OFFSET
1,1
COMMENTS
Finite sums of consecutive even powers are derived from T(n,k) rows and binomial coefficients: Sum_{k=1..n} (2k)^m = Sum_{j=1..m} binomial(n+m+1-j,m+1)*T(m,j).
FORMULA
T(n,k) = Sum_{i = 1..k} (-1)^(k-i)*binomial(n+1,k-i)*(2*i)^n.
a(n) = 2*A257609(n-1). - Robert G. Wilson v, Feb 19 2018
EXAMPLE
The triangle T(n, k) begins:
n\k | 1 2 3 4 5 6 7 8
----+----------------------------------------------------
1 | 2
2 | 4 4
3 | 8 32 8
4 | 16 176 176 16
5 | 32 832 2112 832 32
6 | 64 3648 19328 19328 3648 64
7 | 128 15360 152448 309248 152448 15360 128
8 | 256 63232 1099008 3998464 3998464 1099008 63232 256
...
MATHEMATICA
T[n_, k_] := Sum[(-1)^(k-i)*Binomial[n+1, k-i]*(2*i)^(n), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tony Foster III, Feb 14 2018
STATUS
approved