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

0,4

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.

Links

Table of n, a(n) for n=0..54.

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).

Example

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.

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 *)

Prog

(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)))}

Crossrefs

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

Sequence in context: A292386 A174287 A186826 * A050609 A120870 A010029

Adjacent sequences: A185415 A185416 A185417 * A185419 A185420 A185421

Keyword

nonn,easy,tabl

Author

Peter Bala, Jan 30 2011

Status

approved