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A172021
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Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.
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1
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1, 1, 2, 2, 1, 2, 4, 6, 6, 1, 2, 4, 8, 14, 20, 20, 1, 2, 4, 8, 16, 30, 50, 70, 70, 1, 2, 4, 8, 16, 32, 62, 112, 182, 252, 252, 1, 2, 4, 8, 16, 32, 64, 126, 238, 420, 672, 924, 924, 1, 2, 4, 8, 16, 32, 64, 128, 254, 492, 912, 1584, 2508, 3432, 3432
(list;
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OFFSET
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1,3
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COMMENTS
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Triangle is a (mirrored) interspaced binomial transform of 1^n (see example). - Mark Dols, Jan 24 2010]
T(n,k) is the number of k permutations of n (indistinguishable) objects of type I and n (indistinguishable) objects of type II. - Geoffrey Critzer, Mar 15 2010
Equivalently T(n,k) is the number of words length k from an alphabet of 2 letters with at most n occurrences of each letter. - Giovanni Artico, Aug 24 2013
T(n,k) is also the number of ways k persons can be accommodated into 2 rooms with at most n persons per room. - Giovanni Artico, Aug 24 2013
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LINKS
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FORMULA
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E.g.f. for row n is: ( 1 + x + x^2/2! + ... + x^n/n! )^2. - Geoffrey Critzer, Mar 15 2010
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EXAMPLE
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Triangle begins:
......1
....1,2,2
..1,2,4,6,6
1,2,4,8,14,20,20
Interspaced binomial transform of 1^n:
1...1...1...1...1...1...
..2...2...2...2...2...2.
2...4...4...4...4...4...
..6...8...8...8...8...8.
6.. 14..16..16..16..16..
..20..30..32..32..32..32
20..50..62..64..64..64..
(End)
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MAPLE
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seq(PolynomialTools:-CoefficientList((convert(taylor(exp(x), x, n+1), polynom)^2), x)*~[seq(i!, i=0..2 n)], n=0..10) # Giovanni Artico, Aug 30 2013
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MATHEMATICA
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Table[CoefficientList[Series[(Sum[x^i/i!, {i, 0, m}])^2, {x, 0, 2 m}], x]*Table[n!, {n, 0, 2 m}], {m, 0, 10}] // Grid (* Geoffrey Critzer, Mar 15 2010 *)
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PROG
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(Derive) T(n, k):=POLY_COEFF(SUM(x^i/i!, i, 0, n)^2, x, k)·k!
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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