This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A253286 #32 Nov 12 2024 21:08:59
%S A253286 1,0,1,0,1,1,0,3,2,1,0,13,8,3,1,0,73,44,15,4,1,0,501,304,99,24,5,1,0,
%T A253286 4051,2512,801,184,35,6,1,0,37633,24064,7623,1696,305,48,7,1,0,394353,
%U A253286 261536,83079,18144,3145,468,63,8,1
%N A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!*C(n,n-j)* C(n-1,n-j)*k^j, for n>=0 and k>=0.
%H A253286 Seiichi Manyama, <a href="/A253286/b253286.txt">Antidiagonals n = 0..139, flattened</a>
%H A253286 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A253286 A(n,k) = k*n!*hypergeom([1-n],[2],-k) for n>=1 and 1 for n=0.
%F A253286 Row sums of triangle, Sum_{k=0..n} A(n-k, k) = 1 + A256325(n).
%F A253286 From _Seiichi Manyama_, Feb 03 2021: (Start)
%F A253286 E.g.f. of column k: exp(k*x/(1-x)).
%F A253286 T(n,k) = (2*n+k-2) * T(n-1,k) - (n-1) * (n-2) * T(n-2, k) for n > 1. (End)
%F A253286 From _G. C. Greubel_, Feb 23 2021: (Start)
%F A253286 A(n, k) = k*(n-1)!*LaguerreL(n-1, 1, -k) with A(0, k) = 1.
%F A253286 T(n, k) = k*(n-k-1)!*LaguerreL(n-k-1, 1, -k) with T(n, n) = 1.
%F A253286 T(n, 2) = A052897(n) = A086915(n)/2.
%F A253286 Sum_{k=0..n} T(n, k) = 1 + Sum_{k=0..n-1} (n-k-1)*k!*LaguerreL(k, 1, k-n+1). (End)
%e A253286 Square array starts, A(n,k):
%e A253286 1, 1, 1, 1, 1, 1, 1, ... A000012
%e A253286 0, 1, 2, 3, 4, 5, 6, ... A001477
%e A253286 0, 3, 8, 15, 24, 35, 48, ... A005563
%e A253286 0, 13, 44, 99, 184, 305, 468, ... A226514
%e A253286 0, 73, 304, 801, 1696, 3145, 5328, ...
%e A253286 0, 501, 2512, 7623, 18144, 37225, 68976, ...
%e A253286 0, 4051, 24064, 83079, 220096, 495475, 997056, ...
%e A253286 A000007, A000262, A052897, A255806, ...
%e A253286 Triangle starts, T(n, k) = A(n-k, k):
%e A253286 1;
%e A253286 0, 1;
%e A253286 0, 1, 1;
%e A253286 0, 3, 2, 1;
%e A253286 0, 13, 8, 3, 1;
%e A253286 0, 73, 44, 15, 4, 1;
%e A253286 0, 501, 304, 99, 24, 5, 1;
%p A253286 L := (n, k) -> (n-k)!*binomial(n,n-k)*binomial(n-1,n-k):
%p A253286 A := (n, k) -> add(L(n,j)*k^j, j=0..n):
%p A253286 # Alternatively:
%p A253286 # A := (n, k) -> `if`(n=0,1, simplify(k*n!*hypergeom([1-n],[2],-k))):
%p A253286 for n from 0 to 6 do lprint(seq(A(n,k), k=0..6)) od;
%t A253286 A253286[n_, k_]:= If[k==n, 1, k*(n-k-1)!*LaguerreL[n-k-1, 1, -k]];
%t A253286 Table[A253286[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 23 2021 *)
%o A253286 (PARI) {T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^j*binomial(n-1, j-1)/j!))} \\ _Seiichi Manyama_, Feb 03 2021
%o A253286 (PARI) {T(n, k) = if(n<2, (k-1)*n+1, (2*n+k-2)*T(n-1, k)-(n-1)*(n-2)*T(n-2, k))} \\ _Seiichi Manyama_, Feb 03 2021
%o A253286 (Sage) flatten([[1 if k==n else k*factorial(n-k-1)*gen_laguerre(n-k-1, 1, -k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 23 2021
%o A253286 (Magma) [k eq n select 1 else k*Factorial(n-k-1)*Evaluate(LaguerrePolynomial(n-k-1, 1), -k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 23 2021
%Y A253286 Main diagonal gives A293145.
%Y A253286 Cf. A000262, A001477, A005563, A052897, A226514, A255806, A256325.
%K A253286 tabl,easy,nonn
%O A253286 0,8
%A A253286 _Peter Luschny_, Mar 24 2015