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 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). 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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 kn. - 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

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Last modified July 29 16:52 EDT 2021. Contains 346346 sequences. (Running on oeis4.)