A182579 - OEIS

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).

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, 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, 30, 25, 9, 2, 1, 1, 11, 10, 44, 35, 77, 50, 55, 25, 11, 2

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

A000204(n+1) = sum of n-th row, Lucas numbers;

A000204(n+3) = alternating row sum of n-th row;

A182584(n) = T(2*n,n), central terms;

A000012(n) = T(n,0), left edge;

A040000(n) = T(n,n), right edge;

A054977(n-1) = T(n,1) for n > 0;

A109613(n-1) = T(n,n-1) for n > 0;

A008794(n) = T(n,n-2) for n > 1.

LINKS

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.

FORMULA

T(n+1,2*k+1) = T(n,2*k), T(n+1,2*k) = T(n,2*k-1) + T(n,2*k).

EXAMPLE

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):

(0) 1 1 7 6 14 9 7 2 [A] row 7 prepended by 0

1 1 7 6 14 9 7 2 (0) [B] row 7, 0 appended

1 0 1 0 1 0 1 0 1 [C] 1 and 0 alternating

1 0 7 0 14 0 7 0 0 [D] = [B] multiplied by [C]

1 1 8 7 20 14 16 7 2 [E] = [D] added to [A] = row 8.

The triangle begins: | A000204

1 | 1

1 2 | 3

1 1 2 | 4

1 1 3 2 | 7

1 1 4 3 2 | 11

1 1 5 4 5 2 | 18

1 1 6 5 9 5 2 | 29

1 1 7 6 14 9 7 2 | 47

1 1 8 7 20 14 16 7 2 | 76

1 1 9 8 27 20 30 16 9 2 | 123

1 1 10 9 35 27 50 30 25 9 2 | 199 .

MATHEMATICA

T[_, 0] = 1;

T[n_, n_] /; n > 0 = 2;

T[_, 1] = 1;

T[n_, k_] := T[n, k] = Which[

OddQ[k], T[n - 1, k - 1],

EvenQ[k], T[n - 1, k - 1] + T[n - 1, k]];

Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)

PROG

(Haskell)

a182579 n k = a182579_tabl !! n !! k

a182579_row n = a182579_tabl !! n

a182579_tabl = [1] : iterate (\row ->

zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1, 2]

CROSSREFS

Cf. A065941, A059841.

Sequence in context: A322426 A335438 A145574 * A290737 A056138 A067594

Adjacent sequences: A182576 A182577 A182578 * A182580 A182581 A182582

KEYWORD

nonn,tabl,changed

AUTHOR

Reinhard Zumkeller, May 06 2012

STATUS

approved