A073107 - OEIS

%I #82 Oct 16 2024 20:52:47

%S 1,2,1,5,4,1,16,15,6,1,65,64,30,8,1,326,325,160,50,10,1,1957,1956,975,

%T 320,75,12,1,13700,13699,6846,2275,560,105,14,1,109601,109600,54796,

%U 18256,4550,896,140,16,1,986410,986409,493200,164388,41076,8190,1344,180,18,1

%N Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp((1+y)*x)/(1-x).

%C Triangle is second binomial transform of A008290. - _Paul Barry_, May 25 2006

%C Ignoring signs, n-th row is the coefficient list of the permanental polynomial of the n X n matrix with 2's along the main diagonal and 1's everywhere else (see Mathematica code below). - _John M. Campbell_, Jul 02 2012

%H Seiichi Manyama, <a href="/A073107/b073107.txt">Rows n = 0..139, flattened</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sheffer_sequence">Sheffer sequence</a>.

%F O.g.f. for k-th column is (1/k!)*Sum_{i >= k} i!*x^i/(1-x)^(i+1).

%F For n > 0, T(n, 0) = floor(n!*exp(1)) = A000522(n), T(n, 1) = floor(n!*exp(1) - 1) = A007526(n), T(n, 2) = 1/2!*floor(n!*exp(1) - 1 - n) = A038155(n), T(n, 3) = 1/3!*floor(n!*exp(1) - 1 - n - n*(n - 1)), T(n, 4) = 1/4!*floor(n!*exp(1) - 1 - n - n*(n - 1) - n*(n - 1)*(n - 2)), ... .

%F Row sums give A010842.

%F E.g.f. for k-th column is (x^k/k!)*exp(x)/(1 - x).

%F O.g.f. for k-th row is n!*Sum_{k = 0..n} (1 + x)^k/k!.

%F T(n,k) = Sum_{j = 0..n} binomial(j,k)*n!/j!. - _Paul Barry_, May 25 2006

%F -exp(-x) * Sum_{k=0..n} T(n,k)*x^k = Integral (x+1)^n*exp(-x) dx = -exp(1)*Gamma(n+1,x+1). - _Gerald McGarvey_, Mar 15 2009

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

%F Exponential Riordan array [exp(x)/(1-x),x] belonging to the Appell subgroup, which factorizes in the Appell group as [1/1-x,x]*[exp(x),x] = A094587*A007318.

%F The n-th row polynomial R(n,x) of the triangle satisfies d/dx(R(n,x)) = n*R(n-1,x), as well as R(n,x + y) = Sum {k = 0..n} binomial(n,k)*R(k,x)*y^(n-k). The row polynomials are a Sheffer sequence of Appell type.

%F Matrix inverse of triangle is a signed version of A093375. (End)

%F From _Tom Copeland_, Oct 20 2015: (Start)

%F The raising operator, with D = d/dx, for the row polynomials is RP = x + d{log[e^D/(1-D)]}/dD = x + 1 + 1/(1-D) =  x + 2 + D + D^2 + ..., i.e., RP R(n,x) = R(n+1,x).

%F This operator is the limit as t tends to 1 of the raising operator of the polynomials p(n,x;t) described in A046802, implying R(n,x) = p(n,x;1). Compare with the raising operator of A094587, x + 1/(1-D), and that of signed A093375, x - 1 - 1/(1-D).

%F From the Appell formalism, the row polynomials RI(n,x) of signed A093375 are the umbral inverse of this entry's row polynomials; that is, R(n,RI(.,x)) = x^n = RI(n,R(.,x)) under umbral composition. (End)

%F From _Werner Schulte_, Sep 07 2020: (Start)

%F T(n,k) = (n! / k!) * (Sum_{i=k..n} 1 / (n-i)!) for 0 <= k <= n.

%F T(n,k) = n * T(n-1,k) + binomial(n,k) for 0 <= k <= n with initial values T(0,0) = 1 and T(i,j) = 0 if j < 0 or j > i.

%F T(n,k) = A000522(n-k) * binomial(n,k) for 0 <= k <= n. (End)

%e exp((1 + y)*x)/(1 - x) =

%e   1 +

%e   1/1! * (2 + y) * x +

%e   1/2! * (5 + 4*y + y^2) * x^2 +

%e   1/3! * (16 + 15*y + 6*y^2 + y^3) * x^3 +

%e   1/4! * (65 + 64*y + 30*y^2 + 8*y^3 + y^4) * x^4 +

%e   1/5! * (326 + 325*y + 160*y^2 + 50*y^3 + 10*y^4 + y^5) * x^5 + ...

%e Triangle starts:

%e   [0]     1;

%e   [1]     2,     1;

%e   [2]     5,     4,    1;

%e   [3]    16,    15,    6,    1;

%e   [4]    65,    64,   30,    8,   1;

%e   [5]   326,   325,  160,   50,  10,   1;

%e   [6]  1957,  1956,  975,  320,  75,  12,  1;

%e   [7] 13700, 13699, 6846, 2275, 560, 105, 14, 1;

%p T := (n, k) -> binomial(n,k)*KummerU(k-n, k-n, 1);

%p seq(seq(simplify(T(n, k)), k = 0..n), n=0..8);  # _Peter Luschny_, Oct 16 2024

%t perm[m_List] := With[{v=Array[x,Length[m]]},Coefficient[Times@@(m.v),Times@@v]] ;

%t A[q_] := Array[KroneckerDelta[#1,#2] + 1&,{q,q}] ;

%t n = 1 ; Print[{1}]; While[n < 10, Print[Abs[CoefficientList[perm[A[n] - IdentityMatrix[n] * k], k]]]; n++] (* _John M. Campbell_, Jul 02 2012 *)

%t A073107[n_, k_] := If[n == k, 1, Floor[E*(n - k)!]*Binomial[n, k]];

%t Table[A073107[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Oct 16 2024 *)

%o (SageMath)

%o def T(n, k):

%o     return sum(binomial(j,k) * factorial(n) // factorial(j) for j in range(n+1))

%o for n in range(8): print([T(n, k) for k in range(n+1)])

%o # _Peter Luschny_, Oct 16 2024

%Y Cf. A008290, A008291, A046802, A093375 (unsigned inverse), A094587, A010842 (row sums), A000142 (alternating row sums), A367963 (central terms).

%Y Column k=0..4 give A000522, A007526, A038155, A357479, A357480.

%K easy,nonn,tabl

%O 0,2

%A _Vladeta Jovovic_, Aug 19 2002

%E More terms from _Emeric Deutsch_, Feb 23 2004