A034261 Infinite square array f(a,b) = C(a+b,b+1)*(a*b+a+1)/(b+2), a, b >= 0, read by antidiagonals. Equivalently, triangular array T(n,k) = f(k,n-k), 0 <= k <= n, read by rows.

0, 0, 1, 0, 1, 3, 0, 1, 5, 6, 0, 1, 7, 14, 10, 0, 1, 9, 25, 30, 15, 0, 1, 11, 39, 65, 55, 21, 0, 1, 13, 56, 119, 140, 91, 28, 0, 1, 15, 76, 196, 294, 266, 140, 36, 0, 1, 17, 99, 300, 546, 630, 462, 204, 45, 0, 1, 19, 125, 435, 930, 1302, 1218, 750, 285, 55

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• easy
• nice

Comments

f(h,k) = number of paths consisting of steps from (0,0) to (h,k) using h unit steps right, k+1 unit steps up and 1 unit step down, in some order, with first step not down and no repeated points.

Examples

			Triangle begins:
  0;
  0, 1;
  0, 1, 3;
  0, 1, 5,  6;
  0, 1, 7, 14, 10;
  ...
As a square array,
  [ 0  0  0   0   0 ...]
  [ 1  1  1   1   1 ...]
  [ 3  5  7   9  11 ...]
  [ 6 14 25  39  56 ...]
  [10 30 65 119 196 ...]
  [...      ...     ...]
		

Crossrefs

Cf. A001787 (row sums), A000330(n) = f(n,1).

Cf. A034263, A034264, A034265, A034267 - A034275 for diagonals n -> f(n,n+k), for several fixed k.

Programs

Maple

A034261 := proc(n, k) binomial(n, n-k+1)*(k+(k-1)/(k-n-2)); end proc; # argument indices of the triangle

Mathematica

Flatten[Table[Binomial[n,n-k+1](k+(k-1)/(k-n-2)),{n,0,15},{k,0,n}]] (* Harvey P. Dale, Jan 11 2013 *)

PARI

f(h,k)=binomial(h+k,k+1)*(k*h+h+1)/(k+2)

PARI

tabl(nn) = for (n=0, nn, for (k=0, n, print1(binomial(n, n-k+1)*(k+(k-1)/(k-n-2)), ", ")); print()); \\ Michel Marcus, Mar 20 2015

Formula

Another formula: f(h,k) = binomial(h+k,k+1) + Sum{C(i+j-1, j)*C(h+k-i-j, k-j+1): i=1, 2, ..., h-1, j=1, 2, ..., k+1}

Extensions

Entry revised by N. J. A. Sloane, Apr 21 2000. The formula for f in the definition was found by Michael Somos.

Edited by M. F. Hasler, Nov 08 2017