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A124320
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Triangle read by rows: T(n,k) = k!*binom(n+k-1,k) (n>=0, 0<=k<=n), rising factorial power, Pochhammer's symbol.
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1
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1, 1, 1, 1, 2, 6, 1, 3, 12, 60, 1, 4, 20, 120, 840, 1, 5, 30, 210, 1680, 15120, 1, 6, 42, 336, 3024, 30240, 332640, 1, 7, 56, 504, 5040, 55440, 665280, 8648640, 1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 1, 9, 90, 990, 11880, 154440, 2162160
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums yield A123680. T(n,n)=(2n-1)!/(n-1)!=A000407(n-1).
This is the Pochhammer function which is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1. Also known as the rising factorial power. - Peter Luschny, Jan 09 2011
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REFERENCES
| Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
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LINKS
| NIST Digital Library of Mathematical Functions, Pochhammer's Symbol
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FORMULA
| T(n,k) = GAMMA(n+k)/GAMMA(n) for n>0. - Peter Luschny, Jan 09 2011
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EXAMPLE
| [0] 1
[1] 1, 1
[2] 1, 2, 6
[3] 1, 3, 12, 60
[4] 1, 4, 20, 120, 840
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MAPLE
| T:=proc(n, k) if k<=n then binomial(n+k-1, k)*k! else 0 fi end: for n from 0 to 9 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
A124320 := (n, k)-> `if`(n=0 and k=0, 1, pochhammer(n, k)); seq(print(seq(A124320(n, k), k=0..n)), n=0..5); - Peter Luschny, Jan 09 2011
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MATHEMATICA
| Table[Pochhammer[n, k], {n, 0, 5}, {k, 0, n}] - Peter Luschny, Jan 09 2011
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PROG
| (Sage)
for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] - Peter Luschny, Jan 09 2011
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CROSSREFS
| Cf. A123680, A000407, A068424 (falling factorial power).
Sequence in context: A121601 A122761 A100469 * A156146 A192043 A154584
Adjacent sequences: A124317 A124318 A124319 * A124321 A124322 A124323
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 26 2006
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