A123243 Triangular array: odd: p(k, x) = 2*x*p(k-1, x) + (1-x2)*p(k-2, x), even: p(k, x) = (Sum_{m=0..k} x^m)*p(k-1, x).

1, 1, 1, 2, 1, 2, 3, 3, 3, 1, 12, 16, 17, 18, 6, 12, 28, 45, 63, 69, 69, 57, 41, 24, 6, 120, 268, 434, 613, 672, 684, 570, 410, 240, 60, 120, 388, 822, 1435, 2107, 2791, 3361, 3771, 3891, 3683, 3249, 2636, 1964, 1280, 710, 300, 60, 1680, 5312, 11240, 19656, 28885

Offset

1,4

References

E. S. R. Gopal, Specific Heats at Low Temperatures, Plenum Press, New York, 1966, pages 36-40

S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 110

Links

Table of n, a(n) for n=1..57.

B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.

Formula

odd: p(k, x) = 2*x*p(k-1, x) + (1-x2)*p(k-2, x);

even: p(k, x) = (Sum_{m=0..k} x^m)*p(k-1, x)

Example

{1},
{1, 1},
{2, 1},
{2, 3, 3, 3, 1},
{12, 16, 17, 18, 6},
{12, 28, 45, 63, 69, 69, 57, 41, 24, 6},
{120, 268, 434, 613, 672, 684, 570, 410, 240, 60},
{120, 388, 822, 1435, 2107, 2791, 3361, 3771, 3891, 3683, 3249, 2636, 1964, 1280, 710, 300, 60}

Mathematica

p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = If[Mod[k, 2] == 0, 2*(k - 1)*p[k - 1, x] - x*p[k - 2, x], Sum[x^m, {m, 0, k}]*p[k - 1, x]];
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

Crossrefs

Cf. A008302.

Sequence in context: A244516 A363264 A002339 * A037193 A291615 A361789

Adjacent sequences: A123240 A123241 A123242 * A123244 A123245 A123246

Keyword

nonn,uned,tabf

Author

Roger L. Bagula, Oct 07 2006

Status

approved