|
%I #353 Jan 14 2024 08:57:18
%S 1,1,1,1,3,2,1,7,12,6,1,15,50,60,24,1,31,180,390,360,120,1,63,602,
%T 2100,3360,2520,720,1,127,1932,10206,25200,31920,20160,5040,1,255,
%U 6050,46620,166824,317520,332640,181440,40320,1,511,18660,204630,1020600,2739240,4233600,3780000,1814400,362880
%N Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
%C Let M = n X n matrix with (i,j)-th entry a(n+1-j, n+1-i), e.g., if n = 3, M = [1 1 1; 3 1 0; 2 0 0]. Given a sequence s = [s(0)..s(n-1)], let b = [b(0)..b(n-1)] be its inverse binomial transform and let c = [c(0)..c(n-1)] = M^(-1)*transpose(b). Then s(k) = Sum_{i=0..n-1} b(i)*binomial(k,i) = Sum_{i=0..n-1} c(i)*k^i, k=0..n-1. - _Gary W. Adamson_, Nov 11 2001
%C From _Gary W. Adamson_, Aug 09 2008: (Start)
%C Julius Worpitzky's 1883 algorithm generates Bernoulli numbers.
%C By way of example [Wikipedia]:
%C B0 = 1;
%C B1 = 1/1 - 1/2;
%C B2 = 1/1 - 3/2 + 2/3;
%C B3 = 1/1 - 7/2 + 12/3 - 6/4;
%C B4 = 1/1 - 15/2 + 50/3 - 60/4 + 24/5;
%C B5 = 1/1 - 31/2 + 180/3 - 390/4 + 360/5 - 120/6;
%C B6 = 1/1 - 63/2 + 602/3 - 2100/4 + 3360/5 - 2520/6 + 720/7;
%C ...
%C Note that in this algorithm, odd n's for the Bernoulli numbers sum to 0, not 1, and the sum for B1 = 1/2 = (1/1 - 1/2). B3 = 0 = (1 - 7/2 + 13/3 - 6/4) = 0. The summation for B4 = -1/30. (End)
%C Pursuant to Worpitzky's algorithm and given M = A028246 as an infinite lower triangular matrix, M * [1/1, -1/2, 1/3, ...] (i.e., the Harmonic series with alternate signs) = the Bernoulli numbers starting [1/1, 1/2, 1/6, ...]. - _Gary W. Adamson_, Mar 22 2012
%C From _Tom Copeland_, Oct 23 2008: (Start)
%C G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1 x + (2 + t)*x^2/2! + (6 + 6t + t^2)*x^3/3! + ... gives row polynomials for A090582, the f-polynomials for the permutohedra (see A019538).
%C G(x,t-1) = 1 + 1*x + (1 + t)*x^2 / 2! + (1 + 4t + t^2)*x^3 / 3! + ... gives row polynomials for A008292, the h-polynomials for permutohedra.
%C G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ... gives row polynomials for the present triangle. (End)
%C The Worpitzky triangle seems to be an apt name for this triangle. - _Johannes W. Meijer_, Jun 18 2009
%C If Pascal's triangle is written as a lower triangular matrix and multiplied by A028246 written as an upper triangular matrix, the product is a matrix where the (i,j)-th term is (i+1)^j. For example,
%C 1,0,0,0 1,1,1, 1 1,1, 1, 1
%C 1,1,0,0 * 0,1,3, 7 = 1,2, 4, 8
%C 1,2,1,0 0,0,2,12 1,3, 9,27
%C 1,3,3,1 0,0,0, 6 1,4,16,64
%C So, numbering all three matrices' rows and columns starting at 0, the (i,j) term of the product is (i+1)^j. - Jack A. Cohen (ProfCohen(AT)comcast.net), Aug 03 2010
%C The Fi1 and Fi2 triangle sums are both given by sequence A000670. For the definition of these triangle sums see A180662. The mirror image of the Worpitzky triangle is A130850. - _Johannes W. Meijer_, Apr 20 2011
%C Let S_n(m) = 1^m + 2^m + ... + n^m. Then, for n >= 0, we have the following representation of S_n(m) as a linear combination of the binomial coefficients:
%C S_n(m) = Sum_{i=1..n+1} a(i+n*(n+1)/2)*C(m,i). E.g., S_2(m) = a(4)*C(m,1) + a(5)*C(m,2) + a(6)*C(m,3) = C(m,1) + 3*C(m,2) + 2*C(m,3). - _Vladimir Shevelev_, Dec 21 2011
