A073107 Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp((1+y)*x)/(1-x).
1, 2, 1, 5, 4, 1, 16, 15, 6, 1, 65, 64, 30, 8, 1, 326, 325, 160, 50, 10, 1, 1957, 1956, 975, 320, 75, 12, 1, 13700, 13699, 6846, 2275, 560, 105, 14, 1, 109601, 109600, 54796, 18256, 4550, 896, 140, 16, 1, 986410, 986409, 493200, 164388, 41076, 8190, 1344, 180, 18, 1
Offset: 0
Examples
exp((1 + y)*x)/(1 - x) = 1 + 1/1! * (2 + y) * x + 1/2! * (5 + 4*y + y^2) * x^2 + 1/3! * (16 + 15*y + 6*y^2 + y^3) * x^3 + 1/4! * (65 + 64*y + 30*y^2 + 8*y^3 + y^4) * x^4 + 1/5! * (326 + 325*y + 160*y^2 + 50*y^3 + 10*y^4 + y^5) * x^5 + ... Triangle starts: [0] 1; [1] 2, 1; [2] 5, 4, 1; [3] 16, 15, 6, 1; [4] 65, 64, 30, 8, 1; [5] 326, 325, 160, 50, 10, 1; [6] 1957, 1956, 975, 320, 75, 12, 1; [7] 13700, 13699, 6846, 2275, 560, 105, 14, 1;
Links
- Seiichi Manyama, Rows n = 0..139, flattened
- Wikipedia, Sheffer sequence.
Crossrefs
Programs
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Maple
T := (n, k) -> binomial(n,k)*KummerU(k-n, k-n, 1); seq(seq(simplify(T(n, k)), k = 0..n), n=0..8); # Peter Luschny, Oct 16 2024
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Mathematica
perm[m_List] := With[{v=Array[x,Length[m]]},Coefficient[Times@@(m.v),Times@@v]] ; A[q_] := Array[KroneckerDelta[#1,#2] + 1&,{q,q}] ; n = 1 ; Print[{1}]; While[n < 10, Print[Abs[CoefficientList[perm[A[n] - IdentityMatrix[n] * k], k]]]; n++] (* John M. Campbell, Jul 02 2012 *) A073107[n_, k_] := If[n == k, 1, Floor[E*(n - k)!]*Binomial[n, k]]; Table[A073107[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Oct 16 2024 *) -
SageMath
def T(n, k): return sum(binomial(j,k) * factorial(n) // factorial(j) for j in range(n+1)) for n in range(8): print([T(n, k) for k in range(n+1)]) # Peter Luschny, Oct 16 2024
Formula
O.g.f. for k-th column is (1/k!)*Sum_{i >= k} i!*x^i/(1-x)^(i+1).
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)), ... .
Row sums give A010842.
E.g.f. for k-th column is (x^k/k!)*exp(x)/(1 - x).
O.g.f. for k-th row is n!*Sum_{k = 0..n} (1 + x)^k/k!.
T(n,k) = Sum_{j = 0..n} binomial(j,k)*n!/j!. - Paul Barry, May 25 2006
-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
From Peter Bala, Sep 20 2012: (Start)
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.
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.
Matrix inverse of triangle is a signed version of A093375. (End)
From Tom Copeland, Oct 20 2015: (Start)
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).
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).
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)
From Werner Schulte, Sep 07 2020: (Start)
T(n,k) = (n! / k!) * (Sum_{i=k..n} 1 / (n-i)!) for 0 <= k <= n.
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.
T(n,k) = A000522(n-k) * binomial(n,k) for 0 <= k <= n. (End)
Extensions
More terms from Emeric Deutsch, Feb 23 2004
Comments