A123242 Irregular triangle: p(k, x) = 2*x*p(k-1, x) + (1 - x^2)*p(k-2, x) for even k, p(k, x) = 2*(k-1)*p(k-1, x) - x*p(k-2, x) for odd k.

1, 1, 1, 1, 2, 1, 4, 7, 3, 1, 10, 14, 4, -1, 8, 76, 105, 29, -8, 1, 26, 165, 204, 43, -20, 1, 12, 304, 1904, 2343, 487, -232, 12, 1, 50, 772, 3986, 4564, 750, -506, 44, -1, 16, 788, 12048, 61872, 70681, 11513, -7864, 692, -16, 1, 82, 2347, 28032, 127536, 138126, 17956, -16434, 1889, -76, 1

Offset

1,5

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..62.

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

Formula

p(k, x) = If[Mod[k, 2] == 1, 2*(k - 1)*p(k - 1, x) - x*p(k - 2, x), 2*x*p(k - 1, x) + (1 - x^2)*p(k - 2, x)]

Example

Triangle Starts:
{1},
{1, 1},
{1, 2, 1},
{4, 7, 3},
{1, 10, 14, 4, -1},
{8, 76, 105, 29, -8},
{1, 26, 165, 204, 43, -20, 1}

Mathematica

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

Crossrefs

Sequence in context: A328034 A123360 A072015 * A322941 A217205 A358566

Adjacent sequences: A123239 A123240 A123241 * A123243 A123244 A123245

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Oct 07 2006

Status

approved