A176199 Triangle, read by rows, 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.

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, 1, 6089807, 202236439, 1283008495, 2302094507, 1283008495, 202236439, 6089807, 1

Offset: 0

Author

Roger L. Bagula, Apr 11 2010

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Related triangles dependent on q: A008518 (q=1), A176198 (q=2), A174599 (q=3), this sequence (q=4).

Programs

Magma

m:=13;
R:=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

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

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

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).

Extensions

Edited by G. C. Greubel, Jun 18 2024