A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.

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

Offset: 1

Author

Mark Dols, Jan 22 2010

Keywords

• nonn
• tabf

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

Examples

			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)
		

Crossrefs

Cf. A030662, A171661, A171698, A004070.

Programs

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

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 *)

Formula

E.g.f. for row n is: ( 1 + x + x^2/2! + ... + x^n/n! )^2. - Geoffrey Critzer, Mar 15 2010

Extensions

Definition rewritten by N. J. A. Sloane, Jan 23 2010

More terms from Mark Dols, Jan 24 2010