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A055376 CIK transform of Pascal's triangle A007318. 1
0, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 15, 24, 15, 5, 7, 31, 62, 62, 31, 7, 13, 63, 161, 212, 161, 63, 13, 19, 127, 381, 635, 635, 381, 127, 19, 35, 255, 900, 1785, 2244, 1785, 900, 255, 35, 59, 511, 2044, 4774, 7154, 7154, 4774, 2044, 511, 59, 107, 1023, 4619 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

From Petros Hadjicostas, Dec 06 2017: (Start)

Denote by b(n,k) the number of this sequence (double array) in row n and column k.

The CIK transform of double array (C(n,k): n>=1, k>=0) = (binomial(n,k): n>=1, k>=0), which has bivariate g.f. A(x,y) = Sum_{n>=1, k>=0} C(n,k)*x^n*y^k = x*(1+y)/(1-x*(1+y)), is given by CIK(A(x,y)) = -Sum_{s>=1} (phi(s)/s)*log(1-A(x^s,y^s)). (Unlike in sequence A055891, here, 1 is not added to the formula of the CIK transform. The addition of 1 seems to be arbitrary.)

To find the auxiliary double array (e(n,k): n>=1, k>=0) used in the formula b(n,k) = (1/n)*Sum_{d|gcd(n,k)} phi(d)*e(n/d, k/d), we use the formula E(x,y) = Sum_{n>=1, k>=0} e(n,k)*x^n*y^k = x*(dA(x,y)/dx)/(1-A(x,y)). We may find E(x,y) = x*(1+y)/((1-2x*(1+y))*(1-x*(1+y)), from which we can easily prove that e(n,k) = C(n,k)*(2^n-1).

Letting y=1 in the bivariate g.f. for b(n,k), we get the univariate g.f. for the row sums in A055891. (For this, ignore row 0 here and the 0th element in sequence A055891.)

(End)

LINKS

Table of n, a(n) for n=0..57.

C. G. Bower, Transforms (2)

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

From Petros Hadjicostas, Dec 06 2017: (Start)

Let b(n,k) be the number in row n and column k. We have b(0,0) = 0 and b(n,k) = (1/n)*Sum_{d|gcd(n,k)} phi(d)*C(n/d, k/d)*(2^{n/d}-1) for n>=1 and k>=0. Here, C(n,k) = binomial(n,k).

G.f. for b(n,k): Sum_{n>=1, k>=0} b(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log((1-2*x^s*(1+y^s))/(1-x^s*(1+y^s)).

G.f. for row n>=1: Sum_{k>=0} b(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*(2^{n/d}-1)*(1+y^d)^{n/d}.

G.f. for column k = 0: Sum_{n>=1} b(n,k=0)*x^n = Sum_{s>=1} (phi(s)/s)*log((1-x^s)/(1-2*x^s)) = -x/(1-x) - Sum_{s>=1} (phi(s)/s)*log(1-2*x^s).

G.f. for column k >= 1: Sum_{n>=1} b(n,k)*x^n = Sum_{d|k} (phi(d)/d)*(g_{k/d}(2*x^d) - g_{k/d}(x^d)), where g_k(x) = Sum_{s=0..k-1} C(k-1, s)*(-1)^s/((k-s)*(1-x)^{k-s}).

(End)

EXAMPLE

   0;

   1,   1;

   2,   3,   2;

   3,   7,   7,   3;

   5,  15,  24,  15,   5;

   7,  31,  62,  62,  31,   7;

  13,  63, 161, 212, 161,  63,  13;

  19, 127, 381, 635, 635, 381, 127,  19;

  ...

CROSSREFS

Row sums give A055891.

Sequence in context: A228527 A055375 A091533 * A085215 A076731 A085216

Adjacent sequences:  A055373 A055374 A055375 * A055377 A055378 A055379

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, May 16 2000

STATUS

approved

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Last modified February 22 01:04 EST 2019. Contains 320381 sequences. (Running on oeis4.)