A220883 - OEIS

%I #32 Mar 20 2024 15:10:06

%S 1,1,3,2,12,16,6,55,150,125,24,300,1260,2160,1296,120,1918,11025,

%T 29155,36015,16807,720,14112,103936,376320,716800,688128,262144,5040,

%U 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

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

%C Related to Stirling numbers A008275, A008277.

%H Paolo Xausa, <a href="/A220883/b220883.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of the triangle, flattened).

%H Peter Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1209.5959">Duplicial algebras and Lagrange inversion</a>, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

%F From _Peter Bala_, Nov 16 2015: (Start)

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

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

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

%F 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! + ....

%F 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 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)

%F 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]

%e Triangle begins:

%e     1

%e     1     3

%e     2    12     16

%e     6    55    150    125

%e    24   300   1260   2160   1296

%e   120  1918  11025  29155  36015  16807

%e   720 14112 103936 376320 716800 688128 262144

%e   ...

%p 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

%p # Alternative:

%p T := (n, k) -> (-1)^(n-k)*(n+1)^(k-1)*Stirling1(n, k):

%p seq(print(seq(T(n, k), k=1..n)), n=1..8);

%p # _Peter Luschny_, Mar 20 2024

%t A220883[n_, k_] := (-1)^(n-k)*(n+1)^(k-1)*StirlingS1[n, k];

%t Table[A220883[n, k], {n, 10}, {k, n}] (* _Paolo Xausa_, Mar 19 2024 *)

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

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Dec 29 2012