A174303 A symmetrical triangle: T(n,k) = A008292(n+1, k) * f(n,k), where f(n,k) = 2^k when floor(n/2) >= k, otherwise 2^(n-k).
1, 1, 1, 1, 8, 1, 1, 22, 22, 1, 1, 52, 264, 52, 1, 1, 114, 1208, 1208, 114, 1, 1, 240, 4764, 19328, 4764, 240, 1, 1, 494, 17172, 124952, 124952, 17172, 494, 1, 1, 1004, 58432, 705872, 2499040, 705872, 58432, 1004, 1, 1, 2026, 191360, 3641536, 20965664, 20965664, 3641536, 191360, 2026, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 8, 1; 1, 22, 22, 1; 1, 52, 264, 52, 1; 1, 114, 1208, 1208, 114, 1; 1, 240, 4764, 19328, 4764, 240, 1; 1, 494, 17172, 124952, 124952, 17172, 494, 1; 1, 1004, 58432, 705872, 2499040, 705872, 58432, 1004, 1;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
- Eric Weisstein's World of Mathematics, Eulerian Number
Programs
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Magma
Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >; [[Floor(n/2) ge k select 2^k*Eulerian(n+1,k) else 2^(n-k)*Eulerian(n+1,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 15 2019
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Mathematica
Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1,j]*(k-j+1)^n, {j,0,k+1}]; Table[Eulerian[n+1,m]*If[Floor[n/2] >= m, 2^m, 2^(n-m)], {n,0,10}, {m,0,n} ]//Flatten (* modified by G. C. Greubel, Apr 15 2019 *) -
PARI
{eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n)}; for(n=0,10, for(k=0,n, print1(eulerian(n+1,k)*if(floor(n/2)>=k, 2^k, 2^(n-k)), ", "))) \\ G. C. Greubel, Apr 15 2019 -
Sage
def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1)) def T(n,k): if floor(n/2)>=k: return 2^k*Eulerian(n+1,k) else: return 2^(n-k)*Eulerian(n+1,k) [[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 15 2019
Formula
T(n,k) = Eulerian(n+1, k)*if(floor(n/2) greater than or equal to k then 2^m otherwise 2^(n-k)), where the Eulerian numbers are defined as A008292(n,k).
Extensions
Edited by G. C. Greubel, Apr 15 2019
Comments