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A125103 Triangle read by rows: T(n,k) = binomial(n,k) + 2^k*binomial(n,k+1) (0 <= k <= n).

%I A125103 #16 Nov 30 2019 15:24:28
%S A125103 1,2,1,3,4,1,4,9,7,1,5,16,22,12,1,6,25,50,50,21,1,7,36,95,140,111,38,
%T A125103 1,8,49,161,315,371,245,71,1,9,64,252,616,966,952,540,136,1,10,81,372,
%U A125103 1092,2142,2814,2388,1188,265,1,11,100,525,1800,4242,6972,7890,5880,2605,522,1
%N A125103 Triangle read by rows: T(n,k) = binomial(n,k) + 2^k*binomial(n,k+1) (0 <= k <= n).
%C A125103 Row sums = A094374: (1, 3, 8, 21, 56, ...).
%C A125103 Binomial transform of the infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (1,2,4,8,...) in the subdiagonal.
%H A125103 Harvey P. Dale, <a href="/A125103/b125103.txt">Table of n, a(n) for n = 0..1000</a>
%e A125103 First few rows of the triangle are
%e A125103   1;
%e A125103   2,   1;
%e A125103   3,   4,   1;
%e A125103   4,   9,   7,   1;
%e A125103   5,  16,  22,  12,   1;
%e A125103   6,  25,  50,  50,  21,   1;
%e A125103   7,  36,  95, 140, 111,  38,  1;
%e A125103   ...
%p A125103 T:=(n,k)->binomial(n,k)+2^k*binomial(n,k+1): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A125103 Table[Binomial[n,k]+2^k Binomial[n,k+1],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Nov 30 2019 *)
%o A125103 (PARI) T(n,k) = binomial(n,k) + 2^k*binomial(n,k+1);
%o A125103 matrix(11, 11, n, k, T(n-1,k-1)) \\ _Michel Marcus_, Nov 09 2019
%Y A125103 Cf. A094374.
%K A125103 nonn,tabl
%O A125103 0,2
%A A125103 _Gary W. Adamson_, Nov 20 2006
%E A125103 Edited by _N. J. A. Sloane_, Nov 29 2006