A176199 - OEIS

%I #8 Jun 19 2024 09:32:39

%S 1,1,1,1,35,1,1,329,329,1,1,2535,6811,2535,1,1,18225,103925,103925,

%T 18225,1,1,127435,1384685,2868895,1384685,127435,1,1,881977,17115873,

%U 64568761,64568761,17115873,881977,1,1,6089807,202236439,1283008495,2302094507,1283008495,202236439,6089807,1

%N 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.

%H G. C. Greubel, <a href="/A176199/b176199.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F 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.

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

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     35,        1;

%e   1,    329,      329,        1;

%e   1,   2535,     6811,     2535,        1;

%e   1,  18225,   103925,   103925,    18225,        1;

%e   1, 127435,  1384685,  2868895,  1384685,   127435,      1;

%e   1, 881977, 17115873, 64568761, 64568761, 17115873, 881977,     1;

%t m:=13;

%t p[x_,n_,q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j,0,m+ 2}];

%t f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];

%t T[n_,k_,q_]:= f[n,k,q] - f[n,0,q] + 1;

%t Table[T[n,k,4], {n,0,m}, {k,0,n}]//Flatten

%o (Magma)

%o m:=13;

%o R<x>:=PowerSeriesRing(Integers(), m+2);

%o 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]]) >;

%o f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;

%o T:= func< n,k,q | f(n,k,q) - f(n,0,q) + 1 >; // T = A176199

%o [T(n,k,4): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jun 18 2024

%o (SageMath)

%o m=13

%o 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))

%o def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]

%o def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A176199

%o flatten([[T(n,k,4) for k in range(n+1)] for n in (0..m)]) # _G. C. Greubel_, Jun 18 2024

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

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 11 2010

%E Edited by _G. C. Greubel_, Jun 18 2024