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 A048994 #132 Aug 15 2025 22:42:38
%S A048994 1,0,1,0,-1,1,0,2,-3,1,0,-6,11,-6,1,0,24,-50,35,-10,1,0,-120,274,-225,
%T A048994 85,-15,1,0,720,-1764,1624,-735,175,-21,1,0,-5040,13068,-13132,6769,
%U A048994 -1960,322,-28,1,0,40320,-109584,118124,-67284,22449,-4536,546,-36,1,0,-362880,1026576,-1172700,723680,-269325,63273,-9450,870,-45,1
%N A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.
%C A048994 The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.
%C A048994 Mirror image of the triangle A054654. - _Philippe Deléham_, Dec 30 2006
%C A048994 Also the triangle gives coefficients T(n,k) of x^k in the expansion of C(x,n) = (a(k)*x^k)/n!. - _Mokhtar Mohamed_, Dec 04 2012
%C A048994 From _Wolfdieter Lang_, Nov 14 2018: (Start)
%C A048994 This is the Sheffer triangle of Jabotinsky type (1, log(1 + x)). See the e.g.f. of the triangle below.
%C A048994 This is the inverse Sheffer triangle of the Stirling2 Sheffer triangle A008275.
%C A048994 The a-sequence of this Sheffer triangle (see a W. Lang link in A006232)
%C A048994 is from the e.g.f. A(x) = x/(exp(x) -1) a(n) = Bernoulli(n) = A027641(n)/A027642(n), for n >= 0. The z-sequence vanishes.
%C A048994 The Boas-Buck sequence for the recurrences of columns has o.g.f. B(x) = Sum_{n>=0} b(n)*x^n = 1/((1 + x)*log(1 + x)) - 1/x. b(n) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), b = {-1/2, 5/12, -3/8, 251/720, -95/288, 19087/60480,...}. For the Boas-Buck recurrence of Riordan and Sheffer triangles see the Aug 10 2017 remark in A046521, adapted to the Sheffer case, also for two references. See the recurrence and example below. (End)
%C A048994 Let G(n,m,k) be the number of simple labeled graphs on [n] with m edges and k components. Then T(n,k) = Sum (-1)^m*G(n,m,k). See the Read link below. Equivalently, T(n,k) = Sum mu(0,p) where the sum is over all set partitions p of [n] containing k blocks and mu is the Moebius function in the incidence algebra associated to the set partition lattice on [n]. - _Geoffrey Critzer_, May 11 2024
%D A048994 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D A048994 L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.
%D A048994 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.
%D A048994 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D A048994 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245.
%D A048994 J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%H A048994 T. D. Noe, <a href="/A048994/b048994.txt">Rows n=0..100 of triangle, flattened</a>
%H A048994 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A048994 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry1/barry263.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations</a>, Journal of Integer Sequences, 17 (2014), #14.2.3.
%H A048994 Fatima Zohra Bensaci, Rachid Boumahdi, and Laala Khaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Boumahdi/boumahdi12.html">Finite Sums Involving Fibonacci and Lucas Numbers</a>, J. Int. Seq. (2024). See p. 9.
%H A048994 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling1.html">Stirling numbers of the first kind</a>. [Illustrates the unsigned Stirling cycle numbers A132393.]
%H A048994 Askar Dzhumadil'daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%H A048994 Bill Gosper, <a href="/A008275/a008275.png">Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7</a>.
%H A048994 Gergő Nemes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Nemes/nemes4.html">An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind</a>, J. Int. Seq. 14 (2011), #11.4.8.
%H A048994 A. Hennessy and Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry6/barry161.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials</a>, J. Int. Seq. 14 (2011), #11.8.2 (A-number typo A048894).
%H A048994 NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.8">Stirling Numbers</a>
%H A048994 Ken Ono, Larry Rolen, and Florian Sprung, <a href="http://arxiv.org/abs/1602.00752">Zeta-Polynomials for modular form periods</a>, p. 6, arXiv:1602.00752 [math.NT], 2016.
%H A048994 Ricardo A. Podestá, <a href="http://arxiv.org/abs/1603.09156">New identities for binary Krawtchouk polynomials, binomial coefficients and Catalan numbers</a>, arXiv preprint arXiv:1603.09156 [math.CO], 2016.
%H A048994 Ronald Read, <a href="https://doi.org/10.1016/S0021-9800(68)80087-0">An Introduction to Chromatic Polynomials</a>, Journal of Combinatorial Theory, 4(1968)52-71.
%H A048994 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_numbers_and_exponential_generating_functions">Stirling numbers and exponential generating functions</a>.
%F A048994 s(n, k) = A008275(n,k) for n >= 1, k = 1..n; column k = 0 is {1, repeat(0)}.
%F A048994 s(n, k) = s(n-1, k-1) - (n-1)*s(n-1, k), n, k >= 1; s(n, 0) = s(0, k) = 0; s(0, 0) = 1.
%F A048994 The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0) = a(0, k) = 0; a(0, 0) = 1.
