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A034261
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Triangular array T(n,k) in which n-th row is {f(k,n-k), 0<=k<=n} where f(a,b)=C(a+b,b+1)*(a*b+a+1)/(b+2). Equivalently, square array f(a,b), a>=0, 0<=b<=a, read by antidiagonals.
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26
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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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.
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FORMULA
| Another formula: f(h, k)=C(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}
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EXAMPLE
| 0; 0,1; 0,1,3; 0,1,5,6; 0,1,7,14,10; ...
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MAPLE
| A034261 := proc(n, k); binomial(n+k, k+1)*(k*n+n+1)/(k+2); end;
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PROG
| (PARI) f(h, k)=if(h<0|k<0, 0, binomial(h+k, k+1)*(k*h+h+1)/(k+2))
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CROSSREFS
| f(n, 1)=A000330(n) (square pyramidal numbers).
Sequence in context: A123880 A186363 A143626 * A046778 A119925 A102765
Adjacent sequences: A034258 A034259 A034260 * A034262 A034263 A034264
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Apr 21 2000. The formula for f in the definition was found by Michael Somos.
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