A174599 Triangle T(n, k) = A154646(n,k) - A154646(n,0) + 1, 0 <= k <= n.

1, 1, 1, 1, 22, 1, 1, 145, 145, 1, 1, 780, 2246, 780, 1, 1, 3919, 25144, 25144, 3919, 1, 1, 19202, 243047, 524812, 243047, 19202, 1, 1, 93349, 2168107, 8760511, 8760511, 2168107, 93349, 1, 1, 453592, 18445564, 127880680, 235517062, 127880680, 18445564, 453592, 1

Offset

0,5

Comments

The first and last element of each row of A154646 are reduced to 1 by subtracting a constant from each row.

Links

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

Formula

From G. C. Greubel, Jun 18 2024: (Start)

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 = 3.

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 = 3.

T(n, k) = A154646(n,k) - A154646(n,0) + 1.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,    22,       1;
  1,   145,     145,       1;
  1,   780,    2246,     780,       1;
  1,  3919,   25144,   25144,    3919,       1;
  1, 19202,  243047,  524812,  243047,   19202,     1;
  1, 93349, 2168107, 8760511, 8760511, 2168107, 93349,    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, 3], {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 = A174599
[T(n, k, 3): 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 = A174599
flatten([[T(n, k, 3) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024

Crossrefs

Related triangles dependent on q: A008518 (q=1), A176198 (q=2), this sequence (q=3), A176199 (q=4).

Cf. A154646.

Sequence in context: A040484 A225356 A291072 * A291074 A225076 A022185

Adjacent sequences: A174596 A174597 A174598 * A174600 A174601 A174602

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 23 2010

Status

approved