cp's OEIS Frontend

A130534 Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

%I A130534 #206 Mar 09 2025 12:44:10
%S A130534 1,1,1,2,3,1,6,11,6,1,24,50,35,10,1,120,274,225,85,15,1,720,1764,1624,
%T A130534 735,175,21,1,5040,13068,13132,6769,1960,322,28,1,40320,109584,118124,
%U A130534 67284,22449,4536,546,36,1,362880,1026576,1172700,723680,269325,63273,9450,870,45,1
%N A130534 Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.
%C A130534 This triangle is an unsigned version of the triangle of Stirling numbers of the first kind, A008275, which is the main entry for these numbers. - _N. J. A. Sloane_, Jan 25 2011
%C A130534 Or, triangle T(n,k), 0 <= k <= n, read by rows given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.
%C A130534 Reversal of A094638.
%C A130534 Equals A132393*A007318, as infinite lower triangular matrices. - _Philippe Deléham_, Nov 13 2007
%C A130534 From _Johannes W. Meijer_, Oct 07 2009: (Start)
%C A130534 The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the exponential integrals E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + n*(n+1)/x^2 - n*(n+1)*(n+2)/x^3 + ...), see Abramowitz and Stegun. This formula follows from the general formula for the asymptotic expansion, see A163932. We rewrite E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - ...) and observe that the T(n,m) are the polynomials coefficients in the denominators. Looking at the a(n,m) formula of A028421, A163932 and A163934, and shifting the offset given above to 1, we can write T(n-1,m-1) = a(n,m) = (-1)^(n+m)*Stirling1(n,m), see the Maple program.
%C A130534 The asymptotic expansion leads for values of n from one to eleven to known sequences, see the cross-references. With these sequences one can form the triangles A008279 (right-hand columns) and A094587 (left-hand columns).
%C A130534 See A163936 for information about the o.g.f.s. of the right-hand columns of this triangle.
%C A130534 (End)
%C A130534 The number of elements greater than i to the left of i in a permutation gives the i-th element of the inversion vector. (Skiena-Pemmaraju 2003, p. 69.) T(n,k) is the number of n-permutations that have exactly k 0's in their inversion vector. See evidence in Mathematica code below. - _Geoffrey Critzer_, May 07 2010
%C A130534 T(n,k) counts the rooted trees with k+1 trunks in forests of "naturally grown" rooted trees with n+2 nodes. This corresponds to sums of coefficients of iterated derivatives representing vectors, Lie derivatives, or infinitesimal generators for flow fields and formal group laws. Cf. links in A139605. - _Tom Copeland_, Mar 23 2014
%C A130534 A refinement is A036039. - _Tom Copeland_, Mar 30 2014
%C A130534 From _Tom Copeland_, Apr 05 2014: (Start)
%C A130534 With initial n=1 and row polynomials of T as p(n,x)=x(x+1)...(x+n-1), the powers of x correspond to the number of trunks of the rooted trees of the "naturally-grown" forest referred to above. With each trunk allowed m colors, p(n,m) gives the number of such non-plane colored trees for the forest with each tree having n+1 vertices.
%C A130534 p(2,m) = m + m^2 = A002378(m) = 2*A000217(m) = 2*(first subdiag of |A238363|).
%C A130534 p(3,m) = 2m + 3m^2 + m^3 = A007531(m+2) = 3*A007290(m+2) = 3*(second subdiag A238363).
%C A130534 p(4,m) = 6m + 11m^2 + 6m^3 + m^4 = A052762(m+3) = 4*A033487(m) = 4*(third subdiag).
%C A130534 From the Joni et al. link, p(n,m) also represents the disposition of n distinguishable flags on m distinguishable flagpoles.
%C A130534 The chromatic polynomial for the complete graph K_n is the falling factorial, which encodes the colorings of the n vertices of K_n and gives a shifted version of p(n,m).
%C A130534 E.g.f. for the row polynomials: (1-y)^(-x).
%C A130534 (End)
%C A130534 A relation to derivatives of the determinant |V(n)| of the n X n Vandermonde matrix V(n) in the indeterminates c(1) thru c(n):
%C A130534   |V(n)| = Product_{1<=j<k<=n} (c(j)-c(k)). Let W(n,x) = |V(n)|*(c(1)c(2)...c(n))^x, then p(n,x) = W^(-1)[c(1)d/dc(1)...c(n)d/dc(n)]W. This is a variant of the Cayley identity. See Chervov link, p. 47. - _Tom Copeland_, Apr 10 2014
%C A130534 From _Peter Bala_, Jul 21 2014: (Start)
%C A130534 Let M denote the lower unit triangular array A094587 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
%C A130534   /I_k 0\
%C A130534   \ 0  M/
%C A130534 having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well defined). See the Example section. (End)
%C A130534 For the relation of this rising factorial to the moments of Viennot's Laguerre stories, see the Hetyei link, p. 4. - _Tom Copeland_, Oct 01 2015
%C A130534 Can also be seen as the Bell transform of n! without column 0 (and shifted enumeration). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D A130534 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 93-94.
