A296229 Triangle read by rows: Eulerian triangle that produces sums of even powers.

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

Author

Tony Foster III, Feb 14 2018

Keywords

• nonn
• tabl

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).

Examples

			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
...
		

Crossrefs

Row sums: A000165, A000079, A257609.

Programs

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

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