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A102541 Triangle read by rows, formed from antidiagonals of Losanitsch's triangle. T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).

%I A102541 #38 Dec 29 2024 00:51:24
%S A102541 1,1,1,1,1,1,1,2,1,1,2,2,1,3,4,1,1,3,6,2,1,4,9,6,1,1,4,12,10,3,1,5,16,
%T A102541 19,9,1,1,5,20,28,19,3,1,6,25,44,38,12,1,1,6,30,60,66,28,4,1,7,36,85,
%U A102541 110,66,16,1,1,7,42,110,170,126,44,4,1,8,49,146,255,236,110,20,1,1,8,56
%N A102541 Triangle read by rows, formed from antidiagonals of Losanitsch's triangle. T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).
%C A102541 Row sums A102526 are essentially the same as A001224, A060312 and A068928.
%C A102541 Moving the terms in each column of this triangle, see the example, upwards to row 0 gives Losanitsch's triangle A034851 as a square array. - _Johannes W. Meijer_, Aug 24 2013
%C A102541 The number of ways to cover n-length line by exactly k 2-length segments excluding symmetric covers. - _Philipp O. Tsvetkov_, Nov 08 2013
%C A102541 Also the number of equivalence classes of ways of placing k 2 X 2 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - _Christopher Hunt Gribble_, Feb 16 2014
%C A102541 T(n, k) is the number of irreducible caterpillars with n+3 edges and diameter k+2. - _Christian Barrientos_, Apr 05 2020
%F A102541 T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).
%F A102541 T(n, k) = T(n-1, k) + T(n-2, k-1) - C((n-3)/2-(k-1)/2, (n-3)/2-(k-1)) except when n or k even then T(n, k) = T(n-1, k) + T(n-2, k-1) with T(0, 0) = 1, T(n, 0) = 0 for n<0 and T(n, k) = 0 for k < 0 and k > floor(n/2). - _Johannes W. Meijer_, Aug 24 2013
%e A102541 The first few rows of triangle T(n, k) are:
%e A102541 n/k: 0, 1, 2, 3
%e A102541 0:   1
%e A102541 1:   1
%e A102541 2:   1, 1
%e A102541 3:   1, 1
%e A102541 4:   1, 2, 1
%e A102541 5:   1, 2, 2
%e A102541 6:   1, 3, 4, 1
%e A102541 7:   1, 3, 6, 2
%p A102541 From _Johannes W. Meijer_, Aug 24 2013: (Start)
%p A102541 T := proc(n,k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/2) then return(0) fi: A034851(n-k, k) end: A034851 := proc(n, k) option remember; local t; if k = 0 or k = n then return(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1, (k-1)/2) else t := 0; fi; A034851(n-1, k-1) + A034851(n-1, k)-t; end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..16);  # End first program
%p A102541 T := proc(n,k) option remember: if n < 0 then return(0) fi: if k < 0 or k > floor(n/2) then return(0) fi: if n=0 then return(1) fi: if type(n, even) or type(k, even) then procname(n-1, k) + procname(n-2, k-1) else procname(n-1, k) + procname(n-2, k-1) - binomial((n-3)/2-(k-1)/2, (n-3)/2-(k-1)) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..16); # End second program (End)
%t A102541 t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2;
%t A102541 t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2;
%t A102541 T[n_, k_] := t[n - k, k];
%t A102541 Table[T[n, k], {n, 0, 16}, {k, 0, Quotient[n, 2]}] // Flatten (* _Jean-François Alcover_, Jul 21 2022 *)
%Y A102541 Cf. A034851, A005685, A005688, A068930, A102526, A102543, A192928.
%K A102541 nonn,tabf
%O A102541 0,8
%A A102541 _Gerald McGarvey_, Feb 24 2005
%E A102541 Definition edited, incorrect formula deleted, keyword corrected and extended by _Johannes W. Meijer_, Aug 24 2013