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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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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 = A174599 [T(n,k,3): 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,3], {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 = A174599 flatten([[T(n,k,3) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024
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, n-k) = T(n, k). (End)
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