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
Examples
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)
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
Programs
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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
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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
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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 *) -
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
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)
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