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A182579 Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2*k) = n/(n-k) * binomial(n-k,k), T(n,2*k+1) = (n-1)/(n-1-k) * binomial(n-1-k,k).

%I A182579 #21 Jan 05 2025 19:51:39
%S A182579 1,1,2,1,1,2,1,1,3,2,1,1,4,3,2,1,1,5,4,5,2,1,1,6,5,9,5,2,1,1,7,6,14,9,
%T A182579 7,2,1,1,8,7,20,14,16,7,2,1,1,9,8,27,20,30,16,9,2,1,1,10,9,35,27,50,
%U A182579 30,25,9,2,1,1,11,10,44,35,77,50,55,25,11,2
%N A182579 Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2*k) = n/(n-k) * binomial(n-k,k), T(n,2*k+1) = (n-1)/(n-1-k) * binomial(n-1-k,k).
%C A182579 A000204(n+1) = sum of n-th row, Lucas numbers;
%C A182579 A000204(n+3) = alternating row sum of n-th row;
%C A182579 A182584(n) = T(2*n,n), central terms;
%C A182579 A000012(n) = T(n,0), left edge;
%C A182579 A040000(n) = T(n,n), right edge;
%C A182579 A054977(n-1) = T(n,1) for n > 0;
%C A182579 A109613(n-1) = T(n,n-1) for n > 0;
%C A182579 A008794(n) = T(n,n-2) for n > 1.
%H A182579 Reinhard Zumkeller, <a href="/A182579/b182579.txt">Rows n = 0..150 of triangle, flattened</a>
%H A182579 Henry W. Gould, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-4/gould.pdf">A Variant of Pascal's Triangle</a>, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/4-1/corrections2.pdf">corrections</a>.
%F A182579 T(n+1,2*k+1) = T(n,2*k), T(n+1,2*k) = T(n,2*k-1) + T(n,2*k).
%e A182579 Starting with 2nd row = [1 2] the rows of the triangle are defined recursively without computing explicitely binomial coefficients; demonstrated for row 8, (see also Haskell program):
%e A182579    (0) 1  1  7  6 14  9  7  2      [A]  row 7 prepended by 0
%e A182579     1  1  7  6 14  9  7  2 (0)     [B]  row 7, 0 appended
%e A182579     1  0  1  0  1  0  1  0  1      [C]  1 and 0 alternating
%e A182579     1  0  7  0 14  0  7  0  0      [D]  = [B] multiplied by [C]
%e A182579     1  1  8  7 20 14 16  7  2      [E]  = [D] added to [A] = row 8.
%e A182579 The triangle begins:                 | A000204
%e A182579               1                      |       1
%e A182579              1  2                    |       3
%e A182579             1  1  2                  |       4
%e A182579            1  1  3  2                |       7
%e A182579           1  1  4  3  2              |      11
%e A182579          1  1  5  4  5  2            |      18
%e A182579         1  1  6  5  9  5  2          |      29
%e A182579        1  1  7  6 14  9  7  2        |      47
%e A182579       1  1  8  7 20 14 16  7  2      |      76
%e A182579      1  1  9  8 27 20 30 16  9  2    |     123
%e A182579     1  1 10  9 35 27 50 30 25  9  2  |     199 .
%t A182579 T[_, 0] = 1;
%t A182579 T[n_, n_] /; n > 0 = 2;
%t A182579 T[_, 1] = 1;
%t A182579 T[n_, k_] := T[n, k] = Which[
%t A182579      OddQ[k],  T[n - 1, k - 1],
%t A182579      EvenQ[k], T[n - 1, k - 1] + T[n - 1, k]];
%t A182579 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 01 2021 *)
%o A182579 (Haskell)
%o A182579 a182579 n k = a182579_tabl !! n !! k
%o A182579 a182579_row n = a182579_tabl !! n
%o A182579 a182579_tabl = [1] : iterate (\row ->
%o A182579   zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1,2]
%Y A182579 Cf. A065941, A059841.
%K A182579 nonn,tabl
%O A182579 0,3
%A A182579 _Reinhard Zumkeller_, May 06 2012