199
flatten([[T(n, k, 4) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 11 2010
EXTENSIONS
Edited by G. C. Greubel, Jun 18 2024
STATUS
approved