%D A130534 Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge University Press, 2003, pp. 69-71. [_Geoffrey Critzer_, May 07 2010]
%H A130534 T. D. Noe, <a href="/A130534/b130534.txt">Rows n = 0..50 of triangle, flattened</a>
%H A130534 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, Chapter 5, pp. 227-251. [From _Johannes W. Meijer_, Oct 07 2009]
%H A130534 A. Chervov, <a href="http://arxiv.org/abs/1203.5759">Decomplexification of the Capelli identities and holomorphic factorization</a>, arxiv 1203.5759 [math.QA], Mar 2012. [_Tom Copeland_, Apr 10 2014]
%H A130534 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000007">The number of saliances of the permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000031">The number of cycles in the cycle decomposition of a permutation</a>.
%H A130534 Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Griffiths/griffiths31.html">Generating Functions for Extended Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 17 (2014), #14.6.4.
%H A130534 G. Hetyei, <a href="http://arxiv.org/abs/0909.4352">Meixner polynomials of the second kind and quantum algebras representing su(1,1)</a>, arXiv preprint arXiv:0909.4352 [math.QA], 2009.
%H A130534 S. Joni, G. Rota, and B. Sagan, <a href="http://dx.doi.org/10.1016/0012-365X(81)90219-3">From Sets to Functions: Three Elementary Examples</a>, Discrete Mathematics, vol. 37, no. 2-3, pp. 193-202, 1981. [_Tom Copeland_, Apr 05 2014]
%H A130534 Matthieu Josuat-Verges, <a href="https://arxiv.org/abs/1610.02965">A q-analog of Schläfli and Gould identities on Stirling numbers</a>, Preprint, arXiv:1610.02965 [math.CO], 2016.
%H A130534 Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H A130534 Lucas Sá and Antonio M. García-García, <a href="https://arxiv.org/abs/2104.07647">The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos</a>, arXiv:2104.07647 [hep-th], 2021.
%H A130534 Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-02-2.pdf#page=15">On the ordinal numbers of triangles of generalized special numbers</a>, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
%H A130534 Dennis Walsh, <a href="https://web.archive.org/web/20111208160649/http://frank.mtsu.edu/~dwalsh/STIRLIN1.pdf">A short note on unsigned Stirling numbers</a>
%F A130534 T(0,0) = 1, T(n,k) = 0 if k > n or if n < 0, T(n,k) = T(n-1,k-1) + n*T(n-1,k). T(n,0) = n! = A000142(n). T(2*n,n) = A129505(n+1). Sum_{k=0..n} T(n,k) = (n+1)! = A000142(n+1). Sum_{k=0..n} T(n,k)^2 = A047796(n+1). T(n,k) = |Stirling1(n+1,k+1)|, see A008275. (x+1)(x+2)...(x+n) = Sum_{k=0..n} T(n,k)*x^k. [Corrected by _Arie Bos_, Jul 11 2008]
%F A130534 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, respectively. - _Philippe Deléham_, Nov 13 2007
%F A130534 For k=1..n, let A={a_1,a_2,...,a_k} denote a size-k subset of {1,2,...,n}. Then T(n,n-k) = Sum(Product_{i=1..k} a_i) where the sum is over all subsets A. For example, T(4,1)=50 since 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 = 50. - _Dennis P. Walsh_, Jan 25 2011
%F A130534 The preceding formula means T(n,k) = sigma_{n-k}(1,2,3,..,n) with the (n-k)-th elementary symmetric function sigma with the indeterminates chosen as 1,2,...,n. See the Oct 24 2011 comment in A094638 with sigma called there a. - _Wolfdieter Lang_, Feb 06 2013
%F A130534 From _Gary W. Adamson_, Jul 08 2011: (Start)
%F A130534 n-th row of the triangle = top row of M^n, where M is the production matrix:
%F A130534   1, 1;
%F A130534   1, 2, 1;
%F A130534   1, 3, 3, 1;
%F A130534   1, 4, 6, 4, 1;
%F A130534   ... (End)
%F A130534 Exponential Riordan array [1/(1 - x), log(1/(1 - x))]. Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (n + 1)!/(n + 1 - i)!*T(n-i,k). - _Peter Bala_, Jul 21 2014
