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

1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 57, 57, 27, 7, 1, 361, 361, 175, 51, 9, 1, 2763, 2763, 1353, 413, 83, 11, 1, 24611, 24611, 12125, 3801, 819, 123, 13, 1, 250737, 250737, 123987, 39487, 8857, 1441, 171, 15, 1, 2873041, 2873041, 1424215, 458331, 105489, 18057, 2327, 227, 17, 1

Offset: 0

Author

Peter Bala, Jan 30 2011

Keywords

• nonn
• easy
• tabl

Comments

The table entries T(n,k), n,k>=0, are defined by the recurrence relation:

1)... T(n+1,k) = k*T(n,k-1)+(k+1)*T(n,k+1) with boundary condition T(0,k) = 1.

The first column of the table produces the sequence of Springer numbers A001586.

For similarly defined tables see A185414, A185416 and A185420.

Examples

			Square array begins
n\k|.....0......1.......2.......3........4........5........6
============================================================
..0|.....1......1.......1.......1........1........1........1
..1|.....1......3.......5.......7........9.......11.......13
..2|.....3.....11......27......51.......83......123......171
..3|....11.....57.....175.....413......819.....1441.....2327
..4|....57....361....1353....3801.....8857....18057....33321
..5|...361...2763...12125...39487...105489...244211...507013
..6|..2763..24611..123987..458331..1379003..3569523..8229891
..
Examples of recurrence relation:
T(4,3) = 3801 = 3*T(3,2) + 4*T(3,4) = 3*175 + 4*819;
T(5,1) = 2763 = 1*T(4,0)+ 2*T(4,2) = 1*57 + 2*1353.
		

Crossrefs

Cf. A001586, A185417, A185414, A185416, A185420.

Programs

Maple

# A185418
S := proc(n, x) option remember; description `polynomials S(n, x)`;
if n = 0 then 1 else x*S(n-1,x-1)+(x+1)*S(n-1,x+1) end if end proc:
for n from 0 to 10 do seq(S(n, k), k = 0..10) end do;

Mathematica

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]]];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 22 2021 *)

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)))}

Formula

(1)... T(n,k) = S(n,k) with S(n,x) the polynomials described in A185417.

(2)... First column: T(n,0) = A001586(n).

(3)... Second column: T(n,1) = A001586(n+1).

(4)... Second row: T(1,k) = A005408(k).

(5)... Third row: T(2,k) = A164897(k).