A146967 Triangle read by rows: row n is the expansion of p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(2^(m - 1) +n*m - n + 1)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}].

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 34, 34, 34, 1, 1, 109, 102, 102, 109, 1, 1, 350, 303, 292, 303, 350, 1, 1, 1127, 901, 819, 819, 901, 1127, 1, 1, 3688, 2716, 2296, 2182, 2296, 2716, 3688, 1, 1, 12425, 8420, 6548, 5822, 5822, 6548, 8420, 12425, 1, 1, 43402

Offset

0,5

Links

Table of n, a(n) for n=0..56.

Formula

T(n,0) = T(n,n) = 1. For 0 < m < n, T(n,m) = binomial(n,m) + 2^(n-3)*(2^(m-1)+2^(n-m-1)+n^2-2*n+2). - Jason Yuen, Oct 03 2024

Example

Triangle begins:
  {1}
  {1, 1}
  {1, 4, 1}
  {1, 11, 11, 1}
  {1, 34, 34, 34, 1}
  {1, 109, 102, 102, 109, 1}
  {1, 350, 303, 292, 303, 350, 1}
  {1, 1127, 901, 819, 819, 901, 1127, 1}
  {1, 3688, 2716, 2296, 2182, 2296, 2716, 3688, 1}
  {1, 12425, 8420, 6548, 5822, 5822, 6548, 8420, 12425, 1}
  {1, 43402, 27181, 19320, 15826, 14844, 15826, 19320, 27181, 43402, 1}

Mathematica

p[x_, n_] = If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(2^(m - 1) + n*m - n + 1)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Prog

(PARI) T(n, m) = if(m==0 || m==n, 1, binomial(n, m) + 2^(n-3)*(2^(m-1)+2^(n-m-1)+n^2-2*n+2)) \\ Jason Yuen, Oct 03 2024

Crossrefs

Sequence in context: A008292 A174036 A157221 * A173152 A156049 A192015

Adjacent sequences: A146964 A146965 A146966 * A146968 A146969 A146970

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Nov 03 2008

Status

approved