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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

Table of n, a(n) for n=1..64.

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

Cf. A030662, A171661, A171698, A004070.

Sequence in context: A247495 A230290 A294783 * A325182 A215959 A209555

Adjacent sequences:  A172018 A172019 A172020 * A172022 A172023 A172024

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 December 1 08:44 EST 2021. Contains 349426 sequences. (Running on oeis4.)