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A008292
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Triangle of Eulerian numbers T(n,k) (n>=1, 1 <= k <= n) read by rows.
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207
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1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 66, 26, 1, 1, 57, 302, 302, 57, 1, 1, 120, 1191, 2416, 1191, 120, 1, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013
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OFFSET
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1,5
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COMMENTS
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The indexing used here follows that given in the classic books by Riordan and Comtet. For two other versions see A173018 and A123125. - N. J. A. Sloane, Nov 21 2010
Coefficients of Eulerian polynomials. Number of permutations of n objects with k-1 rises. Number of increasing rooted trees with n+1 nodes and k leaves.
T(n,k)=number of permutations of [n] with k runs. T(n,k)=number of permutations of [n] requiring k readings (see the Knuth reference). T(n,k)=number of permutations of [n] having k distinct entries in its inversion table. - Emeric Deutsch, Jun 09 2004
T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004
Subtriangle for k>=1 and n>=1 of triangle A123125 . - Philippe DELEHAM, Oct 22 2006
Comment from Stefano Zunino, Oct 25 2006: T(n,k)/n! also represents the n-dimensional volume of the portion of the n-dimensional hyper-cube cut by the (n-1)-dimensional hyperplanes x_1 + x_2 + ... x_n = k, x_1 + x_2 + ... x_n = k-1; or, equivalently, it represents the probability that the sum of n independent random variables with uniform distribution between 0 and 1 is between k-1 and k.
[ E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and
[ -P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials and P(n,t) are the polynomials in A131758. - Tom Copeland, Sep 30 2007
Comment from Tom Copeland, Oct 07 2008: (Start)
G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1*x + (2+t)*x^2/2! + (6+6*t+t^2)*x^3/3! + ...
gives row polynomials for A090582-- reverse f-polynomials for the permutohedra (see A019538).
G(x,t-1) = 1 + 1*x + (1+t)*x^2/2! + (1+4*t+t^2)*x^3/3! + ...
gives row polynomials for A008292, the h-polynomials for permutohedra.
G((t+1)*x,-1/(t+1)) = 1 + (1+t)*x + (1+3*t+2*t^2)*x^2/2! + ...
gives row polynomials for A028246.
(End)
A subexceedant function f on [n] is a map f:[n] -> [n] such that 1 <= f(i) <= i for all i, 1 <= i <= n. T(n,k) equals the number of subexceedant functions f of [n] such that the image of f has cardinality k [Mantaci & Rakotondrajao]. Example T(3,2) = 4: if we identify a subexceedant function f with the word f(1)f(2)...f(n) then the subexceedant functions on [3] are 111, 112, 113, 121, 122 and 123 and four of these functions have an image set of cardinality 2. [From Peter Bala, Oct 21 2008]
Further to the comments of T. Copeland above, the n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type A_(n-1). The corresponding f-vectors are the rows of A019538. For example, 1 + 4*x + x^2 = y^2 + 6*y + 6 and 1 + 11*x + 11*x^2 + x^3 = y^3 + 14*y^2 + 36*y + 24, where x = y + 1, give [1,6,6] and [1,14,36,24] as the third and fourth rows of A019538. The Hilbert transform of this triangle (see A145905 for the definition) is A047969. See A060187 for the triangle of Eulerian numbers of type B (the h-vectors of the simplicial complexes dual to permutohedra of type B). See A066094 for the array of h-vectors of type D. For tables of restricted Eulerian numbers see A144696 - A144699. [From Peter Bala, Oct 26 2008]
For a natural refinement of A008292 with connections to compositional inversion and iterated derivatives, see A145271. [From Tom Copeland, Nov 06 2008]
Contribution from Johannes W. Meijer, May 24 2009: (Start)
The polynomials E(z,n) = numer(sum((-1)^(n+1)*k^n*z^(k-1), k=1..infinity)) for n >=1 lead directly to the triangle of Eulerian numbers. (End)
Contribution from Walther Janous (walther.janous(AT)tirol.com), Nov 01 2009: (Start)
The (Eulerian) polynomials e(n,x) = SUM(T(n,k+1)*x^k, k = 0 to n-1) turn out to be also the numerators of the closed-form expressions of the infinite sums
S(p,x):= SUM((j+1)^p*x^j, j = 0 to infinity), that is
S(p,x) = e(p,x)/(1-x)^(p+1), whenever |x| < 1 and p is a positive integer.
