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A039755 Triangle of B-analogs of Stirling numbers of the second kind. 30

%I

%S 1,1,1,1,4,1,1,13,9,1,1,40,58,16,1,1,121,330,170,25,1,1,364,1771,1520,

%T 395,36,1,1,1093,9219,12411,5075,791,49,1,1,3280,47188,96096,58086,

%U 13776,1428,64,1,1,9841,239220,719860,618870,209622,32340,2388,81,1,1

%N Triangle of B-analogs of Stirling numbers of the second kind.

%C Let M = an infinite lower triangular bidiagonal matrix with (1,3,5,7,...) in the main diagonal and (1,1,1,...) in the subdiagonal. n-th row = M^n * [1,0,0,0,...]. - _Gary W. Adamson_, Apr 13 2009

%C From _Peter Bala_, Aug 08 2011: (Start)

%C A type B_n set partition is a partition P of the set {1, 2, ..., n, -1, -2, ..., -n} such that for any block B of P, -B is also a block of P, and there is at most one block, called a zero-block, satisfying B = -B. We call (B, -B) a block pair of P if B is not a zero-block. Then T(n,k) is the number of type B_n set partitions with k block pairs. See [Wang].

%C For example, T(2,1) = 4 since the B_2 set partitions with 1 block pair are {1,2}{-1,-2}, {1,-2}{-1,2}, {1,-1}{2}{-2} and {2,-2}{1}{-1} (the last two partitions contain a zero block).

%C (End)

%C Exponential Riordan array [exp(x), (1/2)*(exp(2*x) - 1)]. Triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis polynomials (x-1)*(x-3)*...*(x-(2*k-1)) of A039757. An example is given below. Inverse array is A039757. Equals matrix product A008277 * A122848. - _Peter Bala_, Jun 23 2014

%C T(n, k) gives also the (dimensionless) volume of the multichoose(k+1, n-k) = binomial(n, k) polytopes of dimension n-k with side lengths from the set {1, 3, ..., 1+2*k}. See the column g.f.s and the complete homogeneous symmetric function formula for T(n, k) below. - _Wolfdieter Lang_, May 26 2017

%C T(n, k) is the number of k-dimensional subspaces (i.e., sets of fixed points like rotation axes and symmetry planes) of the n-cube. See "Sets of fixed points..." in LINKS section. - _Tilman Piesk_, Oct 26 2019

%H G. C. Greubel, <a href="/A039755/b039755.txt">Rows n=0..100 of triangle, flattened</a>

%H V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.

%H Eli Bagno, Riccardo Biagioli, David Garber, <a href="https://arxiv.org/abs/1901.07830">Some identities involving second kind Stirling numbers of types B and D</a>, arXiv:1901.07830 [math.CO], 2019.

%H Peter Bala, <a href="/A035342/a035342.txt">Generalized Dobinski formulas</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry3/barry84r2.html">A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.

%H Sandrine Dasse-Hartaut and Pawel Hitczenko, <a href="http://arxiv.org/abs/1202.3092">Greek letters in random staircase tableaux</a> arXiv:1202.3092v1 [math.CO], 2012.

%H I. Dolgachev and V. Lunts, <a href="http://www.math.lsa.umich.edu/~idolga/luntz94.pdf">A character formula for the representation of the Weyl group in the cohomology of the associated toric variety</a> Journal of Algebra, 168, 741-772, (1994)

%H Thomas Godland, Zakhar Kabluchko, <a href="https://arxiv.org/abs/2009.04186">Projections and angle sums of permutohedra and other polytopes</a>, arXiv:2009.04186 [math.MG], 2020.

%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.

%H L. Liu, Y. Wang, <a href="http://www.arXiv.org/abs/math.CO/0509207">A unified approach to polynomial sequences with only real zeros</a>, arXiv:math/0509207v5 [math.CO], 2005-2006.

%H Shi-Mei Ma, T. Mansour, D. Callan, <a href="http://arxiv.org/abs/1404.0731">Some combinatorial arrays related to the Lotka-Volterra system</a>, arXiv:1404.0731 [math.CO], 2014.

%H E. Munarini, <a href="https://doi.org/10.2298/AADM0901157M">Characteristic, admittance and matching polynomials of an antiregular graph</a>, Appl. Anal. Discrete Math 3 (1) (2009) 157-176.

%H Tillmann Nett, Nadine Nett, Andreas Glöckner, <a href="https://doi.org/10.31234/osf.io/pxwq7">Bayesian Analysis of Processed Information in Decision Making Experiments</a>, FernUniversität (Hagen, Germany), University of Cologne (Germany, 2019).

