A125185 Triangle read by rows: T(n,k) is the coefficient of t^k in the polynomial S(n,t)=[(1+t)(2+t)^n+(1-t)t^n]/2 (0<=k<=n).

1, 1, 2, 2, 4, 3, 4, 10, 9, 4, 8, 24, 28, 16, 5, 16, 56, 80, 60, 25, 6, 32, 128, 216, 200, 110, 36, 7, 64, 288, 560, 616, 420, 182, 49, 8, 128, 640, 1408, 1792, 1456, 784, 280, 64, 9, 256, 1408, 3456, 4992, 4704, 3024, 1344, 408, 81, 10, 512, 3072, 8320, 13440, 14400

Offset

0,3

Comments

Sum of terms in row n = 3^n. The polynomials S(n,t) can be defined recursively by S(0,t)=1, S(n,t)=t^n - t^(n-1) + (2+t)S(n-1,t) for n>=1. S(n,t)=Sum(B(j,t), j=2^n .. 2^(n+1)-1), where B(n,t) are the Stern polynomials, defined by B(0,t)=0, B(1,t)=1, B(2n,t)=tB(n,t), B(2n+1,t)=B(n+1,t)+B(n,t) for n>=1 (see S. Klavzar et al. and A125184). For example, S(2,t)=B(4,t)+B(5,t)+B(6,t)+B(7,t).

Subtriangle of (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

Links

Table of n, a(n) for n=0..59.

S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007), 86-95.

Formula

T(n,k)=2^(n-k-1)*(n+k+1)binomial(n,k)/(n-k+1) if k<n; T(n,n)=n+1. Rec. rel.: T(n,k)=2T(n-1,k)+T(n-1,k-1) if k<=n-2.

G.f.: (1-x)/((1-y*x)(1-(y+2)*x). - Philippe Deléham, Feb 26 2012

Recurrence : T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 26 2012

Sum _ {k, 0<=k<=n} T(n,k)*x^k = A033999(n), A011782(n), A000244(n), A010036(n), A081625(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 26 2012

Example

Triangle starts:
1;
1,2;
2,4,3;
4,10,9,4;
8,24,28,16,5;
16,56,80,60,25,6;
Triangle (0,1,1,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,...) begins :
1
0, 1
0, 1, 2
0, 2, 4, 3
0, 4, 10, 9, 4
0, 8, 24, 28, 16, 5
0, 16, 56, 80, 60, 25, 6

Maple

T:=proc(n, k) if k<n then 2^(n-k-1)*binomial(n, k)*(n+k+1)/(n-k+1) elif k=n then n+1 else 0 fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Crossrefs

Cf. A125184.

Sequence in context: A131816 A223541 A128181 * A274895 A173387 A133438

Adjacent sequences: A125182 A125183 A125184 * A125186 A125187 A125188

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 04 2006

Status

approved