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A296229
Triangle read by rows: Eulerian triangle that produces sums of even powers.
0
2, 4, 4, 8, 32, 8, 16, 176, 176, 16, 32, 832, 2112, 832, 32, 64, 3648, 19328, 19328, 3648, 64, 128, 15360, 152448, 309248, 152448, 15360, 128, 256, 63232, 1099008, 3998464, 3998464, 1099008, 63232, 256, 512, 257024, 7479296, 45175808, 79969280, 45175808, 7479296, 257024, 512, 1024, 1037312, 48988160
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
Row sums: A000165, A000079, A257609.
Sequence in context: A239649 A264190 A006967 * A322175 A298117 A122033
KEYWORD
nonn,tabl
AUTHOR
Tony Foster III, Feb 14 2018
STATUS
approved