A375466 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.

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

Offset

0,8

Links

Table of n, a(n) for n=0..77.

Formula

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

Example

Sequence of polynomials P(n, m) for n = 0, 1, 2, ...:
[0]   1;
[1]   1*m   +         x;
[2]   2*m^2 +     2*m*x +         x^2;
[3]   6*m^3 +   6*m^2*x +     3*m*x^2 +         x^3;
[4]  24*m^4 +  24*m^3*x +  12*m^2*x^2 +     4*m*x^3 +        x^4;
[5] 120*m^5 + 120*m^4*x +  60*m^3*x^2 +  20*m^2*x^3 +    5*m*x^4 +     x^5;
[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;
.
Array of the coefficients of the polynomials for m = 0, 1, 2, ...:
[0] 1, 0, 1,  0,  0, 1,    0,   0,  0, 1,     0,    0,   0,  0, 1, ...  A023531
[1] 1, 1, 1,  2,  2, 1,    6,   6,  3, 1,    24,   24,  12,  4, 1, ...  A094587
[2] 1, 2, 1,  8,  4, 1,   48,  24,  6, 1,   384,  192,  48,  8, 1, ...
[3] 1, 3, 1, 18,  6, 1,  162,  54,  9, 1,  1944,  648, 108, 12, 1, ...
[4] 1, 4, 1, 32,  8, 1,  384,  96, 12, 1,  6144, 1536, 192, 16, 1, ...
[5] 1, 5, 1, 50, 10, 1,  750, 150, 15, 1, 15000, 3000, 300, 20, 1, ...
[6] 1, 6, 1, 72, 12, 1, 1296, 216, 18, 1, 31104, 5184, 432, 24, 1, ...
.
Seen as triangle:
  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, 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, 54, 6, 1,  0;

Maple

# Computes the polynomials depending on the parameter m.
P := (n, m) -> ifelse(m = 0, x^n, n! * m^n * hypergeom([-n], [-n], x/m)):
seq(print(simplify(P(n, m))), n = 0..5);
# Computes the array of coefficients:
P := (n, k, m) -> (n!/k!) * m^(n-k) * x^k:
Arow := (m, len) -> local n, k;
seq(seq(coeff(P(n, k, m), x, k), k = 0..n), n = 0..len):
seq(lprint(Arow(n, 4)), n = 0..6);

Crossrefs

Cf. A094587, A023531.

Sequence in context: A185962 A279928 A297325 * A278528 A257261 A355141

Adjacent sequences: A375463 A375464 A375465 * A375467 A375468 A375469

Keyword

nonn,tabl

Author

Peter Luschny, Aug 17 2024

Status

approved