0,4

The n-th term of the n-th binomial transform of a sequence {b} is given by {d} where d(n) = sum(k=0,n,T(n,k)*b(k)) and T(n,k)=binomial(n,k)*n^(n-k); such diagonals are related to the hyperbinomial transform (A088956). - Paul D. Hanna, Nov 04 2003

T(n,k) gives the number of divisors of A181555(n) with (n-k) distinct prime factors. See also A001221, A146289, A146290, A181567. - Matthew Vandermast, Oct 31 2010

T(n,k) is the number of partial functions on {1,2,...,n} leaving exactly k elements undefined. Row sums = A000169. - Geoffrey Critzer, Jan 08 2012

As a triangular matrix, transforms rows into diagonals in the table of coefficients of successive iterations of x/(1-x). - Paul D. Hanna, Jan 19 2014

Also the number of rooted trees on n+1 labeled vertices in which some specified vertex (say, vertex 1) has k children. - Alan Sokal, Jul 22 2022

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.

Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019 and Discrete Math. 343, 111865 (2020).

T(n,k) = binomial(n, k)*n^(n-k).

E.g.f.: (-LambertW(-y)/y)^x/(1+LambertW(-y)). - Vladeta Jovovic

1

1 1

4 4 1

27 27 9 1

256 256 96 16 1

3125 3125 1250 250 25 1

46656 46656 19440 4320 540 36 1

T:= (n, k)-> binomial(n, k)*n^(n-k): seq(seq(T(n, k), k=0..n), n=0..10);

Prepend[Flatten[ Table[Table[Binomial[n, k] n^(n - k), {k, 0, n}], {n, 1, 8}]], 1] (* Geoffrey Critzer, Jan 08 2012 *)

(PARI) T(n, k)=if(k<0 || k>n, 0, binomial(n, k)*n^(n-k))

(PARI) /* Transforms rows into diagonals in the iterations of x/(1-x): */

{T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x/(1-x+x*O(x^(m+2))))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x; for(i=1, r, F=subst(F, x, x/(1-x+x*O(x^(m+2))))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 19 2014

Cf. A000169, A000312, A089466, A088956, A166900.

Sequence in context: A116866 A126280 A170986 * A136214 A067328 A386676

Adjacent sequences: A071204 A071205 A071206 * A071208 A071209 A071210

easy,nonn,tabl

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

Name edited by Alan Sokal, Jul 22 2022

approved