%F A048994 Triangle (signed) = [0, -1, -1, -2, -2, -3, -3, -4, -4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; Triangle(unsigned) = [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.
%F A048994 Sum_{k=0..n} (-m)^(n-k)*s(n, k) = A000142(n), A001147(n), A007559(n), A007696(n), ... for m = 1, 2, 3, 4, ... .- _Philippe Deléham_, Oct 29 2005
%F A048994 A008275*A007318 as infinite lower triangular matrices. - _Gerald McGarvey_, Aug 20 2009
%F A048994 T(n,k) = n!*[x^k]([t^n]exp(x*log(1+t))). - _Peter Luschny_, Dec 30 2010, updated Jun 07 2020
%F A048994 From _Wolfdieter Lang_, Nov 14 2018: (Start)
%F A048994 Recurrence from the Sheffer a-sequence (see a comment above): s(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j, j)*Bernoulli(j)*s(n-1, k-1+j), for n >= 1 and k >= 1, with s(n, 0) = 0 if n >= 1, and s(0,0) = 1.
%F A048994 Boas-Buck type recurrence for column k: s(n, k) = (n!*k/(n - k))*Sum_{j=k..n-1} b(n-1-j)*s(j, k)/j!, for n >= 1 and k = 0..n-1, with input s(n, n) = 1. For sequence b see the Boas-Buck comment above. (End)
%F A048994 T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A271705(n,j)*A216294(j,k). - _Mélika Tebni_, Feb 23 2023
%e A048994 Triangle begins:
%e A048994 n\k 0 1 2 3 4 5 6 7 8 9 ...
%e A048994 0 1
%e A048994 1 0 1
%e A048994 2 0 -1 1
%e A048994 3 0 2 -3 1
%e A048994 4 0 -6 11 -6 1
%e A048994 5 0 24 -50 35 -10 1
%e A048994 6 0 -120 274 -225 85 -15 1
%e A048994 7 0 720 -1764 1624 -735 175 -21 1
%e A048994 8 0 -5040 13068 -13132 6769 -1960 322 -28 1
%e A048994 9 0 40320 -109584 118124 -67284 22449 -4536 546 -36 1
%e A048994 ... - _Wolfdieter Lang_, Aug 22 2012
%e A048994 ------------------------------------------------------------------
%e A048994 From _Wolfdieter Lang_, Nov 14 2018: (Start)
%e A048994 Recurrence: s(5,2)= s(4, 1) - 4*s(4, 2) = -6 - 4*11 = -50.
%e A048994 Recurrence from the a- and z-sequences: s(6, 3) = 2*(1*1*(-50) + 3*(-1/2)*35 + 6*(1/6)*(-10) + 10*0*1) = -225.
%e A048994 Boas-Buck recurrence for column k = 3, with b = {-1/2, 5/12, -3/8, ...}:
%e A048994 s(6, 3) = 6!*((-3/8)*1/3! + (5/12)*(-6)/4! + (-1/2)*35/5!) = -225. (End)
%p A048994 A048994:= proc(n,k) combinat[stirling1](n,k) end: # _R. J. Mathar_, Feb 23 2009
%p A048994 seq(print(seq(coeff(expand(k!*binomial(x,k)),x,i),i=0..k)),k=0..9); # _Peter Luschny_, Jul 13 2009
%p A048994 A048994_row := proc(n) local k; seq(coeff(expand(pochhammer(x-n+1,n)), x,k), k=0..n) end: # _Peter Luschny_, Dec 30 2010
%t A048994 Table[StirlingS1[n, m], {n, 0, 9}, {m, 0, n}] (* _Peter Luschny_, Dec 30 2010 *)
%o A048994 (PARI) a(n,k) = if(k<0 || k>n,0, if(n==0,1,(n-1)*a(n-1,k)+a(n-1,k-1)))
%o A048994 (PARI) trg(nn)=for (n=0, nn-1, for (k=0, n, print1(stirling(n,k,1), ", ");); print();); \\ _Michel Marcus_, Jan 19 2015
%o A048994 (Maxima) create_list(stirling1(n,k),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */
%o A048994 (Haskell)
%o A048994 a048994 n k = a048994_tabl !! n !! k
%o A048994 a048994_row n = a048994_tabl !! n
%o A048994 a048994_tabl = map fst $ iterate (\(row, i) ->
%o A048994 (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 0)
%o A048994 -- _Reinhard Zumkeller_, Mar 18 2013
%Y A048994 See especially A008275 which is the main entry for this triangle. A132393 is an unsigned version, and A008276 is another version.
%Y A048994 Cf. A008277, A039814-A039817, A048993, A084938.
%Y A048994 A000142(n) = Sum_{k=0..n} |s(n, k)| for n >= 0.
%Y A048994 Row sums give A019590(n+1).
%Y A048994 Cf. A002209, A027641/A027642, A216294, A271705.
%K A048994 sign,tabl,nice
%O A048994 0,8
%A A048994 _N. J. A. Sloane_
%E A048994 Offset corrected by _R. J. Mathar_, Feb 23 2009
%E A048994 Formula corrected by _Philippe Deléham_, Sep 10 2009