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

%I A296229 #46 Apr 15 2018 16:45:27
%S A296229 2,4,4,8,32,8,16,176,176,16,32,832,2112,832,32,64,3648,19328,19328,
%T A296229 3648,64,128,15360,152448,309248,152448,15360,128,256,63232,1099008,
%U A296229 3998464,3998464,1099008,63232,256,512,257024,7479296,45175808,79969280,45175808,7479296,257024,512,1024,1037312,48988160
%N A296229 Triangle read by rows: Eulerian triangle that produces sums of even powers.
%C A296229 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).
%F A296229 T(n,k) = Sum_{i = 1..k} (-1)^(k-i)*binomial(n+1,k-i)*(2*i)^n.
%F A296229 a(n) = 2*A257609(n-1). - _Robert G. Wilson v_, Feb 19 2018
%e A296229 The triangle T(n, k) begins:
%e A296229 n\k |   1     2       3       4       5       6     7   8
%e A296229 ----+----------------------------------------------------
%e A296229   1 |   2
%e A296229   2 |   4     4
%e A296229   3 |   8    32       8
%e A296229   4 |  16   176     176      16
%e A296229   5 |  32   832    2112     832      32
%e A296229   6 |  64  3648   19328   19328    3648      64
%e A296229   7 | 128 15360  152448  309248  152448   15360   128
%e A296229   8 | 256 63232 1099008 3998464 3998464 1099008 63232 256
%e A296229 ...
%t A296229 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
%Y A296229 Row sums: A000165, A000079, A257609.
%K A296229 nonn,tabl
%O A296229 1,1
%A A296229 _Tony Foster III_, Feb 14 2018