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A112492
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Triangle from inverse scaled Pochhammer symbols.
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10
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1, 1, 1, 1, 3, 1, 1, 7, 11, 1, 1, 15, 85, 50, 1, 1, 31, 575, 1660, 274, 1, 1, 63, 3661, 46760, 48076, 1764, 1, 1, 127, 22631, 1217776, 6998824, 1942416, 13068, 1, 1, 255, 137845, 30480800, 929081776, 1744835904, 104587344, 109584, 1, 1, 511, 833375
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| This expansion is based on the partial fraction identity: 1/product(x+j,j=1..m)= (1 + sum(((-1)^j)*binomial(m,j)*x/(x+j),j=1..m))/m!, e.g., p. 37 of the Ch. Jordan reference.
Another version of this triangle (without a column of 1's) is A008969.
The column sequences are, for m=1..10: A000012 (powers of 1), A000225, A001240, A001241, A001242, A111886-A111888.
From Gottfried Helms, Dec 11 2001. (Start)
The triangle occurs as U-factor in the LDU-decomposition of the matrix M defined by m(r,c)=1/(1+r)^c (r,c,beginning at 0).
Then
a(r,c)= m(r,c) * (1+r)!^(c-r)
An explicite expansion based on this can be made by defining a "recursice harmonic number" (rhn). (This representation is just a heuristic pattern-interpretation, no analytic proof yet available).
Consider
h(k,0)=1 for k>0 as rhn of order zero(0).
Then consider
h(1,1)=1*h(1,0)
h(2,1)=1*h(1,0) + 1/2*h(2,0)
h(3,1)=1*h(1,0) + 1/2*h(2,0) + 1/3*h(3,0) = h(2,1)+1/3*h(3,0)
...
and recursively
h(1,r)=1*h(1,r-1)
h(2,r)=1*h(1,r-1) + 1/2*h(2,r-1)
h(3,r)=1*h(1,r-1) + 1/2*h(2,r-1) + 1/3*h(3,r-1) = h(2,r)+1/3*h(3,r-1)
...
h(k,r)=h(k-1,r)+1/k*h(k,r-1)
then the upper triangular triangle A:=a(r,c) for c-r>0
a(r,c) = h(r,c-r) *(1+r)!^(c-r)
(End)
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REFERENCES
| Charles Jordan, Calculus of Finite Differences, Chelsea, 1965.
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LINKS
| W. Lang, First 10 rows.
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FORMULA
| G.f. for column m>=1: (x^m)/product(1-m!*x/j, j=1..m).
a(n, m)= -(m!^(n-m+1))*sum(((-1)^j)*binomial(m, j)/j^(n-m+1), j=1..m), m>=1. a(n, m)=0 if n+1<m.
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PROG
| (PARI): {h(n, recurse=1) = if(recurse == 0, return(1)); ;
return( sum(k=0, n, h(k, recurse-1) / (1+k) )); }
a(r, c) = h(r-1, c-r) * r!^(c-r) - Gottfried Helms, Dec 11 2001
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CROSSREFS
| Row sums give A111885.
Sequence in context: A176791 A075440 A137470 * A049290 A147990 A134567
Adjacent sequences: A112489 A112490 A112491 * A112493 A112494 A112495
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005
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