(Note the inconsistent use of T(n,k) in the section listing the formula section. I adhere tacitly to the first one.) (End)
If n is an odd prime, then all numbers of the (n-2)-th and (n-1)-th rows are in the progression k*n+1.-Vladimir Shevelev, Jul 01 2011.
The Eulerian triangle is an element of the formula for the r-th successive summation of sum(k^j,k=1..n)which appears to be sum(T(j,k-1) * binomial(j-k+n+r,j+r), k=1..n). [From Gary Detlefs, Nov 11 2001]
Li and Wong show that T(n,k) counts the combinatorially inequivalent star polygons with n+1 vertices and sum of angles (2*k-n-1)*Pi. An equivalent formulation is: define the total sign change S(p) of a permutation p in the symmetric group S_n to be equal to sum {i = 1..n} sign(p(i)-p(i+1)), where we take p(n+1) = p(1). T(n,k) gives the number of permutations q in S_(n+1) with q(1) = 1 and S(q) = 2*k-n-1. For example, T(3,2) = 4 since in S_4 the permutations (1243), (1324), (1342) and (1423) have total sign change 0. - Peter Bala, Dec 27 2011
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REFERENCES
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L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
J. Desarmenien and D. Foata, The signed Eulerian numbers. Discrete Math. 99 (1992), no. 1-3, 49-58.
Ehrenborg & Readdy (J Comb. Theory, Series A 81, 121-126).
D. Foata, Distributions eule'riennes et mahoniennes sur le groupe des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254; 2nd. ed., p. 268.
Hauser, Herwig; Koutschan, Christoph. Multivariate linear recurrences and power series division. Discrete Math. 312 (2012), no. 24, 3553--3560. MR2979485.
P. Hitczenko and S. Janson, Weighted random staircase tableaux, arXiv preprint arXiv:1212.5498, 2012.
A. Kerber and K.-J. Thuerlings, Eulerian numbers, Foulkes characters and Lefschetz characters of S_n, Seminaire Lotharingien, Vol. 8 (1984), 31-36.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1998, Vol. 3, p. 47, (exercise 5.1.4 Nr. 20) and p. 605 (solution).
Lehmer, D. H. Generalized Eulerian numbers. J. Combin. Theory Ser.A 32 (1982), no. 2, 195--215. MR0654621 (83k:10026).
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO].
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654.
P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.
Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page http://math.ucsd.edu/~remmel/
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
J. Sack and H. Ulfarsson, Refined inversion statistics on permutations, Arxiv preprint arXiv:1106.1995, 2011.
R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Figure M3416, Academic Press, 1995.
R. Sprugnoli, Alternating Weighted Sums of Inverses of Binomial Coefficients, J. Integer Sequences, 15 (2012), #12.6.3..
S. Tanimoto, A study of Eulerian numbers by means of an operator on permutations, Europ. J. Combin., 24 (2003), 33-43.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973, see p. 208.
D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012; http://www.kazntu.kz/sites/default/files/20121221ND_Eleusizov.pdf.
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LINKS
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T. D. Noe, Rows 1 to 100 of triangle, flattened.
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, Arxiv preprint 2011.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, Arxiv preprint 2011.
D. Barsky, Analyse p-adique et suites classiques de nombres, Sem. Loth. Comb. B05b (1981) 1-21.
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
L. Carlitz, Eulerian numbers and operators, Collectanea Mathematica 24:2 (1973), pp. 175-200.
Mircea I. Cirnu, Determinantal formulas for sum of generalized arithmetic-geometric series, Boletin de la Asociacion Matematica Venezolana, Vol. XVIII, No. 1 (2011), p. 13.
J. Desarmenien and D. Foata, The signed Eulerian Numbers
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]
D. Foata, M. Schutzenberger. - Theorie Geometrique des Polynomes Euleriens, Lecture Notes in Math., no.138, Springer Verlag 1970. [From Peter Bala, Oct 26 2008]
Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432
Ira Gessel, The Smith College diploma problem.
