A156130 A triangle sequence from a sum: t0(n,m)=(2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m]); t1(n,k)=Sum[(-1)^j *t0[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=If[n == 0, 1, t1(n, k) + t1(n, n - k)]

1, 2, 2, 4, 4, 4, 53, 28, 28, 53, 246, 138, 102, 138, 246, 4207, 1775, 814, 814, 1775, 4207, 60922, 25758, 10035, 5654, 10035, 25758, 60922, 1114857, 457387, 168720, 63929, 63929, 168720, 457387, 1114857, 10706482, 4934925, 2063635, 791789

Offset

0,2

Comments

Row sums are:

{1, 4, 12, 162, 870, 13592, 199084, 3609786, 37432146, 1169080936,

28322293344,...}.

This sequence is equivalent to the Eulerian numbers sum for the symmetrical q partitions.

Links

Table of n, a(n) for n=0..39.

Formula

t0(n,m)=(2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m]); t1(n,k)=Sum[(-1)^j *t0[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=If[n == 0, 1, t1(n, k) + t1(n, n - k)]

Example

{1},
{2, 2},
{4, 4, 4},
{53, 28, 28, 53},
{246, 138, 102, 138, 246},
{4207, 1775, 814, 814, 1775, 4207},
{60922, 25758, 10035, 5654, 10035, 25758, 60922},
{1114857, 457387, 168720, 63929, 63929, 168720, 457387, 1114857},
{10706482, 4934925, 2063635, 791789, 438484, 791789, 2063635, 4934925, 10706482},
{357283571, 146408798, 55255747, 18896106, 6696246, 6696246, 18896106, 55255747, 146408798, 357283571},
{8544327694, 3575342129, 1376619624, 477863600, 150928577, 72130096, 150928577, 477863600, 1376619624, 3575342129, 8544327694}

Mathematica

Clear[t, t0, n, k];
t0[n_, m_] = (2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m]);
t[n_, k_] = Sum[(-1)^j *t0[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
Table[Table[If[n == 0, 1, t[n, k] + t[n, n - k]], {k, 0, n}], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A211882 A284432 A284435 * A123756 A357256 A300755

Adjacent sequences: A156127 A156128 A156129 * A156131 A156132 A156133

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 04 2009

Status

approved