This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A220883 #32 Mar 20 2024 15:10:06
%S A220883 1,1,3,2,12,16,6,55,150,125,24,300,1260,2160,1296,120,1918,11025,
%T A220883 29155,36015,16807,720,14112,103936,376320,716800,688128,262144,5040,
%U A220883 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 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.
%C A220883 Related to Stirling numbers A008275, A008277.
%H A220883 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 A220883 Peter Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>
%H A220883 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 A220883 From _Peter Bala_, Nov 16 2015: (Start)
%F A220883 E.g.f.: A(x,t) = x + (1 + 3*t)*x^2/2! + (1 + 4*t)*(2 + 4*t)*x^3/3! + ....
%F A220883 The function F(x,t) := 1 + t*A(x,t) has several nice properties:
%F A220883 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 A220883 F(x,t)*(1 - x*F(x,t))^t = 1.
%F A220883 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 A220883 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 A220883 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 A220883 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 A220883 Triangle begins:
%e A220883 1
%e A220883 1 3
%e A220883 2 12 16
%e A220883 6 55 150 125
%e A220883 24 300 1260 2160 1296
%e A220883 120 1918 11025 29155 36015 16807
%e A220883 720 14112 103936 376320 716800 688128 262144
%e A220883 ...
%p A220883 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 A220883 # Alternative:
%p A220883 T := (n, k) -> (-1)^(n-k)*(n+1)^(k-1)*Stirling1(n, k):
%p A220883 seq(print(seq(T(n, k), k=1..n)), n=1..8);
%p A220883 # _Peter Luschny_, Mar 20 2024
%t A220883 A220883[n_, k_] := (-1)^(n-k)*(n+1)^(k-1)*StirlingS1[n, k];
%t A220883 Table[A220883[n, k], {n, 10}, {k, n}] (* _Paolo Xausa_, Mar 19 2024 *)
%Y A220883 Cf. A220884, A008275, A008277, A056856, A260687.
%K A220883 nonn,tabl
%O A220883 1,3
%A A220883 _N. J. A. Sloane_, Dec 29 2012