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Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.
4

%I #12 Feb 20 2022 12:48:24

%S 1,1,1,3,4,2,13,21,18,6,73,136,156,96,24,501,1045,1460,1260,600,120,

%T 4051,9276,15030,16320,11160,4320,720,37633,93289,170142,219450,

%U 192360,108360,35280,5040,394353,1047376,2107448,3116736,3294480,2405760,1149120,322560,40320

%N Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

%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.

%H G. C. Greubel, <a href="/A059114/b059114.txt">Rows n = 0..100 of the triangle, flattened</a>

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

%F E.g.f. for T(n, k) = (x/(1-x))^k * exp(x/(x-1)).

%F T(n, k)= Sum_{i=0..n} L'(n,i) * ( Product_{j=1..k} (i-j+1) ).

%F T(n, 0) = A000262(n).

%F T(n, 1) = A052852(n).

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

%F T(n, k) = n! * k! * Sum_{j=0..n} binomial(j, k)*binomial(n-1, j-1)/j!.

%F T(n, k) = n! * Laguerre(n-k, k-1, -1).

%F T(n, k) = n!*binomial(n-1, k-1)*Hypergeometric1F1([k-n], [k], -1) with T(n, 0) = Hypergeometric2F0([1-n, -n], [], 1). (End)

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 3, 4, 2;

%e 13, 21, 18, 6;

%e 73, 136, 156, 96, 24;

%e 501, 1045, 1460, 1260, 600, 120;

%e ...;

%e E.g.f. for T(n, 2) = (x/(1-x))^2*e^(x/(x-1)) = x^2 + 3*x^3 + 13/2*x^4 + 73/6*x^5 + 167/8*x^6 + 4051/120*x^7 + ...

%t Table[n!*LaguerreL[n-k, k-1, -1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 23 2021 *)

%o (Sage) flatten([[factorial(n)*gen_laguerre(n-k, k-1, -1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 23 2021

%o (Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n-k, k-1), -1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 23 2021

%o (PARI) T(n, k) = n! * pollaguerre(n-k, k-1, -1); \\ _Michel Marcus_, Feb 23 2021

%Y Cf. A000262, A052852, A052897, A059110.

%Y Row sums give A059115. Alternating row sums give A288268.

%K easy,nonn,tabl

%O 0,4

%A _Vladeta Jovovic_, Jan 04 2001