A375466 - OEIS

%I #6 Aug 21 2024 06:01:09

%S 1,1,0,1,1,1,1,2,1,0,1,3,1,2,0,1,4,1,8,2,1,1,5,1,18,4,1,0,1,6,1,32,6,

%T 1,6,0,1,7,1,50,8,1,48,6,0,1,8,1,72,10,1,162,24,3,1,1,9,1,98,12,1,384,

%U 54,6,1,0,1,10,1,128,14,1,750,96,9,1,24,0

%N Array read by ascending antidiagonals of triangles read by rows: the coefficients of the polynomials n! * m^(n-k) * x^k * A094587(n, k), for m >= 0.

%F T(n, m, k) = [x^k] n! * m^n * hypergeom([-n], [-n], x/m)), for n > 0.

%e Sequence of polynomials P(n, m) for n = 0, 1, 2, ...:

%e [0]   1;

%e [1]   1*m   +         x;

%e [2]   2*m^2 +     2*m*x +         x^2;

%e [3]   6*m^3 +   6*m^2*x +     3*m*x^2 +         x^3;

%e [4]  24*m^4 +  24*m^3*x +  12*m^2*x^2 +     4*m*x^3 +        x^4;

%e [5] 120*m^5 + 120*m^4*x +  60*m^3*x^2 +  20*m^2*x^3 +    5*m*x^4 +     x^5;

%e [6] 720*m^6 + 720*m^5*x + 360*m^4*x^2 + 120*m^3*x^3 + 30*m^2*x^4 + 6*m*x^5 + x^6;

%e .

%e Array of the coefficients of the polynomials for m = 0, 1, 2, ...:

%e [0] 1, 0, 1,  0,  0, 1,    0,   0,  0, 1,     0,    0,   0,  0, 1, ...  A023531

%e [1] 1, 1, 1,  2,  2, 1,    6,   6,  3, 1,    24,   24,  12,  4, 1, ...  A094587

%e [2] 1, 2, 1,  8,  4, 1,   48,  24,  6, 1,   384,  192,  48,  8, 1, ...

%e [3] 1, 3, 1, 18,  6, 1,  162,  54,  9, 1,  1944,  648, 108, 12, 1, ...

%e [4] 1, 4, 1, 32,  8, 1,  384,  96, 12, 1,  6144, 1536, 192, 16, 1, ...

%e [5] 1, 5, 1, 50, 10, 1,  750, 150, 15, 1, 15000, 3000, 300, 20, 1, ...

%e [6] 1, 6, 1, 72, 12, 1, 1296, 216, 18, 1, 31104, 5184, 432, 24, 1, ...

%e .

%e Seen as triangle:

%e   1;

%e   1, 0;

%e   1, 1, 1;

%e   1, 2, 1,  0;

%e   1, 3, 1,  2,  0;

%e   1, 4, 1,  8,  2, 1;

%e   1, 5, 1, 18,  4, 1,   0;

%e   1, 6, 1, 32,  6, 1,   6,  0;

%e   1, 7, 1, 50,  8, 1,  48,  6, 0;

%e   1, 8, 1, 72, 10, 1, 162, 24, 3, 1;

%e   1, 9, 1, 98, 12, 1, 384, 54, 6, 1,  0;

%p # Computes the polynomials depending on the parameter m.

%p P := (n, m) -> ifelse(m = 0, x^n, n! * m^n * hypergeom([-n], [-n], x/m)):

%p seq(print(simplify(P(n, m))), n = 0..5);

%p # Computes the array of coefficients:

%p P := (n, k, m) -> (n!/k!) * m^(n-k) * x^k:

%p Arow := (m, len) -> local n, k;

%p seq(seq(coeff(P(n, k, m), x, k), k = 0..n), n = 0..len):

%p seq(lprint(Arow(n, 4)), n = 0..6);

%Y Cf. A094587, A023531.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Aug 17 2024