%H T. Piesk, Sets of fixed points of permutations of the n-cube: n = <a href="https://commons.wikimedia.org/wiki/Category:Cube_subspaces_(image_set)">3</a> and <a href="https://commons.wikimedia.org/wiki/Category:Tesseract_subspaces_(image_set)">4</a>.

%H R. Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

%H D. G. L. Wang, <a href="http://arxiv.org/abs/1108.1264">The Limiting Distribution of the Number of Block Pairs in Type B Set Partitions</a>, arXiv:1108.1264v1 [math.CO], 2011.

%F E.g.f. row polynomials: exp(x + y/2 * (exp(2*x) - 1)).

%F T(n,k) = T(n-1,k-1) + (2*k+1)*T(n-1,k) with T(0,k) = 1 if k=0 and 0 otherwise. Sum_{k=0..n} T(n,k) = A007405(n). - _R. J. Mathar_, Oct 30 2009; corrected by _Joshua Swanson_, Feb 14 2019

%F T(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2*j+1)^n.

%F T(n,k) = (1/(2^k*k!)) * A145901(n,k). - _Peter Bala_

%F The row polynomials R(n,x) satisfy the Dobinski-type identity:

%F R(n,x) = exp(-x/2)*Sum_{k >= 0} (2*k+1)^n*(x/2)^k/k!, as well as the recurrence equation R(n+1,x) = (1+x)*R(n,x)+2*x*R'(n,x). The polynomial R(n,x) has all real zeros (apply [Liu et al., Theorem 1.1] with f(x) = R(n,x) and g(x) = R'(n,x)). The polynomials R(n,2*x) are the row polynomials of A154537. - _Peter Bala_, Oct 28 2011

%F Let f(x) = exp((1/2)*exp(2*x)+x). Then the row polynomials R(n,x) are given by R(n,exp(2*x)) = (1/f(x))*(d/dx)^n(f(x)). Similar formulas hold for A008277, A105794, A111577, A143494 and A154537. - _Peter Bala_, Mar 01 2012

%F From _Peter Bala_, Jul 20 2012: (Start)

%F The o.g.f. for the n-th diagonal (with interpolated zeros) is the rational function D^n(x), where D is the operator x/(1-x^2)*d/dx. For example, D^3(x) = x*(1+8*x^2+3*x^4)/(1-x^2)^5 = x + 13*x^3 + 58*x^5 + 170*x^7 + ... . See A214406 for further details.

%F An alternative formula for the o.g.f. of the n-th diagonal is exp(-x/2)*(Sum_{k >= 0} (2*k+1)^(k+n-1)*(x/2*exp(-x))^k/k!).

%F (End)

%F From _Tom Copeland_, Dec 31 2015: (Start)

%F T(n,m) = Sum_{i=0..n-m} 2^(n-m-i)*binomial(n,i)*St2(n-i,m), where St2(n,k) are the Stirling numbers of the second kind, A048993 (also A008277). See p. 755 of Dolgachev and Lunts.

%F The relation of this entry's e.g.f. above to that of the Bell polynomials, Bell_n(y), of A048993 establishes this formula from a binomial transform of the normalized Bell polynomials, NB_n(y) = 2^n Bell_n(y/2); that is, e^x exp[(y/2)(e^(2x)-1)] = e^x exp[x*2*Bell.(y/2)] = exp[x(1+NB.(y))] = exp(x*P.(y)), so the row polynomials of this entry are given by P_n(y) = [1+NB.(y)]^n = Sum_{k=0..n} C(n,k) NB_k(y) = Sum_{k=0..n} 2^k C(n,k) Bell_k(y/2).

%F The umbral compositional inverses of the Bell polynomials are the falling factorials Fct_n(y) = y! / (y-n)!; i.e., Bell_n(Fct.(y)) = y^n = Fct_n(Bell.(y)). Since P_n(y) = [1+2Bell.(y/2)]^n, the umbral inverses are determined by [1 + 2 Bell.[ 2 Fct.[(y-1)/2] / 2 ] ]^n = [1 + 2 Bell.[ Fct.[(y-1)/2] ] ]^n = [1+y-1]^n = y^n. Therefore, the umbral inverse sequence of this entry's row polynomials is the sequence IP_n( y) = 2^n Fct_n[(y-1)/2] = (y-1)(y-3) .. (y-2n+1) with IP_0(y) = 1 and, from the binomial theorem, with e.g.f. exp[x IP.(y)]= exp[ x 2Fct.[(y-1)/2] ] = (1+2x)^[(y-1)/2] = exp[ [(y-1)/2] log(1+2x) ].

