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