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

1,1

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).

Links

Table of n, a(n) for n=1..48.

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

Adjacent sequences: A296226 A296227 A296228 * A296230 A296231 A296232

Keyword

nonn,tabl

Author

Tony Foster III, Feb 14 2018

Status

approved