%F (End)

%F Let B(n,k)=T(n,k)*((2*k)!)/(2^k*k!) and P(n,x) = Sum_{k=0..n} B(n,k)*x^(2*k+1). Then (1) P(n+1,x) = (x+x^3)*P'(n,x) for n>=0, and (2) Sum_{n>=0} B(n,k)/(n!)*t^n = binomial(2*k,k)*exp(t)*(exp(2*t)-1)^k/4^k for k>=0, and (3) Sum_{n>=0} t^n* P(n,x)/(n!)=x*exp(t)/sqrt(1+x^2-x^2*exp(2*t)). - _Werner Schulte_, Dec 12 2016

%F From _Wolfdieter Lang_, May 26 2017: (Start)

%F G.f. column k: x^k/Product_{j=0..k} (1 - (1+2*j)*x), k >= 0.

%F T(n, k) = h^{(k+1)}_{n-k}, the complete homogeneous symmetric function of degree n-k of the k+1 symbols a_j = 1 + 2*j, j=0, 1, ..., k. (End)

%e Triangle T(n,k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0: 1

%e 1: 1 1

%e 2: 1 4 1

%e 3: 1 13 9 1

%e 4: 1 40 58 16 1

%e 5: 1 121 330 170 25 1

%e 6: 1 364 1771 1520 395 36 1

%e 7: 1 1093 9219 12411 5075 791 49 1

%e 8: 1 3280 47188 96096 58086 13776 1428 64 1

%e 9: 1 9841 239220 719860 618870 209622 32340 2388 81 1

%e 10: 1 29524 1205941 5278240 6289690 2924712 630042 68160 3765 100 1

%e ... reformatted and extended by _Wolfdieter Lang_, May 26 2017

%e The sequence of row polynomials of A214406 begins [1, 1+x, 1+8*x+3*x^2, ...]. The o.g.f.'s for the diagonals of this triangle thus begin

%e 1/(1-x) = 1 + x + x^2 + x^3 + ...

%e (1+x)/(1-x)^3 = 1 + 4*x + 9*x^2 + 16*x^3 + ...

%e (1+8*x+3*x^2)/(1-x)^5 = 1 + 13*x + 58*x^2 + 170*x^3 + ... . - _Peter Bala_, Jul 20 2012

%e Connection constants: x^3 = 1 + 13*(x-1) + 9*(x-1)*(x-3) + (x-1)*(x-3)*(x-5). Hence row 3 = [1,13,9,1]. - _Peter Bala_, Jun 23 2014

%e Complete homogeneous symmetric functions: T(3, 1) = h^{(2)}_2 = 1^2 + 3^2 + 1^1*3^1 = 13. The three 2D polytopes are two squares and a rectangle. T(3, 2) = h^{(3)}_1 = 1^1 + 3^1 + 5^1 = 9. The 1D polytopes are three lines. - _Wolfdieter Lang_, May 26 2017

%e T(4, 3) = 16 is the number of 3-dimensional subspaces (mirror hyperplanes) of the 4-cube. (These are 4 cubes and 12 cuboids.) See "Sets of fixed points..." in LINKS section. - _Tilman Piesk_, Oct 26 2019

%p A039755 := proc(n,k) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1,k-1)+(2*k+1)*procname(n-1,k) ; fi; end: seq(seq(A039755(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Oct 30 2009

%t t[n_, k_] = Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!); Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[1 ;; 56]]

%t (* _Jean-François Alcover_, Jun 09 2011, after _Peter Bala_ *)

%o (PARI) T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp(x+y/2*(exp(2*x+x*O(x^n))-1)),n),k))

%o (MAGMA) [[(&+[(-1)^(k-j)*(2*j+1)^n*Binomial(k, j): j in [0..k]])/( 2^k*Factorial(k)): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Feb 14 2019

%o (Sage) [[sum((-1)^(k-j)*(2*j+1)^n*binomial(k, j) for j in (0..k))/( 2^k*factorial(k)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Feb 14 2019

%Y Cf. A154537, A214406, A039756, A039757, A122848, A008277, A048993.

%K nonn,tabl

%O 0,5

%A Ruedi Suter (suter(AT)math.ethz.ch)

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Last modified April 15 08:18 EDT 2021. Contains 342977 sequences. (Running on oeis4.)