A059110 - OEIS

%I #46 Apr 17 2021 03:40:42

%S 1,1,1,3,4,1,13,21,9,1,73,136,78,16,1,501,1045,730,210,25,1,4051,9276,

%T 7515,2720,465,36,1,37633,93289,85071,36575,8015,903,49,1,394353,

%U 1047376,1053724,519456,137270,20048,1596,64,1,4596553,12975561

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

%C 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.

%C Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - _Paul Barry_, Apr 28 2007

%C From _Wolfdieter Lang_, Jun 22 2017: (Start)

%C 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).

%C 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)

%H Muniru A Asiru, <a href="/A059110/b059110.txt">Rows n=0..50 of triangle, flattened</a>

%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

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

%F From _Wolfdieter Lang_, Jun 22 2017: (Start)

%F 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).

%F 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).

%F 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.

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

%F (End)

%F 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

%F From _G. C. Greubel_, Feb 23 2021: (Start)

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

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

%e The triangle T = A007318*A271703 starts:

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

%e 0:        1

%e 1:        1        1

%e 2:        3        4        1

%e 3:       13       21        9       1

%e 4:       73      136       78      16       1

%e 5:      501     1045      730     210      25      1

%e 6:     4051     9276     7515    2720     465     36     1

%e 7:    37633    93289    85071   36575    8015    903    49    1

%e 8:   394353  1047376  1053724  519456  137270  20048  1596   64  1

%e 9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1

%e ... reformatted. - _Wolfdieter Lang_, Jun 22 2017

%e 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 + ...

%e From _Wolfdieter Lang_, Jun 22 2017: (Start)

%e The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...

%e 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.

%e 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).

%e 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)

%p Lprime := proc(n,i)

%p     if n = 0 and i = 0 then

%p         1;

%p     elif k = 0 then

%p         0 ;

%p     else

%p         n!/i!*binomial(n-1,i-1) ;

%p     end if;

%p end proc:

%p A059110 := proc(n,k)

%p     add(Lprime(n,i)*binomial(i,k),i=0..n) ;

%p end proc: # _R. J. Mathar_, Mar 15 2013

%t (* First program *)

%t 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 *)

%t (* Second program *)

%t A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];

%t Table[A059110[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 23 2021 *)

%o (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

%o (Sage)

%o def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)

%o flatten([[A059110(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 23 2021

%o (Magma)

%o A059110:= func< n,k | n eq 0 select 1 else Factorial(n-1)*Binomial(n,k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;

%o [A059110(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 23 2021

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

%K easy,nonn,tabl

%O 0,4

%A _Vladeta Jovovic_, Jan 04 2001