Jim Haglund and Mirko Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations.
Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom
A. R. Kr\"auter, \"Uber die Permanente gewisser zirkul\"arer Matrizen...
M-H. Li and N-C. Wong, Sums of angles of star polygons and the Eulerian Numbers, Southeast Asian Bulletin of Mathematics 2004.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11
R. Mantaci and F. Rakotondrajao, A permutation representation that knows what "Eulerian" means, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001)
S. Parker, The Combinatorics of Functional Composition and Inversion, Dissertation, Brandeis Univ. (1993) [From Tom Copeland, Nov 10 2008]
A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.
Eric Weisstein's World of Mathematics, Eulerian Number, Euler's Number Triangle, Eulerian Number
L. K. Williams, Enumeration of totally positive Grassmann cells, arxiv:math.CO/0307271
Index entries for sequences related to rooted trees
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FORMULA
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T(n, k) = k * T(n-1, k) + (n-k+1) * T(n-1, k-1), T(1, 1) = 1.
T(n, k) = Sum (-1)^j * (k-j)^n * C(n+1, j), j=0..k.
Row sums = n! = A000142(n) unless n=0. - Michael Somos, Mar 17 2011
E.g.f. A(x, q) = Sum_{n>0} (Sum_{k=1..n} T(n, k) * q^k) * x^n / n! = q * ( e^(q*x) - e^x ) / ( q*e^x - e^(q*x) ) satisfies dA / dx = (A + 1) * (A + q). - Michael Somos, Mar 17 2011
For a column listing, n-th term: T(c, n)=c^(n+c-1)+sum(i=1, c-1, (-1)^i/i!*(c-i)^(n+c-1)*prod(j=1, i, n+c+1-j)). - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002
Four characterizations of Eulerian numbers T(i, n) from John Robertson (jpr2718(AT)aol.com), Sep 02, 2002: (Start)
1. T(0, n)=1 for n>=1, T(i, 1)=0 for i>=1, T(i, n) = (n-i)T(i-1, n-1) + (i+1)T(i, n-1).
2. T(i, n) = sum_{j=0}^{i} (-1)^j (n+1 combin j) (i-j+1)^n for n>=1, i>=0.
3. Let C_n be the unit cube in R^n with vertices (e_1, e_2, ..., e_n) where each e_i is 0 or 1 and all 2^n combinations are used. Then T(i, n)/n! is the volume of C_n between the hyperplanes x_1 + x_2 + ... + x_n = i and x_1 + x_2 + ... + x_n = i+1. Hence T(i, n)/n! is the probability that i <= X_1 + X_2 + ... + X_n < i+1 where the X_j are independent uniform [0, 1] distributions. - See Ehrenborg & Readdy reference.
4. Let f(i, n) = T(i, n)/n!. The f(i, n) are the unique coefficients so that (1/(r-1)^(n+1)) sum_{i=0}^{n-1} f(i, n) r^{i+1} = sum_{j=0}^{infinity} (j^n)/(r^j) whenever n>=1 and abs(r)>1. (End)
O.g.f. for n-th row: (1-x)^(n+1)*polylog(-n, x)/x. - Vladeta Jovovic, Sep 02 2002
Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] (positive integers interspersed with 0's) where DELTA is Deleham's operator defined in A084938.
Sum_{k = 1..n} T(n, k)*2^k = A000629(n) . - Philippe Deléham, Jun 05 2004
From Tom Copeland, Oct 10 2007: (Start)
Bell(n,x) = sum(j=0,...,n) S2(n,j) * x^j = sum(j=0,...,n) E(n,j) * Lag(n,-x, j-n) = sum(j=0,...,n) [ E(n,j)/n! ] * [ n!*Lag(n,-x, j-n) ] = sum(j=0,...,n) E(n,j) * C(Bell(.,x)+j,n) umbrally where Bell(n,x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; Lag(n,x,m), the associated Laguerre polynomials of order m; and C(x,y) = x!/[ y!*(x-y)! ].