%C Given the set X = [1..n] and 1 <= k <= n, then a(n,k) is the number of sets T of size k of subset S of X such that S is either empty or else contains 1 and another element of X and such that any two elemements of T are either comparable or disjoint. - _Michael Somos_, Apr 20 2013
%C Working with the row and column indexing starting at -1, a(n,k) gives the number of k-dimensional faces in the first barycentric subdivision of the standard n-dimensional simplex (apply Brenti and Welker, Lemma 2.1). For example, the barycentric subdivision of the 2-simplex (a triangle) has 1 empty face, 7 vertices, 12 edges and 6 triangular faces giving row 4 of this triangle as (1,7,12,6). Cf. A053440. - _Peter Bala_, Jul 14 2014
%C See A074909 and above g.f.s for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - _Tom Copeland_, Nov 14 2014
%C An e.g.f. G(x,t) = exp[P(.,t)x] = 1/t - 1/[t+(1-t)(1-e^(-xt^2))] = (1-t) * x + (-2t + 3t^2 - t^3) * x^2/2! + (6t^2 - 12t^3 + 7t^4 - t^5) * x^3/3! + ... for the shifted, reverse, signed polynomials with the first element nulled, is generated by the infinitesimal generator g(u,t)d/du = [(1-u*t)(1-(1+u)t)]d/du, i.e., exp[x * g(u,t)d/du] u eval. at u=0 generates the polynomials. See A019538 and the G. Rzadkowski link below for connections to the Bernoulli and Eulerian numbers, a Ricatti differential equation, and a soliton solution to the KdV equation. The inverse in x of this e.g.f. is Ginv(x,t) = (-1/t^2)*log{[1-t(1+x)]/[(1-t)(1-tx)]} = [1/(1-t)]x + [(2t-t^2)/(1-t)^2]x^2/2 + [(3t^2-3t^3+t^4)/(1-t)^3]x^3/3 + [(4t^3-6t^4+4t^5-t^6)/(1-t)^4]x^4/4 + ... . The numerators are signed, shifted A135278 (reversed A074909), and the rational functions are the columns of A074909. Also, dG(x,t)/dx = g(G(x,t),t) (cf. A145271). (Analytic G(x,t) added, and Ginv corrected and expanded on Dec 28 2015.) - _Tom Copeland_, Nov 21 2014
%C The operator R = x + (1 + t) + t e^{-D} / [1 + t(1-e^(-D))] = x + (1+t) + t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... contains an e.g.f. of the reverse row polynomials of the present triangle, i.e., A123125 * A007318 (with row and column offset 1 and 1). Umbrally, R^n 1 = q_n(x;t) = (q.(0;t)+x)^n, with q_m(0;t) = (t+1)^(m+1) - t^(m+1), the row polynomials of A074909, and D = d/dx. In other words, R generates the Appell polynomials associated with the base sequence A074909. For example, R 1 = q_1(x;t) = (q.(0;t)+x) = q_1(0;t) + q__0(0;t)x = (1+2t) + x, and R^2 1 = q_2(x;t) = (q.(0;t)+x)^2 = q_2(0:t) + 2q_1(0;t)x + q_0(0;t)x^2 = 1+3t+3t^2 + 2(1+2t)x + x^2. Evaluating the polynomials at x=0 regenerates the base sequence. With a simple sign change in R, R generates the Appell polynomials associated with A248727. - _Tom Copeland_, Jan 23 2015
%C For a natural refinement of this array, see A263634. - _Tom Copeland_, Nov 06 2015
%C From _Wolfdieter Lang_, Mar 13 2017: (Start)
%C The e.g.f. E(n, x) for {S(n, m)}_{m>=0} with S(n, m) = Sum_{k=1..m} k^n, n >= 0, (with undefined sum put to 0) is exp(x)*R(n+1, x) with the exponential row polynomials R(n, x) = Sum_{k=1..n} a(n, k)*x^k/k!. E.g., e.g.f. for n = 2, A000330: exp(x)*(1*x/1!+3*x^2/2!+2*x^3/3!).
%C The o.g.f. G(n, x) for {S(n, m)}_{m >=0} is then found by Laplace transform to be G(n, 1/p) = p*Sum_{k=1..n} a(n+1, k)/(p-1)^(2+k).
%C Hence G(n, x) = x/(1 - x)^(n+2)*Sum_{k=1..n} A008292(n,k)*x^(k-1).
