|
%I #16 Dec 01 2021 08:31:39
%S 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 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 30,25,9,2,1,1,11,10,44,35,77,50,55,25,11,2
%N 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 A000204(n+1) = sum of n-th row, Lucas numbers;
%C A000204(n+3) = alternating row sum of n-th row;
%C A182584(n) = T(2*n,n), central terms;
%C A000012(n) = T(n,0), left edge;
%C A040000(n) = T(n,n), right edge;
%C A054977(n-1) = T(n,1) for n > 0;
%C A109613(n-1) = T(n,n-1) for n > 0;
%C A008794(n) = T(n,n-2) for n > 1.
%H Reinhard Zumkeller, <a href="/A182579/b182579.txt">Rows n = 0..150 of triangle, flattened</a>
%H Henry W. Gould, <a href="http://www.fq.math.ca/3-4.html"> A Variant of Pascal's Triangle</a>, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 261ff.
%F 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 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 . (0) 1 1 7 6 14 9 7 2 [A] row 7 prepended by 0
%e . 1 1 7 6 14 9 7 2 (0) [B] row 7, 0 appended
%e . 1 0 1 0 1 0 1 0 1 [C] 1 and 0 alternating
%e . 1 0 7 0 14 0 7 0 0 [D] = [B] multiplied by [C]
%e . 1 1 8 7 20 14 16 7 2 [E] = [D] added to [A] = row 8.
%e The triangle begins: | A000204
%e . 1 | 1
%e . 1 2 | 3
%e . 1 1 2 | 4
%e . 1 1 3 2 | 7
%e . 1 1 4 3 2 | 11
%e . 1 1 5 4 5 2 | 18
%e . 1 1 6 5 9 5 2 | 29
%e . 1 1 7 6 14 9 7 2 | 47
%e . 1 1 8 7 20 14 16 7 2 | 76
%e . 1 1 9 8 27 20 30 16 9 2 | 123
%e . 1 1 10 9 35 27 50 30 25 9 2 | 199 .
%t T[_, 0] = 1;
%t T[n_, n_] /; n > 0 = 2;
%t T[_, 1] = 1;
%t T[n_, k_] := T[n, k] = Which[
%t OddQ[k], T[n - 1, k - 1],
%t EvenQ[k], T[n - 1, k - 1] + T[n - 1, k]];
%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 01 2021 *)
%o (Haskell)
%o a182579 n k = a182579_tabl !! n !! k
%o a182579_row n = a182579_tabl !! n
%o a182579_tabl = [1] : iterate (\row ->
%o zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1,2]
%Y Cf. A065941, A059841.
%K nonn,tabl
%O 0,3
%A _Reinhard Zumkeller_, May 06 2012
|