For x = 0, the equation gives sum(j=0,...,n) E(n,j) * C(j,n) = 1 for n = 0 and 0 for all other n. By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*C(y,n), for x in the equation, the Worpitzky identity is obtained; y^n = sum(j=0,...,n) E(n,j) * C(y+j,n) .
Note that E(n,j)/n! = E(n,j)/ {sum(k=0,..,n)E(n,k)}. Also [n!*Lag(n,-1, j-n)] is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to the equation for x=1; n!*Bell(n,1) = n!*sum(j=0,...,n) S2(n,j) = sum(j=0,...,n) {E(n,j) * [n!*Lag(n,-1, j-n)]} . (End)
From the relations between the h- and f-polynomials of permutohedra and reciprocals of e.g.f.s described in A049019: (t-1)[(t-1)d/dx]^n 1/(t-exp(x)) evaluated at x=0 gives the n-th Eulerian row polynomial in t and the n-th row polynomial in (t-1) of A019538 and A090582. From the Comtet and Copeland references in A139605: [(t+exp(x)-1)d/dx]^(n+1) x gives pairs of the Eulerian polynomials in t as the coefficients of x^0 and x^1 in its Taylor series expansion in x. [From Tom Copeland, Oct 05 2008]
G.f: 1/(1-x/(1-x*y/1-2x/(1-2x*y/(1-3x/(1-3x*y/(1-... (continued fraction). [From Paul Barry, Mar 24 2010]
If n is odd prime, then the following consecutive 2*n+1 terms are 1 modulo n: a((n-1)*(n-2)/2+i), i=0,...,2*n. This chain of terms is maximal in the sense that neither the previous term nor the following one are 1 modulo n. [Vladimir Shevelev, Jul 01 2011]
From Peter Bala, Sept 29 2011: (Start)
For k = 0,1,2,... put G(k,x,t) := x -(1+2^k*t)*x^2/2 +(1+2^k*t+3^k*t^2)*x^3/3-(1+2^k*t+3^k*t^2+4^k*t^3)*x^4/4+.... Then the series reversion of G(k,x,t) with respect to x gives an e.g.f. for the present table when k = 0 and for A008517 when k = 1.
The e.g.f. B(x,t) := compositional inverse with respect to x of G(0,x,t) = (exp(x)-exp(x*t))/(exp(x*t)-t*exp(x)) = x+(1+t)*x^2/2!+(1+4*t+t^2)*x^3/3!+... satisfies the autonomous differential equation dB/dx = (1+B)*(1+t*B) = 1 + (1+t)*B + t*B^2.
Applying [Bergeron et al., Theorem 1] gives a combinatorial interpretation for the Eulerian polynomials: A(n,t) counts plane increasing trees on n vertices where each vertex has outdegree <= 2, the vertices of outdegree 1 come in 1+t colors and the vertices of outdegree 2 come in t colors. An example is given below. Cf. A008517. Applying [Dominici, Theorem 4.1] gives the following method for calculating the Eulerian polynomials: Let f(x,t) = (1+x)*(1+t*x) and let D be the operator f(x,t)*d/dx. Then A(n+1,t) = D^n(f(x,t)) evaluated at x = 0.
[End]
With e.g.f. A(x,t)= G[x,(t-1)]-1 in Copeland's 2008 comment, the compositional inverse is Ainv(x,t)= ln[t-(t-1)/(1+x)]/(t-1). - Tom Copeland, Oct 11 2011
T(2*n+1,n+1)= (2*n+2)*T(2*n,n). (E.g. 66=6*11, 2416=8*302...) [From Gary Detlefs, Nov 11 2011]
E.g.f.: (y -1)/(y - exp( (y-1)*x )). - Geoffrey Critzer, Nov 10 2012
From Peter Bala, Mar 12 2013: (Start)
Let {A(n,x)}n>=1 denote the sequence of Eulerian polynomials beginning [1, 1 + x, 1 + 4*x + x^2, ...]. Given two complex numbers a and b, the polynomial sequence defined by R(n,x) := (x+b)^n*A(n+1,(x+a)/(x+b)), n >= 0, satisfies the recurrence equation R(n+1,x) = d/dx((x+a)*(x+b)*R(n,x)). These polynomials give the row generating polynomials for several triangles in the database including A019538 (a = 0, b = 1), A156992 (a = 1, b = 1), A185421 (a = (1+i)/2, b = (1-i)/2), A185423 (a = exp(i*Pi/3), b = exp(-i*Pi/3)) and A185896 (a = i, b = -i).