%C E.g., n=2: G(2, 1/p) = p*(1/(p-1)^2 + 3/(p-1)^3 + 2/(p-1)^4) = p^2*(1+p)/(p-1)^4; hence G(2, x) = x*(1+x)/(1-x)^4.
%C This works also backwards: from the o.g.f. to the e.g.f. of {S(n, m)}_{m>=0}. (End)
%C a(n,k) is the number of k-tuples of pairwise disjoint and nonempty subsets of a set of size n. - _Dorian Guyot_, May 21 2019
%C From _Rajesh Kumar Mohapatra_, Mar 16 2020: (Start)
%C a(n-1,k) is the number of chains of length k in a partially ordered set formed from subsets of an n-element set ordered by inclusion such that the first term of the chains is either the empty set or an n-element set.
%C Also, a(n-1,k) is the number of distinct k-level rooted fuzzy subsets of an n-set ordered by set inclusion. (End)
%C The relations on p. 34 of Hasan (also p. 17 of Franco and Hasan) agree with the relation between A019538 and this entry given in the formula section. - _Tom Copeland_, May 14 2020
%C T(n,k) is the size of the Green's L-classes in the D-classes of rank (k-1) in the semigroup of partial transformations on an (n-1)-set. - _Geoffrey Critzer_, Jan 09 2023
%C T(n,k) is the number of strongly connected binary relations on [n] that have period k (A367948) and index 1. See Theorem 5.4.25(6) in Ki Hang Kim reference. - _Geoffrey Critzer_, Dec 07 2023
%D Ki Hang Kim, Boolean Matrix Theory and Applications, Marcel Dekker, New York and Basel (1982).
%H Seiichi Manyama, <a href="/A028246/b028246.txt">Table of n, a(n) for n = 1..10000</a>
%H V. S. Abramovich, <a href="http://kvant.mccme.ru/1973/05/summy_odinakovyh_stepenej_natu.htm">Power sums of natural numbers</a>, Kvant, no. 5 (1973), 22-25. (in Russian)
%H Peter Bala, <a href="/A131689/a131689.pdf">Deformations of the Hadamard product of power series</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H H. Belbachir, M. Rahmani, and B. Sury, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Rahmani/rahmani3.html">Sums Involving Moments of Reciprocals of Binomial Coefficients</a>, J. Int. Seq. 14 (2011) #11.6.6.
%H Hacene Belbachir and Mourad Rahmani, <a href="http://www.emis.de/journals/JIS/VOL15/Sury/sury42.html">Alternating Sums of the Reciprocals of Binomial Coefficients</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.2.8.
%H F. Brenti and V. Welker, <a href="http://arxiv.org/abs/math/0606356">f-vectors of barycentric subdivisions</a>, arXiv:math/0606356v1 [math.CO], Math. Z., 259(4), 849-865, 2008.
%H Patibandla Chanakya and Putla Harsha, <a href="https://arxiv.org/abs/1808.08699">Generalized Nested Summation of Powers of Natural Numbers</a>, arXiv:1808.08699 [math.NT], 2018. See Table 1.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H Colin Defant, <a href="https://arxiv.org/abs/2004.11367">Troupes, Cumulants, and Stack-Sorting</a>, arXiv:2004.11367 [math.CO], 2020.
%H E. Delucchi, A. Pixton and L. Sabalka. <a href="http://arxiv.org/abs/1002.3201">Face vectors of subdivided simplicial complexes</a> arXiv:1002.3201v3 [math.CO], Discrete Mathematics, Volume 312, Issue 2, January 2012, Pages 248-257.
%H G. H. E Duchamp, N. Hoang-Nghia, and A. Tanasa, <a href="http://arxiv.org/abs/1207.6522">A word Hopf algebra based on the selection/quotient principle</a>, arXiv:1207.6522 [math.CO], 2012-2013; Séminaire Lotharingien de Combinatoire 68 (2013), Article B68c.
%H M. Dukes and C. D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%H Nick Early, <a href="https://arxiv.org/abs/1810.03246">Honeycomb tessellations and canonical bases for permutohedral blades</a>, arXiv:1810.03246 [math.CO], 2018.
%H S. Franco and A. Hasan, <a href="https://arxiv.org/abs/1904.07954">Graded Quivers, Generalized Dimer Models and Toric Geometry </a>, arXiv preprint arXiv:1904.07954 [hep-th], 2019
%H A. Hasan, <a href="https://academicworks.cuny.edu/gc_etds/3321/">Physics and Mathematics of Graded Quivers</a>, dissertation, Graduate Center, City University of New York, 2019.
