A059110 Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 73, 136, 78, 16, 1, 501, 1045, 730, 210, 25, 1, 4051, 9276, 7515, 2720, 465, 36, 1, 37633, 93289, 85071, 36575, 8015, 903, 49, 1, 394353, 1047376, 1053724, 519456, 137270, 20048, 1596, 64, 1, 4596553, 12975561

Offset

0,4

Comments

L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.

Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007

From Wolfdieter Lang, Jun 22 2017: (Start)

The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).

The a- and z-sequences of this Sheffer (aka exponential Riordan) matrix are a = [1,1,repeat(0)] and z(n) = (-1)^(n+1)*A028310(n)/A000027(n-1) with e.g.f. ((1+x)/x)*(1-exp(-x)). For a- and z-sequences see a W. Lang link under A006232 with references. (End)

Links

Muniru A Asiru, Rows n=0..50 of triangle, flattened

Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Index entries for sequences related to Laguerre polynomials

Formula

E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.

From Wolfdieter Lang, Jun 22 2017: (Start)

E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).

Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).

Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.

General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.

(End)

P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018

From G. C. Greubel, Feb 23 2021: (Start)

T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.

Sum_{k=0..n} T(n, k) = A052897(n). (End)

Example

The triangle T = A007318*A271703 starts:
n\m       0        1        2       3       4      5     6    7  8 9 ...
0:        1
1:        1        1
2:        3        4        1
3:       13       21        9       1
4:       73      136       78      16       1
5:      501     1045      730     210      25      1
6:     4051     9276     7515    2720     465     36     1
7:    37633    93289    85071   36575    8015    903    49    1
8:   394353  1047376  1053724  519456  137270  20048  1596   64  1
9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - Wolfdieter Lang, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From Wolfdieter Lang, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)

Maple

Lprime := proc(n, i)
    if n = 0 and i = 0 then
        1;
    elif k = 0 then
        0 ;
    else
        n!/i!*binomial(n-1, i-1) ;
    end if;
end proc:
A059110 := proc(n, k)
    add(Lprime(n, i)*binomial(i, k), i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013

Mathematica

(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

Prog

(GAP) Concatenation([1], Flat(List([1..10], n->List([0..n], m->Sum([0..n], i-> Factorial(n)/Factorial(i)*Binomial(n-1, i-1)*Binomial(i, m)))))); # Muniru A Asiru, Jul 25 2018
(Sage)
def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021
(Magma)
A059110:= func< n, k | n eq 0 select 1 else Factorial(n-1)*Binomial(n, k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Crossrefs

Cf. A000262, A007318, A008297, A052852, A052897, A271703, A271705.

Sequence in context: A110506 A114189 A200659 * A100326 A303728 A321627

Adjacent sequences: A059107 A059108 A059109 * A059111 A059112 A059113

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Jan 04 2001

Status

approved