A114700 -id:A114700 - cp's OEIS frontend

A116466 Unsigned row sums of triangle A114700.

1, 2, 2, 4, 2, 4, 4, 8, 10, 20, 32, 64, 112, 224, 408, 816, 1514, 3028, 5680, 11360, 21472, 42944, 81644, 163288, 311896, 623792, 1196132, 2392264, 4602236, 9204472, 17757184, 35514368, 68680170, 137360340, 266200112, 532400224, 1033703056

Offset: 0

Paul D. Hanna, Feb 19 2006

nonn

Both triangles A112555 and A114700 have the property that the m-th matrix power of the triangles satisfy T^m = I + m*(T - I). So it is curious that the row squared sums of A112555 is a bisection of the unsigned row sums of A114700.

Cf. A112556, A112555, A114700.

CoefficientList[Series[(1 + 2*x)*(2*(1 + x^2)/(1 - x^2) + x^2/(1 - 4*x^2)^(1/2))/(2 + x^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Feb 21 2017 *)

a(n)=local(x=X+X*O(X^n)); polcoeff((1+2*x)*(2*(1+x^2)/(1-x^2)+x^2/(1-4*x^2)^(1/2))/(2+x^2),n,X)

/* a(n) as the unsigned row sums of A114700 */ a(n)=sum(k=0,n,abs(polcoeff(polcoeff(1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y+x*O(x^n)+y*O(y^k))/(1-x*y),n,x),k,y)))

G.f.: (1+2*x)*( 2*(1+x^2)/(1-x^2) + x^2/(1-4*x^2)^(1/2) )/(2+x^2). Also, a(2*n+1) = 2*a(2*n), a(2*n) = A112556(n), where A112556 equals the row squared sums of triangle A112555.

A116467 Row squared sums of triangle A114700: a(n) = Sum_{k=0..n} A114700(n,k)^2.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 8, 18, 52, 164, 544, 1852, 6416, 22520, 79888, 285938, 1031316, 3744628, 13676384, 50209860, 185190400, 685888736, 2549873880, 9511787452, 35592025904, 133559834064, 502494233128, 1895089009088, 7162963968712

Offset: 0

Paul D. Hanna, Feb 19 2006

nonn

Cf. A114700 (triangle), A116466 (unsigned row sums).

{a(n)=sum(k=0,n,polcoeff(polcoeff(1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y+x*O(x^n)+y*O(y^k))/(1-x*y),n,x),k,y)^2)}

G.f.: A(x) = 2*(1-6*x+2*x^2+6*x^3+3*x^4)/(2-10*x-x^2)/(1-x)^2 + x*(2+x)*(1-4*x)/(2-10*x-x^2)/2*deriv(A(x)).

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A116466 Unsigned row sums of triangle A114700.

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A116467 Row squared sums of triangle A114700: a(n) = Sum_{k=0..n} A114700(n,k)^2.

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