A099964 Triangle read by rows: The n-th row is constructed by forming the partial sums of the previous row, reading from the right and if n is a triangular number repeating the final term.

1, 1, 1, 1, 2, 2, 3, 3, 3, 6, 8, 8, 14, 17, 17, 31, 39, 39, 39, 78, 109, 126, 126, 235, 313, 352, 352, 665, 900, 1026, 1026, 1926, 2591, 2943, 2943, 2943, 5886, 8477, 10403, 11429, 11429, 21832, 30309, 36195, 39138, 39138, 75333, 105642, 127474, 138903

Offset

0,5

Comments

...

Links

Reinhard Zumkeller, Rows n = 0..500 of triangle, flattened

Example

Triangle begins
     1;
     1,    1;
     1,    2,
     2,    3,    3;
     3,    6,    8,
     8,   14,   17,
    17,   31,   39,   39;
    39,   78,  109,  126,
   126,  235,  313,  352,
   352,  665,  900, 1026,
  1026, 1926, 2591, 2943, 2943;

Maple

with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n, p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j], j=1..i): vector(n, q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j], j=1..i) else sum(a[j], j=1..n) fi end: vector(n+1, q) end: tr:={seq(n*(n+1)/2, n=1..30)}: R[0]:=vector(1, 1): for n from 1 to 15 do if member(n, tr)=false then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 15 do evalm(R[n]) od; # Emeric Deutsch, Nov 16 2004

Mathematica

triQ[n_] := Reduce[ n == k(k+1)/2, k, Integers] =!= False; row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = (ro = Accumulate[ Reverse[ row[n-1]]]; If[triQ[n], Append[ ro, Last[ro] ], ro]); Flatten[ Table[ row[n], {n, 0, 13}]](* Jean-François Alcover, Nov 24 2011 *)

Prog

(Haskell)
a099964 n k = a099964_tabf !! n !! k
a099964_row n = a099964_tabf !! n
a099964_tabf = scanl f [1] $ tail a010054_list where
   f row t = if t == 1 then row' ++ [last row'] else row'
           where row' = scanl1 (+) $ reverse row
-- Reinhard Zumkeller, May 02 2012

Crossrefs

First column (and row sums) gives A099965. Cf. A099966, A099968.

If an extra term is added to /every/ row we get A008282. Cf. A099959, A099961.

Cf. A010054.

Sequence in context: A115733 A025496 A099959 * A369302 A363826 A094440

Adjacent sequences: A099961 A099962 A099963 * A099965 A099966 A099967

Keyword

nonn,tabf,nice,easy

Author

N. J. A. Sloane, Nov 13 2004

Extensions

More terms from Emeric Deutsch, Nov 16 2004

Status

approved