A293985 - OEIS

%I #37 Jan 27 2025 11:39:10

%S 1,1,1,1,2,3,1,3,7,13,1,4,13,34,73,1,5,21,73,209,501,1,6,31,136,501,

%T 1546,4051,1,7,43,229,1045,4051,13327,37633,1,8,57,358,1961,9276,

%U 37633,130922,394353,1,9,73,529,3393,19081,93289,394353,1441729,4596553

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

%H Seiichi Manyama, <a href="/A293985/b293985.txt">Antidiagonals n = 0..139, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

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

%F A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.

%F A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.

%F From _Seiichi Manyama_, Jan 25 2025: (Start)

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

%F A(n,k) = n! * LaguerreL(n, k-1, -1). (End)

%e Square array begins:

%e     1,    1,    1,    1,     1, ... A000012;

%e     1,    2,    3,    4,     5, ... A000027;

%e     3,    7,   13,   21,    31, ... A002061;

%e    13,   34,   73,  136,   229, ... A135859;

%e    73,  209,  501, 1045,  1961, ...

%e   501, 1546, 4051, 9276, 19081, ...

%e Antidiagonal rows begin as:

%e   1;

%e   1, 1;

%e   1, 2,  3;

%e   1, 3,  7, 13;

%e   1, 4, 13, 34,  73;

%e   1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021

%t t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];

%t T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 09 2021 *)

%o (Sage)

%o @CachedFunction

%o def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )

%o def T(n,k): return t(k,n-k)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021

%o (Magma)

%o function t(n,k)

%o   if n eq 0 then return 1;

%o   else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);

%o   end if; return t;

%o end function;

%o [t(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021

%Y Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.

%Y Main diagonal gives A152059.

%Y Similar table: A086885, A088699, A176120.

%K nonn,tabl,changed

%O 0,5

%A _Seiichi Manyama_, Oct 21 2017