cp's OEIS Frontend

A059328 Table T(n,k) = T(n - 1,k) + T(n,k - 1) + T(n - 1,k)*T(n,k - 1) starting with T(0,0)=1, read by antidiagonals.

%I A059328 #24 Apr 30 2025 16:20:32
%S A059328 1,1,1,1,3,1,1,7,7,1,1,15,63,15,1,1,31,1023,1023,31,1,1,63,32767,
%T A059328 1048575,32767,63,1,1,127,2097151,34359738367,34359738367,2097151,127,
%U A059328 1,1,255,268435455,72057594037927935,1180591620717411303423,72057594037927935,268435455,255,1
%N A059328 Table T(n,k) = T(n - 1,k) + T(n,k - 1) + T(n - 1,k)*T(n,k - 1) starting with T(0,0)=1, read by antidiagonals.
%C A059328 In binary representation T(n,k) is the concatenation of T(n-1,k-1) and T(n-1,k), 0<k<n. - _Reinhard Zumkeller_, Jan 23 2003
%H A059328 G. C. Greubel, <a href="/A059328/b059328.txt">Table of n, a(n) for the first 14 rows, flattened</a>
%F A059328 T(n, k) = 2^C(n+k, n)-1; a(n) = 2^A007318(n)-1.
%F A059328 If U(n, k) := 1 + T(n, k), then U(n, k) = U(n-1, k) * U(n, k-1). - _Michael Somos_, Jan 07 2017
%e A059328 Triangle T(n,k) begins:
%e A059328   1;
%e A059328   1,  1;
%e A059328   1,  3,     1;
%e A059328   1,  7,     7,       1;
%e A059328   1, 15,    63,      15,     1;
%e A059328   1, 31,  1023,    1023,    31,  1;
%e A059328   1, 63, 32767, 1048575, 32767, 63, 1;
%e A059328   ...
%t A059328 Table[2^(Binomial[n, k]) - 1, {n, 0, 5}, {k, 0, n}] (* _G. C. Greubel_, Jan 07 2017 *)
%o A059328 (Python)
%o A059328 from math import comb, isqrt
%o A059328 def A059328(n): return (1<<comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2)))-1 # _Chai Wah Wu_, Apr 30 2025
%Y A059328 Columns k=0-2 give: A000012, A000225, A126883(n-1).
%Y A059328 Row sums give A306020.
%Y A059328 Cf. A007318.
%K A059328 nonn,tabl
%O A059328 0,5
%A A059328 _Henry Bottomley_, Jan 26 2001