A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)*P[n-1]+x*P[n-2].

0, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 20, 20, 8, 1, 1, 10, 34, 50, 34, 10, 1, 1, 12, 52, 104, 104, 52, 12, 1, 1, 14, 74, 190, 258, 190, 74, 14, 1, 1, 16, 100, 316, 552, 552, 316, 100, 16, 1, 1, 18, 130, 490, 1058, 1362, 1058, 490, 130, 18, 1, 1, 20, 164

Offset

0,5

Comments

A variant of A008288 (they satisfy the same recurrence).

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Formula

P[n](x) = (x+1) * ( ((x+1+sqrt(x^2+6x+1))/2)^n - ((x+1-sqrt(x^2+6x+1))/2)^n ) / sqrt(x^2+6x+1) - Max Alekseyev, Mar 10 2008

P[n](x) = (x+1) * (sqrt(x)*I)^(n-1) * U[n-1](-I*(x+1)/sqrt(x)/2), where U[n](t) is Chebyshev polynomial of the 2nd kind. - Max Alekseyev, Mar 10 2008

Example

Triangle begins:
0
1, 1
1, 2, 1
1, 4, 4, 1
1, 6, 10, 6, 1
1, 8, 20, 20, 8, 1
1, 10, 34, 50, 34, 10, 1
1, 12, 52, 104, 104, 52, 12, 1
1, 14, 74, 190, 258, 190, 74, 14, 1
1, 16, 100, 316, 552, 552, 316, 100, 16, 1

Maple

P[0]:=0;
P[1]:=x+1;
for n from 2 to 14 do
P[n]:=expand((x+1)*P[n-1]+x*P[n-2]);
lprint(P[n]);
lprint(seriestolist(series(P[n], x, 200)));
od:

Mathematica

t[n_, k_] := 2^(1-n)*Binomial[n, k]*Sum[Binomial[n, 2*m+1]*HypergeometricPFQ[{-k, -m, k-n}, {1/2-n/2, -n/2}, -1], {m, 0, (n-1)/2}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, after Max Alekseyev *)

Prog

(PARI) { T(n, k) = sum(m=0, (n-1)\2, binomial(n, 2*m+1) * sum(j=0, m, binomial(m, j) * binomial(n-2*j, k-j) * 2^(2*j+1-n) ) ) } - Max Alekseyev, Mar 10 2008
(Haskell)
a128966 n k = a128966_tabl !! n !! k
a128966_row n = a128966_tabl !! n
a128966_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([0], [1, 1])
-- Reinhard Zumkeller, Jul 20 2013

Crossrefs

Cf. A163271 (row sums), A110170 (central terms).

Cf. A102413.

Sequence in context: A156580 A157528 A132731 * A055907 A259698 A274643

Adjacent sequences: A128963 A128964 A128965 * A128967 A128968 A128969

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 10 2007

Status

approved