%H H. Hasse, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002372398">Ein Summierungsverfahren für die Riemannsche Zeta-Reihe</a>, Math. Z. 32, 458-464 (1930).
%H Guy Louchard, Werner Schachinger, and Mark Daniel Ward, <a href="https://arxiv.org/abs/2203.14773">The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis</a>, arXiv:2203.14773 [math.PR], 2022. See p. 5.
%H Shi-Mei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p11">A family of two-variable derivative polynomials for tangent and secant</a>, El J. Combinat. 20(1) (2013), P11.
%H Richard J. Mathar, <a href="https://arxiv.org/abs/2308.14154">Integrals Associated with the Digamma Integral Representation</a>, arXiv:2308.14154 [math.GM], 2023. See p. 5.
%H Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H A. Riskin and D. Beckwith, <a href="http://www.jstor.org/stable/2975362">Problem 10231</a>, Amer. Math. Monthly, 102 (1995), 175-176.
%H G. Rzadkowski, <a href="http://dx.doi.org/10.1142/S1402925110000635">Bernoulli numbers and solitons revisited</a>, Journal of Nonlinear Mathematical Physics, Volume 17, Issue 1, 2010.
%H John K. Sikora, <a href="https://arxiv.org/abs/1806.00887">On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles</a>, arXiv:1806.00887 [math.NT], 2018.
%H G. J. Simmons, <a href="http://www.jstor.org/stable/2689153">A combinatorial problem associated with a family of combination locks</a>, Math. Mag., 37 (1964), 127-132 (but there are errors). The triangle is on page 129.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Sam Vandervelde, <a href="https://doi.org/10.1080/00029890.2018.1408347">The Worpitzky Numbers Revisited</a>, Amer. Math. Monthly, 125:3 (2018), 198-206.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Barycentric_subdivision">Barycentric subdivision</a>
%H David C. Wood, <a href="http://www.cs.kent.ac.uk/pubs/1992/110/content.pdf">The computation of polylogarithms</a> (2014).
%F E.g.f.: -log(1-y*(exp(x)-1)). - _Vladeta Jovovic_, Sep 28 2003
%F a(n, k) = S2(n, k)*(k-1)! where S2(n, k) is a Stirling number of the second kind (cf. A008277). Also a(n,k) = T(n,k)/k, where T(n, k) = A019538.
%F Essentially same triangle as triangle [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deléham's operator defined in A084938, but the notation is different.
%F Sum of terms in n-th row = A000629(n) - _Gary W. Adamson_, May 30 2005
%F The row generating polynomials P(n, t) are given by P(1, t)=t, P(n+1, t) = t(t+1)(d/dt)P(n, t) for n >= 1 (see the Riskin and Beckwith reference). - _Emeric Deutsch_, Aug 09 2005
%F From _Gottfried Helms_, Jul 12 2006: (Start)
%F Delta-matrix as can be read from H. Hasse's proof of a connection between the zeta-function and Bernoulli numbers (see link below).
%F Let P = lower triangular matrix with entries P[row,col] = binomial(row,col).
%F Let J = unit matrix with alternating signs J[r,r]=(-1)^r.
%F Let N(m) = column matrix with N(m)(r) = (r+1)^m, N(1)--> natural numbers.
%F Let V = Vandermonde matrix with V[r,c] = (r+1)^c.
%F V is then also N(0)||N(1)||N(2)||N(3)... (indices r,c always beginning at 0).
%F Then Delta = P*J * V and B' = N(-1)' * Delta, where B is the column matrix of Bernoulli numbers and ' means transpose, or for the single k-th Bernoulli number B_k with the appropriate column of Delta,
%F B_k = N(-1)' * Delta[ *,k ] = N(-1)' * P*J * N(k).
%F Using a single column instead of V and assuming infinite dimension, H. Hasse showed that in x = N(-1) * P*J * N(s), where s can be any complex number and s*zeta(1-s) = x.
%F His theorem reads: s*zeta(1-s) = Sum_{n>=0..inf} (n+1)^-1*delta(n,s), where delta(n,s) = Sum_{j=0..n} (-1)^j * binomial(n,j) * (j+1)^s.
