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A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row. 1
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; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Rows sum up to A030662
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
LINKS
FORMULA
E.g.f. for row n is: ( 1 + x + x^2/2! + ... + x^n/n! )^2. - Geoffrey Critzer, Mar 15 2010
EXAMPLE
Triangle begins:
......1
....1,2,2
..1,2,4,6,6
1,2,4,8,14,20,20
From Mark Dols, Jan 24 2010: (Start)
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)
MAPLE
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
MATHEMATICA
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 *)
PROG
(Derive) T(n, k):=POLY_COEFF(SUM(x^i/i!, i, 0, n)^2, x, k)·k!
TABLE(VECTOR(T(v, u), u, 0, 2·v), v, 0, 10) # Giovanni Artico, Aug 30 2013
CROSSREFS
Sequence in context: A247495 A230290 A294783 * A325182 A215959 A209555
KEYWORD
nonn,tabf
AUTHOR
Mark Dols, Jan 22 2010
EXTENSIONS
Definition rewritten by N. J. A. Sloane, Jan 23 2010
More terms from Mark Dols, Jan 24 2010
STATUS
approved

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Last modified March 4 23:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)