cp's OEIS Frontend

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.

%I A034261 #41 Feb 10 2025 13:08:47
%S A034261 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,
%T A034261 1,13,56,119,140,91,28,0,1,15,76,196,294,266,140,36,0,1,17,99,300,546,
%U A034261 630,462,204,45,0,1,19,125,435,930,1302,1218,750,285,55
%N 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.
%C A034261 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.
%F A034261 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}
%e A034261 Triangle begins:
%e A034261   0;
%e A034261   0, 1;
%e A034261   0, 1, 3;
%e A034261   0, 1, 5,  6;
%e A034261   0, 1, 7, 14, 10;
%e A034261   ...
%e A034261 As a square array,
%e A034261   [ 0  0  0   0   0 ...]
%e A034261   [ 1  1  1   1   1 ...]
%e A034261   [ 3  5  7   9  11 ...]
%e A034261   [ 6 14 25  39  56 ...]
%e A034261   [10 30 65 119 196 ...]
%e A034261   [...      ...     ...]
%p A034261 A034261 := proc(n, k) binomial(n, n-k+1)*(k+(k-1)/(k-n-2)); end proc; # argument indices of the triangle
%t A034261 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 *)
%o A034261 (PARI) f(h,k)=binomial(h+k,k+1)*(k*h+h+1)/(k+2)
%o A034261 (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
%Y A034261 Cf. A001787 (row sums), A000330(n) = f(n,1).
%Y A034261 Cf. A034263, A034264, A034265, A034267 - A034275 for diagonals n -> f(n,n+k), for several fixed k.
%K A034261 nonn,tabl,easy,nice
%O A034261 0,6
%A A034261 _Clark Kimberling_
%E A034261 Entry revised by _N. J. A. Sloane_, Apr 21 2000. The formula for f in the definition was found by _Michael Somos_.
%E A034261 Edited by _M. F. Hasler_, Nov 08 2017