A064861 Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.

1, 1, 2, 1, 3, 2, 1, 5, 8, 4, 1, 6, 13, 12, 4, 1, 8, 25, 38, 28, 8, 1, 9, 33, 63, 66, 36, 8, 1, 11, 51, 129, 192, 168, 80, 16, 1, 12, 62, 180, 321, 360, 248, 96, 16, 1, 14, 86, 304, 681, 1002, 968, 592, 208, 32, 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32, 1, 17

Offset

0,3

Comments

When A064861 is regarded as a triangle read by rows, this is [1,0,-1,0,0,0,0,0,0,...] DELTA [2,-1,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 14 2008

Links

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.

C. de Jesús Pita Ruiz Velasco, Convolution and Sulanke Numbers, JIS 13 (2010) 10.1.8.

R. A. Sulanke, Problem 10894, Amer. Math. Monthly 108, (2001), p. 770.

Formula

G.f.: Sum_{m>=0} Sum_{n>=0} a_{m, n}*t^m*s^n = A(t,s) = (1+2*t+s)/(1-2*t^2-s^2-3*s*t).

Example

Table begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  2,  3,  5,   6,   8,  9, 11, ...
  2,  8, 13,  25,  33, 51, ...
  4, 12, 38,  63, 129, ...
  4, 28, 66, 192, ...

Maple

A064861 := proc(n, k) option remember; if n = 1 then 1; elif k = 0 then 0; else procname(n, k-1)+(3/2-1/2*(-1)^(n+k))*procname(n-1, k); fi; end;
seq(seq(A064861(i, j-i), i=1..j-1), j=1..19);

Mathematica

max = 12; se = Series[(1 + 2*x + y*x)/(1 - 2*x^2 - y^2*x^2 - 3*y*x^2), {x, 0, max}, {y, 0, max}]; cc = CoefficientList[se, {x, y}]; Flatten[ Table[ cc[[n, k]], {n, 1, max}, {k, n, 1, -1}]] (* Jean-François Alcover, Oct 21 2011, after g.f. *)

Prog

(PARI) a(n, m)=if(n<0 || m<0, 0, polcoeff(polcoeff((1+2*x+y*x)/(1-2*x^2-y^2*x^2-3*y*x^2)+O(x^(n+m+1)), n+m), m))
(Haskell)
a064861 n k = a064861_tabl !! n !! k
a064861_row n = a064861_tabl !! n
a064861_tabl = map fst $ iterate f ([1], 2) where
f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 3 - z)
-- Reinhard Zumkeller, May 01 2014

Crossrefs

Cf. central Delannoy numbers a(n,n) = A001850(n), Delannoy numbers (same main diagonal): a(n,n) = A008288(n,n), a(n-1,n)=A002003(n), a(n,n+1)=A002002(n), a(n,1)=A058582(n), apparently a(n,n+2)=A050151(n).

Sequence in context: A061260 A152097 A119442 * A305299 A308701 A191528

Adjacent sequences: A064858 A064859 A064860 * A064862 A064863 A064864

Keyword

nonn,tabl,nice

Author

Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10 2001

Status

approved