1,5

Row sums are:

{1, 2, 2, -3, -17, -24, 52, 287, 197, -2202, -6674};

The idea here is that the Pascal triangle Binomial heights

in the limit give a very normal/ Gaussian-like curve,

so that these sums would,in the limit of large n

as this linear sum, be more Hermite than other linear sums.

The x^0 constants are, first column:

{1, 1, -1, -5, -3, 21, 43, -97, -455, 361, 4951}

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43.

P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].

{1},

{1, 1},

{-1, 2, 1},

{-5, -2, 3, 1},

{-3, -16, -3, 4, 1},

{21, -12, -35, -4, 5, 1},

{43, 104, -33, -64, -5, 6, 1},

{-97, 246, 315, -74, -105, -6, 7,1},

{-455, -656, 859, 752, -145, -160, -7, 8, 1},

{361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1},

{4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1}

P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = Sum[Binomial[n, m]*P[x, m], {m, 0, n}]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P2[x, n], x], {n, 0, 10}]; Flatten[a]

Cf. A137286.

Adjacent sequences: A136642 A136643 A136644 this_sequence A136646 A136647 A136648

Sequence in context: A065274 A136262 A090003 this_sequence A091381 A127156 A144019

uned,tabl,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 01 2008