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A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.

%I A350771 #16 May 04 2024 19:41:58
%S A350771 0,1,1,3,4,3,7,12,12,7,15,32,36,32,15,31,80,100,100,80,31,63,192,270,
%T A350771 280,270,192,63,127,448,714,770,770,714,448,127,255,1024,1848,2128,
%U A350771 2100,2128,1848,1024,255,511,2304,4680,5880,5796,5796,5880,4680,2304,511,1023,5120,11610,16080,16380,15624,16380,16080,11610,5120,1023
%N A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.
%C A350771 The elements in T(n,k) result from the product of each element of A350770(n,k) and binomial(n-1,k).
%H A350771 Andrew Howroyd, <a href="/A350771/b350771.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A350771 Ambrosio Valencia-Romero and P. T. Grogan, <a href="https://doi.org/10.1371/journal.pone.0301394">The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination</a>, PLOS ONE 19(4): e0301394 (<a href="https://doi.org/10.1371/journal.pone.0301394.s001">S1 Appendix</a>).
%F A350771 T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k).
%e A350771 Triangle begins:
%e A350771      0;
%e A350771      1,    1;
%e A350771      3,    4,     3;
%e A350771      7,   12,    12,     7;
%e A350771     15,   32,    36,    32,    15;
%e A350771     31,   80,   100,   100,    80,    31;
%e A350771     63,  192,   270,   280,   270,   192,    63;
%e A350771    127,  448,   714,   770,   770,   714,   448,   127;
%e A350771    255, 1024,  1848,  2128,  2100,  2128,  1848,  1024,   255;
%e A350771    511, 2304,  4680,  5880,  5796,  5796,  5880,  4680,  2304,  511;
%e A350771   1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023;
%e A350771 ...
%p A350771 T := n -> local k; seq((2^(n - k - 1) + 2^k - 2)*binomial(n - 1, k), k = 0 .. n - 1);
%p A350771 seq(T(n), n = 1 .. 11);
%o A350771 (PARI) T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k) \\ _Andrew Howroyd_, Jan 05 2024
%Y A350771 Column k=0 gives A000225(n-1).
%Y A350771 Row sums give A056182(n-1) = 2*A001047(n-1).
%K A350771 nonn,tabl
%O A350771 1,4
%A A350771 _Ambrosio Valencia-Romero_, Jan 14 2022