%F (End)
%F a(n,k) = k*a(n-1,k) + (k-1)*a(n-1,k-1) with a(n,1) = 1 and a(n,n) = (n-1)!. - _Johannes W. Meijer_, Jun 18 2009
%F Rephrasing the Meijer recurrence above: Let M be the (n+1)X(n+1) bidiagonal matrix with M(r,r) = M(r,r+1) = r, r >= 1, in the two diagonals and the rest zeros. The row a(n+1,.) of the triangle is row 1 of M^n. - _Gary W. Adamson_, Jun 24 2011
%F From _Tom Copeland_, Oct 11 2011: (Start)
%F With e.g.f.. A(x,t) = G[(t+1)x,-1/(t+1)]-1 (from 2008 comment) = -1 + 1/[1-(1+t)(1-e^(-x))] = (1+t)x + (1+3t+2t^2)x^2/2! + ..., the comp. inverse in x is
%F B(x,t)= -log(t/(1+t)+1/((1+t)(1+x))) = (1/(1+t))x - ((1+2t)/(1+t)^2)x^2/2 + ((1+3t+3t^2)/(1+t)^3)x^3/3 + .... The numerators are the row polynomials of A074909, and the rational functions are (omitting the initial constants) signed columns of the re-indexed Pascal triangle A007318.
%F Let h(x,t)= 1/(dB/dx) = (1+x)(1+t(1+x)), then the row polynomial P(n,t) = (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t), with P(1,t)=1+t. (Series added Dec 29 2015.)(End)
%F Let <n,k> denote the Eulerian numbers A173018(n,k), then T(n,k) = Sum_{j=0..n} <n,j>*binomial(n-j,n-k). - _Peter Luschny_, Jul 12 2013
%F Matrix product A007318 * A131689. The n-th row polynomial R(n,x) = Sum_{k >= 1} k^(n-1)*(x/(1 + x))^k, valid for x in the open interval (-1/2, inf). Cf A038719. R(n,-1/2) = (-1)^(n-1)*(2^n - 1)*Bernoulli(n)/n. - _Peter Bala_, Jul 14 2014
%F a(n,k) = A141618(n,k) / C(n,k-1). - _Tom Copeland_, Oct 25 2014
%F For the row polynomials, A028246(n,x) = A019538(n-1,x) * (1+x). - _Tom Copeland_, Dec 28 2015
%F A248727 = A007318*(reversed A028246) = A007318*A130850 = A007318*A123125*A007318 = A046802*A007318. - _Tom Copeland_, Nov 14 2016
%F n-th row polynomial R(n,x) = (1+x) o (1+x) o ... o (1+x) (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See example E11 in the Bala link. - _Peter Bala_, Jan 12 2018
%F From _Dorian Guyot_, May 21 2019: (Start)
%F Sum_{i=0..k} binomial(k,i) * a(n,i) = (k+1)^n.
%F Sum_{k=0..n} a(n,k) = 2*A000670(n).
%F (End)
%F With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of this entry, A028246, are given by x^n * A_n(1 + 1/x;0). Other specializations of A_n(x;y) give A046802, A090582, A119879, A130850, and A248727. - _Tom Copeland_, Jan 24 2020
%F The row generating polynomials R(n,x) = Sum_{i=1..n} a(n,i) * x^i satisfy the recurrence equation R(n+1,x) = R(n,x) + Sum_{k=0..n-1} binomial(n-1,k) * R(k+1,x) * R(n-k,x) for n >= 1 with initial value R(1,x) = x. - _Werner Schulte_, Jun 17 2021
%e The triangle a(n, k) starts:
%e n\k 1 2 3 4 5 6 7 8 9
%e 1: 1
%e 2: 1 1
%e 3: 1 3 2
%e 4: 1 7 12 6
%e 5: 1 15 50 60 24
%e 6: 1 31 180 390 360 120
%e 7: 1 63 602 2100 3360 2520 720
%e 8: 1 127 1932 10206 25200 31920 20160 5040
%e 9: 1 255 6050 46620 166824 317520 332640 181440 40320
%e ... [Reformatted by _Wolfdieter Lang_, Mar 26 2015]
%e -----------------------------------------------------
%e Row 5 of triangle is {1,15,50,60,24}, which is {1,15,25,10,1} times {0!,1!,2!,3!,4!}.
%e From _Vladimir Shevelev_, Dec 22 2011: (Start)
%e Also, for power sums, we have
%e S_0(n) = C(n,1);
%e S_1(n) = C(n,1) + C(n,2);
%e S_2(n) = C(n,1) + 3*C(n,2) + 2*C(n,3);
%e S_3(n) = C(n,1) + 7*C(n,2) + 12*C(n,3) + 6*C(n,4);
%e S_4(n) = C(n,1) + 15*C(n,2) + 50*C(n,3) + 60*C(n,4) + 24*C(n,5); etc.
