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A162741 Fibonacci-Pascal triangle; same as Pascal triangle, but beginning another Pascal triangle to the right of each row starting at row 2.

%I A162741 #25 Jan 05 2025 19:51:39
%S A162741 1,1,1,1,1,2,2,1,1,1,3,4,3,2,1,1,1,4,7,7,5,3,2,1,1,1,5,11,14,12,8,5,3,
%T A162741 2,1,1,1,6,16,25,26,20,13,8,5,3,2,1,1,1,7,22,41,51,46,33,21,13,8,5,3,
%U A162741 2,1,1,1,8,29,63,92,97,79,54,34,21,13,8,5,3,2,1,1
%N A162741 Fibonacci-Pascal triangle; same as Pascal triangle, but beginning another Pascal triangle to the right of each row starting at row 2.
%C A162741 Intertwined Pascal-triangles;
%C A162741 the first five rows seen as numbers in decimal representation: row(n) = 110*row(n-1) + 1. - corrected by _Reinhard Zumkeller_, Jul 16 2013
%H A162741 Reinhard Zumkeller, <a href="/A162741/b162741.txt">Rows n = 1..100 of triangle, flattened</a>
%H A162741 Richard L. Ollerton and Anthony G. Shannon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-2/ollerton.pdf">Some properties of generalized Pascal squares and triangles</a>, Fib. Q., 36 (1998), 98-109. See Table 3.
%H A162741 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A162741 T(n,k) = T(n-1,k-1) + T(n-1,k), T(n,1)=1 and for n>1: T(n,2*n-2) = T(n,2*n-1)=1. - _Reinhard Zumkeller_, Jul 16 2013
%e A162741 .                                           1
%e A162741 .                                       1,  1, 1
%e A162741 .                                   1,  2,  2, 1, 1
%e A162741 .                               1,  3,  4,  3, 2, 1, 1
%e A162741 .                           1,  4,  7,  7,  5, 3, 2, 1, 1
%e A162741 .                       1,  5, 11, 14, 12,  8, 5, 3, 2, 1, 1
%e A162741 .                   1,  6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1,1
%e A162741 .               1,  7, 22, 41, 51, 46, 33, 21,13, 8, 5, 3, 2,1,1
%e A162741 .           1,  8, 29, 63, 92, 97, 79, 54, 34,21,13, 8, 5, 3,2,1,1
%e A162741 .       1,  9, 37, 92,155,189,176,133, 88, 55,34,21,13, 8, 5,3,2,1,1
%e A162741 .    1,10, 46,129,247,344,365,309,221,143, 89,55,34,21,13, 8,5,3,2,1,1
%e A162741 . 1,11,56,175,376,591,709,674,530,364,232,144,89,55,34,21,13,8,5,3,2,1,1 .
%t A162741 T[_, 1] = 1; T[n_, k_] /; k == 2*n-2 || k == 2*n-1 = 1; T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k]; Table[T[n, k], {n, 1, 9}, {k, 1, 2*n-1}] // Flatten (* _Jean-François Alcover_, Oct 30 2017, after _Reinhard Zumkeller_ *)
%o A162741 (Haskell)
%o A162741 a162741 n k = a162741_tabf !! (n-1) !! (k-1)
%o A162741 a162741_row n = a162741_tabf !! (n-1)
%o A162741 a162741_tabf = iterate
%o A162741    (\row -> zipWith (+) ([0] ++ row ++ [0]) (row ++ [0,1])) [1]
%o A162741 -- _Reinhard Zumkeller_, Jul 16 2013
%Y A162741 Cf. A005408 (row length), A000225 (row sums), A000045 (central terms), A007318, A136431.
%Y A162741 Cf. A021113. - _Mark Dols_, Jul 18 2009
%Y A162741 Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A228074.
%K A162741 nonn,tabf
%O A162741 1,6
%A A162741 _Mark Dols_, Jul 12 2009, Jul 19 2009