OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 10, 46, 370, 2646, 29338, 285238, 4029658, ...}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Eulerian Number
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).
EXAMPLE
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;
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 15 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 15 2019
STATUS
approved