%e (End)
%e For X = [1,2,3], the sets T are {{}}, {{},{1,2}}, {{},{1,3}}, {{},{1,2,3}}, {{},{1,2},{1,2,3}}, {{},{1,3},{1,2,3}} and so a(3,1)=1, a(3,2)=3, a(3,3)=2. - _Michael Somos_, Apr 20 2013
%p a := (n,k) -> add((-1)^(k-i)*binomial(k,i)*i^n, i=0..k)/k;
%p seq(print(seq(a(n,k),k=1..n)),n=1..10);
%p T := (n,k) -> add(eulerian1(n,j)*binomial(n-j,n-k), j=0..n):
%p seq(print(seq(T(n,k),k=0..n)),n=0..9); # _Peter Luschny_, Jul 12 2013
%t a[n_, k_] = Sum[(-1)^(k-i) Binomial[k,i]*i^n, {i,0,k}]/k; Flatten[Table[a[n, k], {n, 10}, {k, n}]] (* _Jean-François Alcover_, May 02 2011 *)
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, n! * polcoeff( (x / log(1 + x + x^2 * O(x^n) ))^(n+1), n-k))}; /* _Michael Somos_, Oct 02 2002 */
%o (PARI) {T(n,k) = stirling(n,k,2)*(k-1)!}; \\ _G. C. Greubel_, May 31 2019
%o (Sage)
%o def A163626_row(n) :
%o x = polygen(ZZ,'x')
%o A = []
%o for m in range(0, n, 1) :
%o A.append((-x)^m)
%o for j in range(m, 0, -1):
%o A[j - 1] = j * (A[j - 1] - A[j])
%o return list(A[0])
%o for i in (1..7) : print(A163626_row(i)) # _Peter Luschny_, Jan 25 2012
%o (Sage) [[stirling_number2(n,k)*factorial(k-1) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, May 30 2019
%o (Magma) [[StirlingSecond(n,k)*Factorial(k-1): k in [1..n]]: n in [1..10]]; // _G. C. Greubel_, May 30 2019
%o (GAP) Flat(List([1..10], n-> List([1..n], k-> Stirling2(n,k)* Factorial(k-1) ))) # _G. C. Greubel_, May 30 2019
%o (Python) # Assuming offset (n, k) = (0, 0).
%o def T(n, k):
%o if k > n: return 0
%o if k == 0: return 1
%o return k*T(n - 1, k - 1) + (k + 1)*T(n - 1, k)
%o for n in range(9):
%o print([T(n, k) for k in range(n + 1)]) # _Peter Luschny_, Apr 26 2022
%Y Dropping the column of 1's gives A053440.
%Y Without the k in the denominator (in the definition), we get A019538. See also the Stirling number triangle A008277.
%Y Cf. A087127, A087107, A087108, A087109, A087110, A087111, A084938 A075263.
%Y Row sums give A000629(n-1) for n >= 1.
%Y Cf. A027642, A002445. - _Gary W. Adamson_, Aug 09 2008
%Y Appears in A161739 (RSEG2 triangle), A161742 and A161743. - _Johannes W. Meijer_, Jun 18 2009
%Y Binomial transform is A038719. Cf. A131689.
%Y Cf. A007318, A008292, A046802, A074909, A090582, A123125, A130850, A135278, A141618, A145271, A163626, A248727, A263634.
%Y Cf. A119879.
%Y From _Rajesh Kumar Mohapatra_, Mar 29 2020: (Start)
%Y A000007(n-1) (column k=1), A000225(n-1) (column k=2), A028243(n-1) (column k=3), A028244(n-1) (column k=4), A028245(n-1) (column k=5), for n > 0.
%Y Diagonal gives A000142(n-1), for n >=1.
%Y Next-to-last diagonal gives A001710,
%Y Third, fourth, fifth, sixth, seventh external diagonal respectively give A005460, A005461, A005462, A005463, A005464. (End)
%K nonn,easy,nice,tabl
%O 1,5
%A _N. J. A. Sloane_, Doug McKenzie (mckfam4(AT)aol.com)
%E Definition corrected by Li Guo, Dec 16 2006
%E Typo in link corrected by _Johannes W. Meijer_, Oct 17 2009
%E Error in title corrected by _Johannes W. Meijer_, Sep 24 2010
%E Edited by _M. F. Hasler_, Oct 29 2014
|
