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A185418 Square array, read by antidiagonals, used to recursively calculate the Springer numbers A001586.

%I A185418 #19 Jan 05 2025 23:40:14
%S A185418 1,1,1,3,3,1,11,11,5,1,57,57,27,7,1,361,361,175,51,9,1,2763,2763,1353,
%T A185418 413,83,11,1,24611,24611,12125,3801,819,123,13,1,250737,250737,123987,
%U A185418 39487,8857,1441,171,15,1,2873041,2873041,1424215,458331,105489,18057,2327,227,17,1
%N A185418 Square array, read by antidiagonals, used to recursively calculate the Springer numbers A001586.
%C A185418 The table entries T(n,k), n,k>=0, are defined by the recurrence relation:
%C A185418 1)... T(n+1,k) = k*T(n,k-1)+(k+1)*T(n,k+1) with boundary condition T(0,k) = 1.
%C A185418 The first column of the table produces the sequence of Springer numbers A001586.
%C A185418 For similarly defined tables see A185414, A185416 and A185420.
%F A185418 (1)... T(n,k) = S(n,k) with S(n,x) the polynomials described in A185417.
%F A185418 (2)... First column: T(n,0) = A001586(n).
%F A185418 (3)... Second column: T(n,1) = A001586(n+1).
%F A185418 (4)... Second row: T(1,k) = A005408(k).
%F A185418 (5)... Third row: T(2,k) = A164897(k).
%e A185418 Square array begins
%e A185418 n\k|.....0......1.......2.......3........4........5........6
%e A185418 ============================================================
%e A185418 ..0|.....1......1.......1.......1........1........1........1
%e A185418 ..1|.....1......3.......5.......7........9.......11.......13
%e A185418 ..2|.....3.....11......27......51.......83......123......171
%e A185418 ..3|....11.....57.....175.....413......819.....1441.....2327
%e A185418 ..4|....57....361....1353....3801.....8857....18057....33321
%e A185418 ..5|...361...2763...12125...39487...105489...244211...507013
%e A185418 ..6|..2763..24611..123987..458331..1379003..3569523..8229891
%e A185418 ..
%e A185418 Examples of recurrence relation:
%e A185418 T(4,3) = 3801 = 3*T(3,2) + 4*T(3,4) = 3*175 + 4*819;
%e A185418 T(5,1) = 2763 = 1*T(4,0)+ 2*T(4,2) = 1*57 + 2*1353.
%p A185418 # A185418
%p A185418 S := proc(n, x) option remember; description `polynomials S(n, x)`;
%p A185418 if n = 0 then 1 else x*S(n-1,x-1)+(x+1)*S(n-1,x+1) end if end proc:
%p A185418 for n from 0 to 10 do seq(S(n, k), k = 0..10) end do;
%t A185418 T[n_, k_] := T[n, k] = If[n<0 || k<0, 0, If[n == 0, 1, k T[n-1, k-1] + (k+1)*T[n-1, k+1]]];
%t A185418 Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 22 2021 *)
%o A185418 (PARI) {T(n,k)=if(n<0||k<0,0,if(n==0,1,k*T(n-1,k-1)+(k+1)*T(n-1,k+1)))}
%Y A185418 Cf. A001586, A185417, A185414, A185416, A185420.
%K A185418 nonn,easy,tabl
%O A185418 0,4
%A A185418 _Peter Bala_, Jan 30 2011