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Expansion of (1 - t)*(1 + t)^x.
2

%I #12 Jul 02 2022 16:09:05

%S 1,-1,1,0,-3,1,0,5,-6,1,0,-14,23,-10,1,0,54,-105,65,-15,1,0,-264,574,

%T -435,145,-21,1,0,1560,-3682,3199,-1330,280,-28,1,0,-10800,27180,

%U -26124,12649,-3360,490,-36,1,0,85680,-227196,236312,-128205,40089,-7434,798,-45,1

%N Expansion of (1 - t)*(1 + t)^x.

%C Cf. A105793. Inverse array is A227343.

%C The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1 - t and B(t) = log(1 + t). Thus the row polynomials of this triangle form a Sheffer sequence for the pair (2 - exp(t), exp(t) - 1) (see Roman, p. 17).

%C Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the basis polynomials x_[n] as a linear combination of the monomial polynomials x^k, that is, x_[n] = Sum_{k = 0..n} T(n,k) x^k. An example is given below.

%D S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ShefferSequence.html">Sheffer Sequence</a>

%F T(n,k) = Stirling1(n,k) - n*Stirling1(n-1,k).

%F E.g.f.: (1 - t)*(1 + t)^x = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + (5*x - 6*x^2 + x^3)*t^3/3! + ....

%F E.g.f. for column k: (1/k!)*(1 - t)*(log(1 + t))^k.

%F The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = Sum_{k = 0..n} binomial(n,k)*y_(k)*R(n-k,x), where y_(k) is the falling factorial. As a particular case we have the identity R(n,x + 1) - R(n,x) = n*R(n-1,x) for n >= 1.

%e Triangle begins:

%e n\k| 0 1 2 3 4 5 6 7 8 9

%e ===|==================================================================

%e 0 | 1

%e 1 | -1, 1;

%e 2 | 0, -3, 1;

%e 3 | 0, 5, -6, 1;

%e 4 | 0, -14, 23, -10, 1;

%e 5 | 0, 54, -105, 65, -15, 1;

%e 6 | 0, -264, 574, -435, 145, -21, 1;

%e 7 | 0, 1560, -3682, 3199, -1330, 280, -28, 1;

%e 8 | 0, -10800, 27180, -26124, 12649, -3360, 490, -36, 1;

%e 9 | 0, 85680, -227196, 236312, -128205, 40089, -7434, 798, -45, 1;

%e ...

%e Connection constants. Row 4 = [0, -14 ,23, -10, 1]:

%e -14*x + 23*x^2 - 10*x^3 + x^4 = x*(x-1)*(x-2)*(x-7) = x_[4].

%Y Cf. A048994, A105793, A227343 (matrix inverse).

%K sign,tabl,easy

%O 0,5

%A _Peter Bala_, Jul 11 2013