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A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

%I A238303 #12 Jan 19 2025 11:03:43
%S A238303 1,1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,
%T A238303 2,2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2,
%U A238303 2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.
%C A238303 Row sums are A005408(n).
%C A238303 Diagonals sums are A109613(n).
%C A238303 Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
%C A238303 Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
%C A238303 Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
%C A238303 Sum_{k=0..n} k*T(n,k) = A002378(n).
%C A238303 Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
%C A238303 Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).
%H A238303 Antti Karttunen, <a href="/A238303/b238303.txt">Table of n, a(n) for n = 0..22154; the first 210 rows of triangle</a>
%F A238303 T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.
%e A238303 Triangle begins:
%e A238303 1;
%e A238303 1, 2;
%e A238303 1, 2, 2;
%e A238303 1, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A238303 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A238303 ...
%o A238303 (PARI) A238303(n) = (2-ispolygonal(n,3)); \\ _Antti Karttunen_, Jan 19 2025
%Y A238303 Cf. Diagonals: A040000.
%Y A238303 Cf. Columns: A000012, A007395.
%Y A238303 First differences of A001614.
%K A238303 easy,nonn,tabl
%O A238303 0,3
%A A238303 _Philippe Deléham_, Feb 24 2014
%E A238303 Data section extended to a(104) by _Antti Karttunen_, Jan 19 2025