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

%I #5 Apr 11 2013 02:57:13

%S 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 7,63,35,105,21,21,1,1,24,28,168,70,168,28,24,1,3,9,108,84,378,126,

%U 252,36,27,1,1,30,45,360,210,756,210,360,45,30,1

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

%C 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 TABLE ENTRIES

%F (1)... T(n,k) = (2 - (-1)^(n-k))*binomial(n,k).

%F GENERATING FUNCTION

%F (2)... exp(x*t)*(2*exp(t)-exp(-t)) = 1 + (3+x)*t + (1+6*x+x^2)*t^2/2!

%F + ....

%F The e.g.f. can also be written as

%F (3)... exp(x*t)/G(-t), where G(t) = exp(t)/(2-exp(2*t)) is the e.g.f.

%F for A080253.

%F MISCELLANEOUS

%F The row polynomials form an Appell sequence of polynomials.

%F Row sums = A151821.

%e Triangle begins

%e =================================================

%e n\k|..0.....1.....2.....3.....4.....5.....6.....7

%e =================================================

%e 0..|..1

%e 1..|..3.....1

%e 2..|..1.....6.....1

%e 3..|..3.....3.....9.....1

%e 4..|..1....12.....6....12.....1

%e 5..|..3.....5....30....10....15.....1

%e 6..|..1....18....15....60....15....18.....1

%e 7..|..3.....7....63....35...105....21....21.....1

%e ...

%p #A162315

%p T:=(n, k)->(2-(-1)^(n-k))*binomial(n,k):

%p for n from 0 to 10 do seq(T(n,k), k = 0..n) od;

%Y A007318, A151821 (row sums), A080253, A124846, A162313 (unsigned matrix inverse).

%K easy,nonn,tabl

%O 0,2

%A _Peter Bala_, Jul 01 2009

%E Row sums corrected by _Peter Bala_, Apr 01 2010