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A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

%I A059114 #12 Feb 20 2022 12:48:24
%S A059114 1,1,1,3,4,2,13,21,18,6,73,136,156,96,24,501,1045,1460,1260,600,120,
%T A059114 4051,9276,15030,16320,11160,4320,720,37633,93289,170142,219450,
%U A059114 192360,108360,35280,5040,394353,1047376,2107448,3116736,3294480,2405760,1149120,322560,40320
%N A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.
%C A059114 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 A059114 G. C. Greubel, <a href="/A059114/b059114.txt">Rows n = 0..100 of the triangle, flattened</a>
%H A059114 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A059114 E.g.f. for T(n, k) = (x/(1-x))^k * exp(x/(x-1)).
%F A059114 T(n, k)= Sum_{i=0..n} L'(n,i) * ( Product_{j=1..k} (i-j+1) ).
%F A059114 T(n, 0) = A000262(n).
%F A059114 T(n, 1) = A052852(n).
%F A059114 From _G. C. Greubel_, Feb 23 2021: (Start)
%F A059114 T(n, k) = n! * k! * Sum_{j=0..n} binomial(j, k)*binomial(n-1, j-1)/j!.
%F A059114 T(n, k) = n! * Laguerre(n-k, k-1, -1).
%F A059114 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 A059114 Triangle begins as:
%e A059114     1;
%e A059114     1,    1;
%e A059114     3,    4,    2;
%e A059114    13,   21,   18,    6;
%e A059114    73,  136,  156,   96,  24;
%e A059114   501, 1045, 1460, 1260, 600, 120;
%e A059114   ...;
%e A059114 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 A059114 Table[n!*LaguerreL[n-k, k-1, -1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 23 2021 *)
%o A059114 (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 A059114 (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 A059114 (PARI) T(n, k) = n! * pollaguerre(n-k, k-1, -1); \\ _Michel Marcus_, Feb 23 2021
%Y A059114 Cf. A000262, A052852, A052897, A059110.
%Y A059114 Row sums give A059115. Alternating row sums give A288268.
%K A059114 easy,nonn,tabl
%O A059114 0,4
%A A059114 _Vladeta Jovovic_, Jan 04 2001