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A162315 Triangular array 2*P - P^-1, where P is Pascal's triangle A007318.

%I A162315 #5 Apr 11 2013 02:57:13
%S A162315 1,3,1,1,6,1,3,3,9,1,1,12,6,12,1,3,5,30,10,15,1,1,18,15,60,15,18,1,3,
%T A162315 7,63,35,105,21,21,1,1,24,28,168,70,168,28,24,1,3,9,108,84,378,126,
%U A162315 252,36,27,1,1,30,45,360,210,756,210,360,45,30,1
%N A162315 Triangular array 2*P - P^-1, where P is Pascal's triangle A007318.
%C A162315 Row reversed version of A124846. For the signless version of the inverse array and its connection with sums of powers of odd integers see A162313.
%F A162315 TABLE ENTRIES
%F A162315 (1)... T(n,k) = (2 - (-1)^(n-k))*binomial(n,k).
%F A162315 GENERATING FUNCTION
%F A162315 (2)... exp(x*t)*(2*exp(t)-exp(-t)) = 1 + (3+x)*t + (1+6*x+x^2)*t^2/2!
%F A162315 + ....
%F A162315 The e.g.f. can also be written as
%F A162315 (3)... exp(x*t)/G(-t), where G(t) = exp(t)/(2-exp(2*t)) is the e.g.f.
%F A162315 for A080253.
%F A162315 MISCELLANEOUS
%F A162315 The row polynomials form an Appell sequence of polynomials.
%F A162315 Row sums = A151821.
%e A162315 Triangle begins
%e A162315 =================================================
%e A162315 n\k|..0.....1.....2.....3.....4.....5.....6.....7
%e A162315 =================================================
%e A162315 0..|..1
%e A162315 1..|..3.....1
%e A162315 2..|..1.....6.....1
%e A162315 3..|..3.....3.....9.....1
%e A162315 4..|..1....12.....6....12.....1
%e A162315 5..|..3.....5....30....10....15.....1
%e A162315 6..|..1....18....15....60....15....18.....1
%e A162315 7..|..3.....7....63....35...105....21....21.....1
%e A162315 ...
%p A162315 #A162315
%p A162315 T:=(n, k)->(2-(-1)^(n-k))*binomial(n,k):
%p A162315 for n from 0 to 10 do seq(T(n,k), k = 0..n) od;
%Y A162315 A007318, A151821 (row sums), A080253, A124846, A162313 (unsigned matrix inverse).
%K A162315 easy,nonn,tabl
%O A162315 0,2
%A A162315 _Peter Bala_, Jul 01 2009
%E A162315 Row sums corrected by _Peter Bala_, Apr 01 2010