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

%I A172021 #44 Oct 12 2024 02:00:56
%S A172021 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,
%T A172021 32,62,112,182,252,252,1,2,4,8,16,32,64,126,238,420,672,924,924,1,2,4,
%U A172021 8,16,32,64,128,254,492,912,1584,2508,3432,3432
%N A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.
%C A172021 Rows sum up to A030662.
%C A172021 Triangle is a (mirrored) interspaced binomial transform of 1^n (see example). - _Mark Dols_, Jan 24 2010
%C A172021 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
%C A172021 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
%C A172021 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
%F A172021 E.g.f. for row n is: ( 1 + x + x^2/2! + ... + x^n/n! )^2. - _Geoffrey Critzer_, Mar 15 2010
%e A172021 Triangle begins:
%e A172021 ......1
%e A172021 ....1,2,2
%e A172021 ..1,2,4,6,6
%e A172021 1,2,4,8,14,20,20
%e A172021 From _Mark Dols_, Jan 24 2010: (Start)
%e A172021 Interspaced binomial transform of 1^n:
%e A172021 1...1...1...1...1...1...
%e A172021 ..2...2...2...2...2...2.
%e A172021 2...4...4...4...4...4...
%e A172021 ..6...8...8...8...8...8.
%e A172021 6.. 14..16..16..16..16..
%e A172021 ..20..30..32..32..32..32
%e A172021 20..50..62..64..64..64..
%e A172021 (End)
%p A172021 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
%t A172021 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 *)
%o A172021 (Derive) T(n,k):=POLY_COEFF(SUM(x^i/i!, i, 0, n)^2, x, k)·k!
%o A172021 TABLE(VECTOR(T(v, u), u, 0, 2·v), v, 0, 10)  # _Giovanni Artico_, Aug 30 2013
%Y A172021 Cf. A030662, A171661, A171698, A004070.
%K A172021 nonn,tabf
%O A172021 1,3
%A A172021 _Mark Dols_, Jan 22 2010
%E A172021 Definition rewritten by _N. J. A. Sloane_, Jan 23 2010
%E A172021 More terms from _Mark Dols_, Jan 24 2010