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A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

%I A137153 #20 Jun 09 2025 01:09:06
%S A137153 1,1,1,1,2,1,1,3,4,1,1,4,10,8,1,1,5,20,36,16,1,1,6,35,120,136,32,1,1,
%T A137153 7,56,330,816,528,64,1,1,8,84,792,3876,5984,2080,128,1,1,9,120,1716,
%U A137153 15504,52360,45760,8256,256,1,1,10,165,3432,54264,376992,766480,357760,32896
%N A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).
%C A137153 Matrix inverse is A137156.
%C A137153 T(n,k) is the number of relations between a set of k distinguishable elements and a set of n-k indistinguishable elements. - _Isaac R. Browne_, Jun 04 2025
%H A137153 Paul D. Hanna, <a href="/A137153/b137153.txt">Table of n, a(n) for n = 0..1080; rows 0..45 of flattened triangle.</a>
%e A137153 Triangle begins:
%e A137153   1;
%e A137153   1,  1;
%e A137153   1,  2,   1;
%e A137153   1,  3,   4,    1;
%e A137153   1,  4,  10,    8,     1;
%e A137153   1,  5,  20,   36,    16,      1;
%e A137153   1,  6,  35,  120,   136,     32,      1;
%e A137153   1,  7,  56,  330,   816,    528,     64,      1;
%e A137153   1,  8,  84,  792,  3876,   5984,   2080,    128,     1;
%e A137153   1,  9, 120, 1716, 15504,  52360,  45760,   8256,   256,   1;
%e A137153   1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896, 512, 1;
%e A137153   ...
%t A137153 Table[Binomial[2^k+n-k-1,n-k],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Mar 06 2017 *)
%o A137153 (PARI) {T(n,k)=binomial(2^k+n-k-1,n-k)}
%o A137153 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o A137153 (PARI) {T(n, k) = polcoeff(1/(1-x+x*O(x^n))^(2^k), n-k)}
%o A137153 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A137153 Cf. A137154 (row sums), A137155 (antidiagonal sums), A060690 (central terms); A137156 (matrix inverse).
%Y A137153 Cf. A092056 (same with reflected rows).
%K A137153 nonn,tabl
%O A137153 0,5
%A A137153 _Paul D. Hanna_, Jan 24 2008