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A141290 Triangle read by rows, descending antidiagonals of a (1, 3, 5, ...) * (1, 4, 16, ...) multiplication table.

%I A141290 #20 May 22 2024 21:38:12
%S A141290 1,3,4,5,12,16,7,20,48,64,9,28,80,192,256,11,36,112,320,768,1024,13,
%T A141290 44,144,448,1280,3072,4096,15,52,176,576,1792,5120,12288,16384,17,60,
%U A141290 208,704,2304,7168,20480,49152,65536,19,68,240,832,2816,9216,28672,81920,196608,262144
%N A141290 Triangle read by rows, descending antidiagonals of a (1, 3, 5, ...) * (1, 4, 16, ...) multiplication table.
%C A141290 Binary representation of all terms ends in an even number of zeros (cf. A003159).
%F A141290 From _Stefano Spezia_, May 21 2024: (Start)
%F A141290 G.f. as array: x*y*(1 + y)/((1 - 4*x)*(1 - y)^2).
%F A141290 E.g.f. as array: (exp(4*x) - 1)*(exp(y)*(1 - 2*y) - 1)/4. (End)
%e A141290 Given the multiplication table (1, 3, 5, ...) * (1, 4, 16, ...); i.e., odd numbers as column headings, powers of 4 along the left border:
%e A141290    1,   3,   5,   7, ...
%e A141290    4,  12,  20,  28, ...
%e A141290   16,  48,  80, 112, ...
%e A141290   64, 192, 320, 448, ...
%e A141290   ...
%e A141290 Rows of the triangle = descending antidiagonals of the array, getting:
%e A141290    1;
%e A141290    3,  4;
%e A141290    5, 12,  16;
%e A141290    7, 20,  48,  64;
%e A141290    9, 28,  80, 192,  256;
%e A141290   11, 36, 112, 320,  768, 1024;
%e A141290   13, 44, 144, 448, 1280, 3072,  4096;
%e A141290   15, 52, 176, 576, 1792, 5120, 122288, 16384;
%e A141290   ...
%t A141290 A[n_,k_]:=(2k-1)*4^(n-1); Table[A[k,n-k+1],{n,10},{k,n}]//Flatten (* _Stefano Spezia_, May 21 2024 *)
%Y A141290 Cf. A003159, A141291.
%K A141290 nonn,tabl
%O A141290 1,2
%A A141290 _Gary W. Adamson_, Jun 22 2008
%E A141290 a(14), a(36) corrected by _Peter Munn_, Aug 27 2019