%e A130534 Triangle  T(n,k) begins:
%e A130534 n\k         0        1        2       3       4      5      6     7    8  9 10
%e A130534 n=0:        1
%e A130534 n=1:        1        1
%e A130534 n=2:        2        3        1
%e A130534 n=3:        6       11        6       1
%e A130534 n=4:       24       50       35      10       1
%e A130534 n=5:      120      274      225      85      15      1
%e A130534 n=6:      720     1764     1624     735     175     21      1
%e A130534 n=7:     5040    13068    13132    6769    1960    322     28     1
%e A130534 n=8:    40320   109584   118124   67284   22449   4536    546    36    1
%e A130534 n=9:   362880  1026576  1172700  723680  269325  63273   9450   870   45  1
%e A130534 n=10: 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55  1
%e A130534 [Reformatted and extended by _Wolfdieter Lang_, Feb 05 2013]
%e A130534 T(3,2) = 6 because there are 6 permutations of {1,2,3,4} that have exactly 2 0's in their inversion vector: {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {2, 1, 3, 4},{2, 3, 1, 4}, {2, 3, 4, 1}. The respective inversion vectors are {0, 0, 1}, {0, 1, 0}, {0, 2, 0}, {1, 0, 0}, {2, 0, 0}, {3, 0, 0}. - _Geoffrey Critzer_, May 07 2010
%e A130534 T(3,1)=11 since there are exactly 11 permutations of {1,2,3,4} with exactly 2 cycles, namely, (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), (4)(143), (12)(34), (13)(24), and (14)(23). - _Dennis P. Walsh_, Jan 25 2011
%e A130534 From _Peter Bala_, Jul 21 2014: (Start)
%e A130534 With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
%e A130534   / 1          \/1        \/1        \      / 1           \
%e A130534   | 1  1       ||0 1      ||0 1      |      | 1  1        |
%e A130534   | 2  2  1    ||0 1 1    ||0 0 1    |... = | 2  3  1     |
%e A130534   | 6  6  3 1  ||0 2 2 1  ||0 0 1 1  |      | 6 11  6  1  |
%e A130534   |24 24 12 4 1||0 6 6 3 1||0 0 2 2 1|      |24 50 35 10 1|
%e A130534   |...         ||...      ||...      |      |...          |
%e A130534 (End)
%p A130534 with(combinat): A130534 := proc(n,m): (-1)^(n+m)*stirling1(n+1,m+1) end proc: seq(seq(A130534(n,m), m=0..n), n=0..10); # _Johannes W. Meijer_, Oct 07 2009, revised Sep 11 2012
%p A130534 # The function BellMatrix is defined in A264428.
%p A130534 # Adds (1,0,0,0, ..) as column 0 (and shifts the enumeration).
%p A130534 BellMatrix(n -> n!, 9); # _Peter Luschny_, Jan 27 2016
%t A130534 Table[Table[ Length[Select[Map[ToInversionVector, Permutations[m]], Count[ #, 0] == n &]], {n, 0, m - 1}], {m, 0, 8}] // Grid (* _Geoffrey Critzer_, May 07 2010 *)
%t A130534 rows = 10;
%t A130534 t = Range[0, rows]!;
%t A130534 T[n_, k_] := BellY[n, k, t];
%t A130534 Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o A130534 (Haskell)
%o A130534 a130534 n k = a130534_tabl !! n !! k
%o A130534 a130534_row n = a130534_tabl !! n
%o A130534 a130534_tabl = map (map abs) a008275_tabl
%o A130534 -- _Reinhard Zumkeller_, Mar 18 2013
%Y A130534 See A008275, which is the main entry for these numbers; A094638 (reversed rows).
%Y A130534 Diagonals: A000012 A000217 A000914 A001303 A000915 A053567 A112002 A191685. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.
%Y A130534 From _Johannes W. Meijer_, Oct 07 2009: (Start)
%Y A130534 Row sums equal A000142.
%Y A130534 The asymptotic expansions lead to A000142 (n=1), A000142(n=2; minus a(0)), A001710 (n=3), A001715 (n=4), A001720 (n=5), A001725 (n=6), A001730 (n=7), A049388 (n=8), A049389 (n=9), A049398 (n=10), A051431 (n=11), A008279 and A094587.
%Y A130534 Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.
%Y A130534 (End)
%Y A130534 Cf. A136662.
%K A130534 nonn,tabl
%O A130534 0,4
%A A130534 _Philippe Deléham_, Aug 09 2007