A231171 G.f.: A(x,y) = Sum_{n>=0} x^n * Product_{k=1..n} (k + x*y) / (1 + k*x*y), as a triangle read by rows.

1, 1, 2, 6, -3, 24, -25, 6, 120, -190, 90, -12, 720, -1526, 1095, -300, 24, 5040, -13356, 12915, -5490, 960, -48, 40320, -128052, 156100, -90930, 25500, -3000, 96, 362880, -1341936, 1975708, -1469265, 576660, -113040, 9240, -192, 3628800, -15303024, 26413100, -23958711, 12184620, -3423000, 486120, -28200, 384

Offset

0,3

Comments

Compare to the identity:

Sum_{n>=0} x^n * Product_{k=1..n} (t*k + x)/(1 + t*k*x) = 1/(1 - t*x - x^2).

Links

Table of n, a(n) for n=0..46.

Formula

Row sums yield the Fibonacci sequence (A000045).

Sum_{k=0..n} T(n,k)(-1)^k = A231172(n) for n>=0.

Sum_{k=0..n} T(n,k)(-2)^k = A231173(n) for n>=0.

Example

G.f.: A(x,y) = 1 + x*(1) + x^2*(2) + x^3*(6 - 3*y) +
x^4*(24 - 25*y + 6*y^2) +
x^5*(120 - 190*y + 90*y^2 - 12*y^3) +
x^6*(720 - 1526*y + 1095*y^2 - 300*y^3 + 24*y^4) +
x^7*(5040 - 13356*y + 12915*y^2 - 5490*y^3 + 960*y^4 - 48*y^5) +
x^8*(40320 - 128052*y + 156100*y^2 - 90930*y^3 + 25500*y^4 - 3000*y^5 + 96*y^6) + ...
Triangle begins:
1;
1;
2;
6, -3;
24, -25, 6;
120, -190, 90, -12;
720, -1526, 1095, -300, 24;
5040, -13356, 12915, -5490, 960, -48;
40320, -128052, 156100, -90930, 25500, -3000, 96;
362880, -1341936, 1975708, -1469265, 576660, -113040, 9240, -192;
3628800, -15303024, 26413100, -23958711, 12184620, -3423000, 486120, -28200, 384; ...
where the g.f. of the n-th diagonal as a power series in z is given by:
Product_{k=1..n} (k + z) / (1 + k*z), for n>=0.

Prog

(PARI) {T(n, k)=polcoeff(polcoeff(sum(m=0, n, x^m*prod(k=1, m, (k+x*y)/(1+k*x*y +x*O(x^n)))), n, x), k, y)}
for(n=0, 10, for(k=0, min(n, min(abs(n-1), abs(n-2))), print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A231172, A231173.

Sequence in context: A302783 A364318 A008306 * A331431 A248120 A144362

Adjacent sequences: A231168 A231169 A231170 * A231172 A231173 A231174

Keyword

tabf,sign

Author

Paul D. Hanna, Nov 05 2013

Status

approved