A156225 Triangle read by rows:e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=(e[n + 1, m]*PartitionsQ[n] + e[n + 1, n - m]*(PartitionsQ[ n - m] + PartitionsQ[m])) - 2.

1, 1, 1, 1, 10, 1, 3, 42, 42, 3, 3, 128, 262, 128, 3, 5, 340, 1810, 1810, 340, 5, 7, 958, 8335, 19326, 8335, 958, 7, 9, 2468, 38635, 140569, 140569, 38635, 2468, 9, 11, 6022, 160686, 970572, 1561898, 970572, 160686, 6022, 11, 15, 15193, 669758, 6372686

Offset

0,5

Comments

Row sums are:

{1, 2, 12, 90, 524, 4310, 37926, 363362, 3836480, 48184504, 643393254,...}.

Links

Table of n, a(n) for n=0..48.

Formula

e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];

t(n,m)=(e[n + 1, m]*PartitionsQ[n] + e[n + 1, n - m]*(PartitionsQ[ n - m] + PartitionsQ[m])) - 2.

Example

{1},
{1, 1},
{1, 10, 1},
{3, 42, 42, 3},
{3, 128, 262, 128, 3},
{5, 340, 1810, 1810, 340, 5},
{7, 958, 8335, 19326, 8335, 958, 7},
{9, 2468, 38635, 140569, 140569, 38635, 2468, 9},
{11, 6022, 160686, 970572, 1561898, 970572, 160686, 6022, 11},
{15, 15193, 669758, 6372686, 17034600, 17034600, 6372686, 669758, 15193, 15},
{19, 38682, 2594827, 37459294, 155809822, 251587966, 155809822, 37459294, 2594827, 38682, 19}

Mathematica

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

Crossrefs

Sequence in context: A111525 A138261 A068126 * A284327 A322219 A010182

Adjacent sequences: A156222 A156223 A156224 * A156226 A156227 A156228

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 06 2009

Status

approved