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 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. 26
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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. LINKS Table of n, a(n) for n=0..65. 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} EXAMPLE 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 ...] [... ... ...] MAPLE A034261 := proc(n, k) binomial(n, n-k+1)*(k+(k-1)/(k-n-2)); end; 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 *) PROG (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 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. Sequence in context: A356777 A254295 A143626 * A046778 A119925 A210663 Adjacent sequences: A034258 A034259 A034260 * A034262 A034263 A034264 KEYWORD nonn,tabl,easy,nice AUTHOR Clark Kimberling 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 STATUS approved

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Last modified August 14 02:14 EDT 2024. Contains 375146 sequences. (Running on oeis4.)