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%I #35 Sep 28 2022 18:52:54
%S 0,1,1,2,3,4,3,5,8,12,4,7,12,20,32,5,9,16,28,48,80,6,11,20,36,64,112,
%T 192,7,13,24,44,80,144,256,448,8,15,28,52,96,176,320,576,1024,9,17,32,
%U 60,112,208,384,704,1280,2304,10,19,36,68,128,240,448,832,1536,2816,5120
%N Triangle read by rows: triangle A062111 reversed.
%H Alois P. Heinz, <a href="/A152920/b152920.txt">Rows n = 0..150, flattened</a>
%F Row sums: (2^n-1)(n+1) = A058877(n). - _R. J. Mathar_, Jan 22 2009
%F T(2n, n) = 3*n*2^(n-1) = 3*A001787(n). - _Philippe Deléham_, Apr 20 2009
%F From _Werner Schulte_, Jul 31 2020: (Start)
%F T(n, k) = (2*n-k) * 2^(k-1) for 0 <= k <= n.
%F G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = t*(1+x-3*x*t) / ((1-t)^2 * (1-2*x*t)^2).
%F Sum_{k=0..n} (-1)^k * binomial(n,k) * T(n,k) = 0 for n >= 0.
%F Sum_{k=0..n} binomial(n,k) * T(n,k) = 2*n * 3^(n-1) for n >= 0.
%F Define the array B(n,p) = (Sum_{k=0..n} binomial(p+k,p) * T(n,k))/(n+p+1) for n >= 0 and p >= 0. Then see the comment of Robert Coquereaux (2014) at A193844. Conjecture: B(n+1,p) = A(n,p). (End)
%F T(n, k) = T(n, k-1) + T(n-1, k-1) for k>=1, T(n,0) = n. - _Alois P. Heinz_, Sep 12 2022
%F From _G. C. Greubel_, Sep 27 2022: (Start)
%F T(n, n-1) = A001792(n).
%F T(2*n-1, n-1) = A053220(n).
%F T(2*n+1, n-1) = 3*A001792(n).
%F T(m*n, n) = (2*m-1)*A001787(n), for m >= 1. (End)
%e Triangle starts:
%e 0;
%e 1, 1;
%e 2, 3, 4;
%e 3, 5, 8, 12;
%e 4, 7, 12, 20, 32;
%e ...
%p A062111 := proc(n,k) (k+n)*2^(k-n-1) ; end: A152920 := proc(n,k) A062111(n-k,n) ; end: for n from 0 to 15 do for k from 0 to n do printf("%d,",A152920(n,k)) ; od: od: # _R. J. Mathar_, Jan 22 2009
%p # second Maple program:
%p T:= proc(n, k) option remember;
%p `if`(k=0, n, T(n, k-1)+T(n-1, k-1))
%p end:
%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Sep 12 2022
%t t[0, k_]:= k; t[n_, k_]:= t[n, k]= t[n-1, k] + t[n-1, k+1];
%t Table[t[n-k, k], {n,0,10}, {k,n,0,-1}]//Flatten (* _Jean-François Alcover_, Sep 11 2016 *)
%o (Magma) [2^k*(n-k/2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 27 2022
%o (SageMath) flatten([[2^(k-1)*(2*n-k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Sep 27 2022
%Y Cf. A053220, A058877 (row sums), A193844, A212697.
%Y Columns and diagonals: A001787, A001792, A034007, A045623, A045891, A111297, A159694, A159695, A159696, A159697.
%K nonn,tabl,easy
%O 0,4
%A _Paul Curtz_, Dec 15 2008
%E Edited by _N. J. A. Sloane_, Dec 19 2008
%E More terms from _R. J. Mathar_, Jan 22 2009
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