A362996 - OEIS

%I #13 May 20 2023 08:22:24

%S 1,3,1,11,14,3,25,46,117,16,137,652,3699,1344,125,49,568,19197,41728,

%T 19375,1296,363,9872,621837,2397184,2084375,334368,16807,761,23664,

%U 5338467,17115136,99109375,7150032,6705993,262144

%N Triangle read by rows. T(n, k) = numerator([x^k] R(n, n, x)), where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

%F T(n, k) =  A362995(n, k) * A362997(n, k) / lcm(1, 2, ..., n+1).

%e The triangle T(n, k) begins:

%e [0]   1;

%e [1]   3,     1;

%e [2]  11,    14,       3;

%e [3]  25,    46,     117,       16;

%e [4] 137,   652,    3699,     1344,      125;

%e [5]  49,   568,   19197,    41728,    19375,    1296;

%e [6] 363,  9872,  621837,  2397184,  2084375,  334368,   16807;

%e [7] 761, 23664, 5338467, 17115136, 99109375, 7150032, 6705993, 262144;

%e .

%e The first few polynomials are:

%e [0]      1

%e [1]      x   +      3/2

%e [2]    3*x^2 +    (14/3)*x   +      11/6

%e [3]   16*x^3 +   (117/4)*x^2 +     (46/3)*x   +     25/12

%e [4]  125*x^4 +  (1344/5)*x^3 +  (3699/20)*x^2 +   (652/15)*x   + 137/60

%e [5] 1296*x^5 + (19375/6)*x^4 + (41728/15)*x^3 + (19197/20)*x^2 + (568/5)*x + 49/20

%o (SageMath)

%o def R(n, k, x):

%o     return add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n

%o            for j in (0..u)) for u in (0..k))

%o def A362996row(n: int) -> list[int]:

%o     return [r.numerator() for r in R(n, n, x).list()]

%o for n in (0..7): print(A362996row(n))

%Y Cf. A362997 (denominator), A001008 (column 0), A000272 (main diagonal), A362995.

%K nonn,tabl,frac

%O 0,2

%A _Peter Luschny_, May 13 2023