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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.

%I A099964 #23 Aug 03 2021 19:05:00
%S A099964 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,
%T A099964 352,352,665,900,1026,1026,1926,2591,2943,2943,2943,5886,8477,10403,
%U A099964 11429,11429,21832,30309,36195,39138,39138,75333,105642,127474,138903
%N 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.
%C A099964 ...
%H A099964 Reinhard Zumkeller, <a href="/A099964/b099964.txt">Rows n = 0..500 of triangle, flattened</a>
%e A099964 Triangle begins
%e A099964      1;
%e A099964      1,    1;
%e A099964      1,    2,
%e A099964      2,    3,    3;
%e A099964      3,    6,    8,
%e A099964      8,   14,   17,
%e A099964     17,   31,   39,   39;
%e A099964     39,   78,  109,  126,
%e A099964    126,  235,  313,  352,
%e A099964    352,  665,  900, 1026,
%e A099964   1026, 1926, 2591, 2943, 2943;
%p A099964 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
%t A099964 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 *)
%o A099964 (Haskell)
%o A099964 a099964 n k = a099964_tabf !! n !! k
%o A099964 a099964_row n = a099964_tabf !! n
%o A099964 a099964_tabf = scanl f [1] $ tail a010054_list where
%o A099964    f row t = if t == 1 then row' ++ [last row'] else row'
%o A099964            where row' = scanl1 (+) $ reverse row
%o A099964 -- _Reinhard Zumkeller_, May 02 2012
%Y A099964 First column (and row sums) gives A099965. Cf. A099966, A099968.
%Y A099964 If an extra term is added to /every/ row we get A008282. Cf. A099959, A099961.
%Y A099964 Cf. A010054.
%K A099964 nonn,tabf,nice,easy
%O A099964 0,5
%A A099964 _N. J. A. Sloane_, Nov 13 2004
%E A099964 More terms from _Emeric Deutsch_, Nov 16 2004