(End)
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EXAMPLE
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Triangle begins:
1;
1,1;
1,4,1;
1,11,11,1;
1,26,66,26,1;
1,57,302,302,57,1;
1,120,1191,2416,1191,120,1,
1,247,4293,15619,15619,4293,247,1,
1,502,14608,88234,156190,88234,14608,502,1,
1,1013,47840,455192,1310354,1310354,455192,47840,1013,1
...
E.g.f. = (q) * x^1 / 1! + (q + q^2) * x^2 / 2! + (q + 4*q^2 + q^3) * x^3 / 3! + ... - Michael Somos, Mar 17 2011
Let n=7. Then the following 2*7+1=15 consecutive terms are 1(mod 7): a(15+i), i=0,...,14. [Vladimir Shevelev, Jul 01 2011]
Row 3: The plane increasing 0-1-2 trees on 3 vertices (with the number of colored vertices shown to the right of a vertex) are
...............................................
....1o.(1+t).........1o.t.........1o.t.........
....|............... /.\........../.\..........
....|.............. /...\......../...\.........
....2o.(1+t)......2o.....3o....3o....2o........
....|..........................................
....|..........................................
....3o.........................................
...............................................
The total number of trees is (1+t)^2+t+t = 1+4*t+t^2.
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MAPLE
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A008292 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1) ; end if; end proc:
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MATHEMATICA
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len = 54; m = Ceiling[2 Sqrt[len]]; t[n_, k_] = Sum[(-1)^j*(k - j)^n*Binomial[n + 1, j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 1, m}, {k, 1, n}]][[1 ;; len]] (* Jean-François Alcover, May 31 2011, after M. Somos *)
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PROG
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(PARI) {T(n, k) = if( k<1 || k>n, 0, if( n==1, 1, k * T(n-1, k) + (n-k+1) * T(n-1, k-1)))} /* Michael Somos, Jul 19 1999 */
(PARI) {T(n, k) = sum( j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))} /* Michael Somos, Jul 19 1999 */
{A008292(c, n)=c^(n+c-1)+sum(i=1, c-1, (-1)^i/i!*(c-i)^(n+c-1)*prod(j=1, i, n+c+1-j))}
(Haskell)
import Data.List (genericLength)
a008292 n k = a008292_tabl !! (n-1) !! (k-1)
a008292_row n = a008292_tabl !! (n-1)
a008292_tabl = iterate f [1] where
f xs = zipWith (+)
(zipWith (*) ([0] ++ xs) (reverse ks)) (zipWith (*) (xs ++ [0]) ks)
where ks = [1 .. 1 + genericLength xs]
-- Reinhard Zumkeller, May 07 2013
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CROSSREFS
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See A173018 and A123125 for other ways to index the entries of this triangle.
Cf. A000142, A014630, A030196, A055325.
Cf. A006551, A025585, A180056, A180057, A169972, A027656 A084938, A049061, A008275, A008277, A087127.
Columns 2 through 8: A000295, A000460, A000498, A000505, A000514, A001243, A001244.
Cf. A062253, noting that A008292 gives the (unsigned) polynomial coefficients of (kn). Also note that taking alternating differences of rows with an odd number of terms (e.g. 1=1, -1+4-1=2, 1-26+66-26+1=16, -1+120-1191+2416-1191+120-1=272 etc.) gives A000182. - Henry Bottomley, Jun 15 2001
Cf. A019538 (f-vectors type A permutohedra), A047969 (Hilbert transform), A060187 (h-vectors type B permutohedra), A066094. [Peter Bala, Oct 26 2008]
Sequence in context: A168287 A221987 A174526 * A174036 A157221 A146967
Adjacent sequences: A008289 A008290 A008291 * A008293 A008294 A008295
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KEYWORD
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nonn,tabl,nice,eigen,core,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Thanks to Michael Somos for additional comments. Further comments from Christian G. Bower, May 12 2000
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STATUS
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approved
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