A220883 Triangle read by rows: row n gives coefficients of expansion of Product_{k = 1..n-1} ((n + 1)*x + k), starting with lowest power.

1, 1, 3, 2, 12, 16, 6, 55, 150, 125, 24, 300, 1260, 2160, 1296, 120, 1918, 11025, 29155, 36015, 16807, 720, 14112, 103936, 376320, 716800, 688128, 262144, 5040, 117612, 1063692, 4934601, 12859560, 19013778, 14880348, 4782969, 40320, 1095840, 11812400, 67284000, 224490000, 453600000, 546000000, 360000000, 100000000, 362880, 11292336, 141896700, 963218080, 3943187325, 10190179923, 16741251450, 16953838770, 9646149645, 2357947691

Offset: 1

Author

N. J. A. Sloane, Dec 29 2012

Keywords

• nonn
• tabl

Comments

Related to Stirling numbers A008275, A008277.

Examples

			Triangle begins:
    1
    1     3
    2    12     16
    6    55    150    125
   24   300   1260   2160   1296
  120  1918  11025  29155  36015  16807
  720 14112 103936 376320 716800 688128 262144
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).
• Peter Bala, Fractional iteration of a series inversion operator
• Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

Crossrefs

Cf. A220884, A008275, A008277, A056856, A260687.

Programs

Maple

seq(seq(coeff(mul((n+1)*t + k, k = 1..n-1), t, i), i = 0..n-1), n = 1 .. 10); # Peter Bala, Nov 16 2015
# Alternative:
T := (n, k) -> (-1)^(n-k)*(n+1)^(k-1)*Stirling1(n, k):
seq(print(seq(T(n, k), k=1..n)), n=1..8);
# Peter Luschny, Mar 20 2024

Mathematica

A220883[n_, k_] := (-1)^(n-k)*(n+1)^(k-1)*StirlingS1[n, k];
Table[A220883[n, k], {n, 10}, {k, n}] (* Paolo Xausa, Mar 19 2024 *)

Formula

From Peter Bala, Nov 16 2015: (Start)

E.g.f.: A(x,t) = x + (1 + 3*t)*x^2/2! + (1 + 4*t)*(2 + 4*t)*x^3/3! + ....

The function F(x,t) := 1 + t*A(x,t) has several nice properties:

F(x,t) = 1/x*Revert( x*(1 - x)^t ) = 1 + t*x + t*(1 + 3*t)*x^2/2! + t*(2 + 12*t + 16*t^2)*x^3/3! + ..., where Revert denotes the series reversion operator with respect to x.

F(x,t)*(1 - x*F(x,t))^t = 1.

F(x,t)^m = 1 + m*t*x + m*t*((m + 2)*t + 1)*x^2/2! + m*t*((m + 3)*t + 1)*((m + 3)*t + 2)*x^3/3! + m*t*((m + 4)*t + 1)*((m + 4)*t + 2)*((m + 4)*t + 3)*x^4/4! + ....

Log(F(x,t)) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.

F(x,t) = G(x,t)^t, where G(x,t) = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ... is the o.g.f. for A260687. (End)

T(n, k) = (-1)^(n-k)*(n+1)^(k-1)*Stirling1(n, k). - Peter Luschny, Mar 01 2021 [Corrected by Paolo Xausa, Mar 19 2024]