A033842 Triangle of coefficients of certain polynomials (exponents in decreasing order).

1, 1, 1, 3, 3, 1, 16, 16, 6, 1, 125, 125, 50, 10, 1, 1296, 1296, 540, 120, 15, 1, 16807, 16807, 7203, 1715, 245, 21, 1, 262144, 262144, 114688, 28672, 4480, 448, 28, 1, 4782969, 4782969, 2125764, 551124, 91854, 10206, 756, 36, 1, 100000000, 100000000, 45000000, 12000000, 2100000, 252000, 21000, 1200, 45, 1

Offset

0,4

Comments

See A049323.

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Wolfdieter Lang, On generalizations of the Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Thierry Lévy, The Number of Prefixes of Minimal Factorisations of a Cycle, The Electronic Journal of Combinatorics, 23(3) (2016), #P3.35.

Formula

a(n, m) = binomial(n+1, m)*(n+1)^(n-m-1), n >= m >= 0 else 0.

Example

Triangle begins:
  {1};
  {1,1};
  {3,3,1};
  {16,16,6,1};
  {125,125,50,10,1};
  ...
E.g. third row {3,3,1} corresponds to polynomial p(2,x)= 3*x^2+3*x+1.

Mathematica

A033842[n_, m_] := Binomial[n + 1, m]*(n + 1)^(n - m - 1);
Table[A033842[n, m], {n, 0, 10}, {m, 0, n}] (* Paolo Xausa, Sep 28 2025 *)

Crossrefs

a(n, 0)= A000272(n+1), n >= 0 (first column), a(n, 1)= A000272(n+1), n >= 1 (second column). p(k-1, -x)/(1-k*x)^k = (-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.

See also A049323.

Sequence in context: A105599 A239895 A106210 * A104417 A121438 A108391

Adjacent sequences: A033839 A033840 A033841 * A033843 A033844 A033845

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Status

approved