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