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 A054654 #50 Feb 16 2025 08:32:42
%S A054654 1,1,0,1,-1,0,1,-3,2,0,1,-6,11,-6,0,1,-10,35,-50,24,0,1,-15,85,-225,
%T A054654 274,-120,0,1,-21,175,-735,1624,-1764,720,0,1,-28,322,-1960,6769,
%U A054654 -13132,13068,-5040,0
%N A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.
%C A054654 Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
%C A054654 Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g., 3!*C(x,3) = x^3-3*x^2+2*x.
%C A054654 The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).
%C A054654 Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938; mirror image of the Stirling-1 triangle A048994. - _Philippe Deléham_, Dec 30 2006
%C A054654 From _Doudou Kisabaka_, Dec 18 2009: (Start)
%C A054654 The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.
%C A054654 The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)
%D A054654 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, table 18:6:1 at page 152.
%H A054654 Reinhard Zumkeller, <a href="/A054654/b054654.txt">Rows n = 0..125 of triangle, flattened</a>
%H A054654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%H A054654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RisingFactorial.html">Rising Factorial</a>.
%H A054654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FallingFactorial.html">FallingFactorial</a>.
%F A054654 n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
%F A054654 (In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
%F A054654 T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - _Peter Luschny_, Feb 21 2021
%e A054654 Matrix begins:
%e A054654 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A054654 0, 1, -1, 2, -6, 24, -120, 720, -5040, ...
%e A054654 0, 0, 1, -3, 11, -50, 274, -1764, 13068, ...
%e A054654 0, 0, 0, 1, -6, 35, -225, 1624, -13132, ...
%e A054654 0, 0, 0, 0, 1, -10, 85, -735, 6769, ...
%e A054654 0, 0, 0, 0, 0, 1, -15, 175, -1960, ...
%e A054654 0, 0, 0, 0, 0, 0, 1, -21, 322, ...
%e A054654 0, 0, 0, 0, 0, 0, 0, 1, -28, ...
%e A054654 0, 0, 0, 0, 0, 0, 0, 0, 1, ...
%e A054654 ...
%e A054654 Triangle begins:
%e A054654 1;
%e A054654 1, 0;
%e A054654 1, -1, 0;
%e A054654 1, -3, 2, 0;
%e A054654 1, -6, 11, -6, 0;
%e A054654 1, -10, 35, -50, 24, 0;
%e A054654 1, -15, 85, -225, 274, -120, 0;
%e A054654 1, -21, 175, -735, 1624, -1764, 720, 0;
%e A054654 ...
%p A054654 a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x,n)),x,n-k),k=0..n) end: # _Peter Luschny_, Nov 28 2010
%p A054654 seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # _Peter Luschny_, Feb 21 2021
%t A054654 row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* _Jean-François Alcover_, Feb 16 2012, after Maple *)
%t A054654 Table[StirlingS1[n,n-k],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Jun 17 2023 *)
%o A054654 (PARI) T(n,k)=polcoeff(n!*binomial(x,n), n-k)
%o A054654 (Haskell)
%o A054654 a054654 n k = a054654_tabl !! n !! k
%o A054654 a054654_row n = a054654_tabl !! n
%o A054654 a054654_tabl = map reverse a048994_tabl
%o A054654 -- _Reinhard Zumkeller_, Mar 18 2014
%Y A054654 Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
%Y A054654 The Stirling2 counterpart is A106800.
%Y A054654 Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.
%K A054654 tabl,sign,easy,nice
%O A054654 0,8
%A A054654 _N. J. A. Sloane_, Apr 18 2000
%E A054654 Additional comments from _Thomas Wieder_, Dec 29 2006
%E A054654 Edited by _N. J. A. Sloane_ at the suggestion of _Eric W. Weisstein_, Jan 20 2008