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A144510
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Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.
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3
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1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 37, 31, 4, 1, 1, 266, 842, 121, 5, 1, 1, 2431, 45296, 18252, 456, 6, 1, 1, 27007, 4061871, 7958726, 405408, 1709, 7, 1, 1, 353522, 546809243, 7528988476, 1495388159, 9268549, 6427, 8, 1, 1, 5329837, 103123135501, 13130817809439, 15467641899285
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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FORMULA
| T(n,k) = (1/k!) Sum_{i_1 = 1 .. n} Sum_{i_2 = 1 .. n} ... Sum_{i_k = 1 .. n} Multinomial( i_1+i_2+...+i_k; i_1, i_2, ..., i_k).
Contribution from Peter Luschny, Apr 26 2011: (Start)
T(n,k) = (1/k!) Sum_{m=k..kn} m! [x^m](e^x Gamma(n+1,x)/Gamma(n+1)-1)^k. Here [x^m]f(x) is the coefficient of x^m in the series expansion of f(x). (End)
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EXAMPLE
| Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, ...
1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, ...
1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ...
1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ...
1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
...
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MAPLE
| b := proc(n, i, k) local r;
option remember;
if n = i then 1;
elif i < n then 0;
elif n < 1 then 0;
else add( binomial(i-1, r)*b(n-1, i-1-r, k), r=0..k);
end if;
end proc;
T:=proc(n, k); add(b(n, i, k), i=0..(k+1)*n); end proc;
Contribution from Peter Luschny, Apr 26 2011: (Start)
A144510 := proc(n, k) local m;
add(m!*coeff(expand((exp(x)*GAMMA(n+1, x)/GAMMA(n+1)-1)^k), x, m), m=k..k*n)/k! end: for row from 1 to 6 do seq(A144510(row, col), col = 0..5) od; (End)
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CROSSREFS
| For the transposed array see A144512.
Rows include A001515, A144416, A144508, A144509.
Columns include A048775, A144511.
Sequence in context: A105291 A025270 A178234 * A143670 A169730 A196832
Adjacent sequences: A144507 A144508 A144509 * A144511 A144512 A144513
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KEYWORD
| nonn,tabl
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AUTHOR
| David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Dec 15 2008, Jan 30 2009
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