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%I #64 Sep 06 2024 08:05:34
%S 1,1,1,2,2,1,4,6,3,1,10,16,12,4,1,26,50,40,20,5,1,76,156,150,80,30,6,
%T 1,232,532,546,350,140,42,7,1,764,1856,2128,1456,700,224,56,8,1,2620,
%U 6876,8352,6384,3276,1260,336,72,9,1,9496,26200,34380,27840,15960,6552,2100,480,90,10,1
%N Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.
%C Triangle related to A000085.
%C Riordan array [exp(x(2+x)/2),x]. - _Paul Barry_, Nov 05 2008
%C Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - _Peter Bala_, May 14 2012
%C T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - _James East_, Aug 17 2015
%H Muniru A Asiru, <a href="/A111062/b111062.txt">Table of n, a(n) for n = 0..1325</a>
%H Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv:1408.2021 [math.GR], 2014.
%H Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, <a href="http://dx.doi.org/10.1016/j.jcta.2014.11.008">Enumeration of idempotents in diagram semigroups and algebras</a>, J. Combin. Theory Ser. A 131 (2015), 119-152.
%H Tom Halverson and Theodore N. Jacobson, <a href="https://arxiv.org/abs/1808.08118">Set-partition tableaux and representations of diagram algebras</a>, arXiv:1808.08118 [math.RT], 2018.
%F Sum_{k>=0} T(m, k)*T(n, k)*k! = T(m+n, 0) = A000085(m+n).
%F Sum_{k=0..n} T(n, k) = A005425(n).
%F Apparently satisfies T(n,m) = T(n-1,m-1) + T(n-1,m) + m * T(n-1,m+1). - _Franklin T. Adams-Watters_, Dec 22 2005
%F T(n,k) = (n!/k!)*Sum_{j=0..n-k} C(j,n-k-j)/(j!*2^(n-k-j)). - _Paul Barry_, Nov 05 2008
%F G.f.: 1/(1-xy-x-x^2/(1-xy-x-2x^2/(1-xy-x-3x^2/(1-xy-x-4x^2/(1-... (continued fraction). - _Paul Barry_, Apr 23 2009
%F T(n,k) = C(n,k)*Sum_{j=0..n-k} C(n-k,j)*(n-k-j-1)!! where m!!=0 if m is even. - _James East_, Aug 17 2015
%F From _Tom Copeland_, Jun 26 2018: (Start)
%F E.g.f.: exp[t*p.(x)] = exp[t + t^2/2] e^(x*t).
%F These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x + 1 + D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
%F The transpose of the production matrix gives a matrix representation of the raising operator R.
%F exp(D + D^2/2) x^n= e^(D^2/2) (1+x)^n = h_n(1+x) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A000085(n) and h_n(x) the modified Hermite polynomials of A099174.
%F A159834 with the e.g.f. exp[-(t + t^2/2)] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator x - 1 - D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)
%e Rows begin:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 4, 6, 3, 1;
%e 10, 16, 12, 4, 1;
%e 26, 50, 40, 20, 5, 1;
%e 76, 156, 150, 80, 30, 6, 1;
%e 232, 532, 546, 350, 140, 42, 7, 1;
%e 764, 1856, 2128, 1456, 700, 224, 56, 8, 1;
%e 2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1;
%e From _Paul Barry_, Apr 23 2009: (Start)
%e Production matrix is:
%e 1, 1,
%e 1, 1, 1,
%e 0, 2, 1, 1,
%e 0, 0, 3, 1, 1,
%e 0, 0, 0, 4, 1, 1,
%e 0, 0, 0, 0, 5, 1, 1,
%e 0, 0, 0, 0, 0, 6, 1, 1,
%e 0, 0, 0, 0, 0, 0, 7, 1, 1,
%e 0, 0, 0, 0, 0, 0, 0, 8, 1, 1 (End)
%e From _Peter Bala_, Feb 12 2017: (Start)
%e The infinitesimal generator has integer entries and begins
%e 0
%e 1 0
%e 1 2 0
%e 0 3 3 0
%e 0 0 6 4 0
%e 0 0 0 10 5 0
%e 0 0 0 0 15 6 0
%e ...
%e and is the generalized exponential Riordan array [x + x^2/2!,x].(End)
%t a[n_] := Sum[(2 k - 1)!! Binomial[n, 2 k], {k, 0, n/2}]; Table[Binomial[n, k] a[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Aug 20 2015, after _Michael Somos_ at A000085 *)
%o (Sage)
%o def A111062_triangle(dim):
%o M = matrix(ZZ,dim,dim)
%o for n in (0..dim-1): M[n,n] = 1
%o for n in (1..dim-1):
%o for k in (0..n-1):
%o M[n,k] = M[n-1,k-1]+M[n-1,k]+(k+1)*M[n-1,k+1]
%o return M
%o A111062_triangle(9) # _Peter Luschny_, Sep 19 2012
%o (GAP) Flat(List([0..10],n->List([0..n],k->(Factorial(n)/Factorial(k))*Sum([0..n-k],j->Binomial(j,n-k-j)/(Factorial(j)*2^(n-k-j)))))); # _Muniru A Asiru_, Jun 29 2018
%Y Cf. A000085, A005425 (row sums), A007318, A013989, A122832, A132440.
%Y Cf. A099174, A133314, A159834 (inverse matrix).
%K easy,nonn,tabl
%O 0,4
%A _Philippe Deléham_, Oct 07 2005
%E Corrected by _Franklin T. Adams-Watters_, Dec 22 2005
%E 10th row added by _James East_, Aug 17 2015