A099960 -id:A099960 - cp's OEIS frontend

A099959 Triangle read by rows: Each row is constructed by forming the partial sums of the previous row, reading from the right and at every other row repeating the final term.

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 6, 8, 8, 14, 17, 17, 17, 34, 48, 56, 56, 104, 138, 155, 155, 155, 310, 448, 552, 608, 608, 1160, 1608, 1918, 2073, 2073, 2073, 4146, 6064, 7672, 8832, 9440, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 38227, 76454, 112608

Offset: 0

N. J. A. Sloane, Nov 13 2004

...

			Triangle begins
   1;
   1,
   1,   1;
   1,   2,
   2,   3,   3;
   3,   6,   8,
   8,  14,  17,  17;
  17,  34,  48,  56,
  56, 104, 138, 155, 155;

Reinhard Zumkeller, Rows n = 0..119 of triangle, flattened

First column (and row sums) gives A099960.

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

a099959 n k = a099959_tabl !! n !! k
a099959_row n = a099959_tabl !! n
a099959_tabl = map snd $ iterate f (False,[1]) where
   f (s,xs) = (not s, if s then zs ++ [last zs] else zs)
     where zs = scanl1 (+) (reverse xs)
-- Reinhard Zumkeller, Dec 28 2011

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: R[0]:=vector(1,1): for n from 1 to 18 do if n mod 2 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 18 do evalm(R[n]) od; # program yields the successive rows # Emeric Deutsch, Nov 16 2004

row[0] = row[1] = {1}; row[n_?OddQ] := Accumulate[ Reverse[ row[n-1] ] ]; row[n_?EvenQ] := (r = Accumulate[ Reverse[ row[n-1] ] ]; AppendTo[r, Last[r] ]); Flatten[ Table[ row[n], {n, 0, 13}]] (* Jean-François Alcover, Dec 16 2011 *)

More terms from Emeric Deutsch, Nov 16 2004

A127969 Inverse of number triangle A127967.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, 0, -2, 1, 2, 0, -3, 0, 1, -3, 0, 5, 0, -3, 1, -8, 0, 13, 0, -6, 0, 1, 17, 0, -28, 0, 14, 0, -4, 1, 56, 0, -92, 0, 45, 0, -10, 0, 1, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, -608

Offset: 0

Paul Barry, Feb 09 2007

First column is (-1)^C(n+1,2)*A099960(n), signed interleaved Genocchi numbers of the first and second kind. Row sums are (1,0,0,0,...).

			Triangle begins
1,
-1, 1,
-1, 0, 1,
1, 0, -2, 1,
2, 0, -3, 0, 1,
-3, 0, 5, 0, -3, 1,
-8, 0, 13, 0, -6, 0, 1,
17, 0, -28, 0, 14, 0, -4, 1,
56, 0, -92, 0, 45, 0, -10, 0, 1,
-155, 0, 255, 0, -126, 0, 30, 0, -5, 1,
-608, 0, 1000, 0, -493, 0, 115, 0, -15, 0, 1,
2073, 0, -3410, 0, 1683, 0, -396, 0, 55, 0, -6, 1,
9440, 0, -15528, 0, 7662, 0, -1799, 0, 245, 0, -21, 0, 1

A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 59, 172, 547, 1886, 7047, 28360, 122675, 567210, 2796999, 14641044, 81191947, 475148678, 2929442263, 18965690560, 128754649699, 914056305794, 6777666961735, 52367331911180, 421188392986843, 3519168714308702, 30519733808467031

Offset: 0

Alois P. Heinz, Oct 26 2017

nonn

			a(3) = 4: 000, 001, 010, 011.
a(4) = 9: 0000, 0001, 0010, 0011, 0100, 0101, 0102, 0110, 0111.
a(5) = 22: 00000, 00001, 00010, 00011, 00100, 00101, 00102, 00110, 00111, 01000, 01001, 01002, 01010, 01011, 01020, 01021, 01022, 01100, 01101, 01102, 01110, 01111.

Alois P. Heinz, Table of n, a(n) for n = 0..550

Cf. A022493, A099960.

b:= proc(n, i, t, u) option remember; `if`(n<1, 1, add(
       b(n-1, j, t+`if`(j>i, 1, 0), `if`(i=j, u, 1-u)),
       j=`if`(u=0, i..t+1, 0..i)))
    end:
a:= n-> b(n-1, 0$3):
seq(a(n), n=0..30);

a(n) = Sum_{j=0..n} binomial(n-1,j) * A099960(n-j).

cp's OEIS Frontend

A099959 Triangle read by rows: Each row is constructed by forming the partial sums of the previous row, reading from the right and at every other row repeating the final term.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A127969 Inverse of number triangle A127967.

Views

Author

Keywords

Comments

Examples

A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula