A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 10, 3, 1, 0, 75, 74, 21, 4, 1, 0, 541, 730, 219, 36, 5, 1, 0, 4683, 9002, 3045, 484, 55, 6, 1, 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1, 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1, 0, 7087261, 45375130, 26857659, 5227236, 544505, 39390, 2359, 136, 9, 1

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).

T(2*n, n) = A094420(n).

Example

Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,      1;
[3] 0, 3,      2,       1;
[4] 0, 13,     10,      3,       1;
[5] 0, 75,     74,      21,      4,      1;
[6] 0, 541,    730,     219,     36,     5,     1;
[7] 0, 4683,   9002,    3045,    484,    55,    6,    1;
[8] 0, 47293,  133210,  52923,   8676,   905,   78,   7,   1;
[9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;

Maple

F := proc(n) option remember; if n = 0 then return 1 fi:
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);

Prog

(SageMath)
@cached_function
def F(n):
    R.<x> = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
for n in range(10): print(Fval(n))

Crossrefs

Written as an array this is A094416 (with missing column 0).

The coefficients of the Fubini polynomials are A131689.

Cf. A094420.

Sequence in context: A184182 A085771 A253286 * A284799 A111106 A370419

Adjacent sequences: A344496 A344497 A344498 * A344500 A344501 A344502

Keyword

nonn,tabl

Author

Peter Luschny, May 21 2021

Status

approved