OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Sum_{k=0..n} T(n, k) = 2^n - n + A033312(n+1) (row sums).
T(n, k) = 2*A141689(n+1,k+1) - 1. - R. J. Mathar, Jan 19 2011
From G. C. Greubel, Dec 31 2024: (Start)
T(n, n-k) = T(n, k).
T(n, 1) = A036563(n+1).
Sum_{k=0..n} (-1)^k * T(n,k) = ((-1)^(n/2)*A000182(n/2 + 1) - 1)*(1 + (-1)^n)/2 + [n=0]. (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 5, 1;
1, 13, 13, 1;
1, 29, 71, 29, 1;
1, 61, 311, 311, 61, 1;
1, 125, 1205, 2435, 1205, 125, 1;
1, 253, 4313, 15653, 15653, 4313, 253, 1;
1, 509, 14635, 88289, 156259, 88289, 14635, 509, 1;
1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021, 1;
MAPLE
MATHEMATICA
Needs["Combinatorica`"];
T[n_, k_, 0]:= Binomial[n, k];
T[n_, k_, 1]:= Eulerian[1 + n, k];
T[n_, k_, q_]:= T[n, k, q] = T[n, k, q-1] + T[n, k, q-2] - 1;
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma)
A176487:= func< n, k | Binomial(n, k) + EulerianNumber(n+1, k) - 1 >;
[A176487(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 31 2024
(SageMath)
# from sage.all import * # (use for Python)
from sage.combinat.combinat import eulerian_number
def A176487(n, k): return binomial(n, k) +eulerian_number(n+1, k) -1
print(flatten([[A176487(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 31 2024
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Apr 19 2010
STATUS
approved