A119951 -id:A119951 - cp's OEIS frontend

A160466 Row sums of the Eta triangle A160464.

-1, -9, -87, -2925, -75870, -2811375, -141027075, -18407924325, -1516052821500, -153801543183750, -18845978136851250, -2744283682352086875, -468435979952504313750, -92643070481933918821875

Offset: 2

Johannes W. Meijer, May 24 2009

It is conjectured that the row sums of the Eta triangle depend on five different sequences.

Two Maple algorithms are given. The first one gives the row sums according to the Eta triangle A160464 and the second one gives the row sums according to our conjecture.

A160464 is the Eta triangle.

Row sum factors A119951, A000466, A043529, A045896 and A160467.

nmax:=15; c(2) := -1/3: for n from 3 to nmax do c(n):=(2*n-2)*c(n-1)/(2*n-1)-1/ ((n-1)*(2*n-1)) end do: for n from 2 to nmax do GCS(n-1) := ln(1/(2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2); p(n):=2^(-GCS(n-1))*(2*n-1)!; ETA(n, 1) := p(n)*c(n) end do: mmax:=nmax: for m from 2 to mmax do ETA(2, m) := 0 end do: for n from 3 to nmax do for m from 2 to mmax do q(n) := (1+(-1)^(n-3)*(floor(ln(n-1)/ln(2)) - floor(ln(n-2)/ln(2)))): ETA(n, m) := q(n)*(-ETA(n-1, m-1)+(n-1)^2*ETA(n-1, m)) end do end do: for n from 2 to nmax do s1(n):=0: for m from 1 to n-1 do s1(n) := s1(n) + ETA(n, m) end do end do: seq(s1(n), n=2..nmax);
# End first program.
nmax:=nmax; A160467 := proc(n): denom(4*(4^n-1)*bernoulli(2*n)/n) end: A043529 := proc(n): ceil(frac(log[2](n+1))+1) end proc: A000466 := proc(n): 4*n^2-1 end proc: A045896 := proc(n): denom((n)/((n+1)*(n+2))) end proc: A119951 := proc(n) : numer(sum(((2*k1)!/(k1!*(k1+1)!))/2^(2*(k1-1)), k1=1..n)) end proc: for n from 1 to nmax do SF(2*n+1):= A000466(n)/A043529(n-1); SF(2*n+2) := A045896(n-1)/A160467(n+1) end do: FF(2):=1: for n from 3 to nmax do FF(n) := SF(n) * FF(n-1) end do: for n from 2 to nmax do s2(n):= (-1)*A119951(n-1)*FF(n) end do: seq(s2(n), n=2..nmax);
# End second program.

Rowsums(n) = (-1) * A119951(n-1) * FF(n) for n >= 2.
FF(n) = SF(n) * FF(n-1) for n >= 3 with FF(2) =1.
SF(2*n) = A045896(n-2) / A160467(n) for n >= 2.
SF(2*n+1) = A000466(n) / A043529(n-1) for n >= 1.

A120069 Denominators of partial sums of a convergent series involving scaled Catalan numbers A000108.

Original entry on oeis.org

1, 2, 16, 32, 128, 256, 4096, 8192, 32768, 65536, 524288, 1048576, 4194304, 8388608, 268435456, 536870912, 2147483648, 4294967296, 34359738368, 68719476736, 274877906944, 549755813888, 8796093022208

Offset: 1

Wolfdieter Lang, Jul 20 2006

For the corresponding numerator sequence see A119951.
The series s:=Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers) converges by Raabe's test. The value for s is 4 (see A119951).
Asymptotically, C(n)/2^(2*(k-1)) ~ 4/(sqrt(Pi)*k^(3/2)) (see Mathworld). The sum of the asymptotic values from k = 1 to infinity is (4/sqrt(Pi))*Zeta(3/2) = 5.895499840 (Maple10, 10 digits).
The partial sums r(n):=Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
For the rationals r(n) see the W. Lang link under A119951.
a(n) appears to be the denominator of Catalan(n)/4^(n-1) but I have no proof of this. - Groux Roland, Dec 11 2010

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000108, A119951 (numerators).

Denominator[Table[Sum[CatalanNumber[k]/2^(2*(k - 1)), {k, 1, n}], {n, 1, 50}]] (* G. C. Greubel, Feb 08 2017 *)

for(n=1,50, print1(denominator(sum(k=1,n, binomial(2*k,k)/((k+1)*2^(2*k-2)))), ", ")) \\ G. C. Greubel, Feb 08 2017

a(n) = denominator(r(n)) with the rationals r(n) defined above.

First comment corrected by Harvey P. Dale, Oct 09 2017

A120088 Numerators of partial sums of a series for sqrt(2).

Original entry on oeis.org

3, 11, 23, 179, 365, 1439, 2911, 46147, 93009, 369605, 743409, 5917879, 11887761, 47365319, 95064943, 3032383331, 6082445497, 24264959593, 48649328861, 388310999293, 778263028691, 3106935548009, 6225306416473, 99433372856743, 199189221750317, 795541400400905

Offset: 0

Wolfdieter Lang, Jul 20 2006

Involving alternating sums over scaled Catalan numbers, A000108(k)/4^k.
From the expansion of sqrt(1+x) = 1 + x*(Sum_{k>=0} C(k)*(-x/4)^k)/2, valid for |x|<=1, one finds for x=+1: sqrt(2) = 1 + (Sum_{k>=0} (-1)^k*C(k)/4^k)/2.
The denominators are given by 2*A120777(n).
The rationals r(n):=1 + (Sum_{k=0..n} (-1)^k*C(k)/4^k)/2, with the Catalan numbers C(n)=A000108(n), are A120088(n)/(2*A120777(n)), n>=0.

			Rationals r(n): [3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, 2911/2048, 46147/32768,...]

G. C. Greubel, Table of n, a(n) for n = 0..1000
W. Lang, Rationals r(n).

For similar partial sums with positive terms (not alternating) see rationals A119951/A120069.

For the partial sums (Sum_{k=0..n} (-1)^k*C(k)/4^k) see A120788(n)/A120777(n).

[Numerator(1 + (&+[(-1/4)^k*Binomial(2*k,k)/(2*(k+1)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, Mar 27 2018

r[n_]:= 1+Sum[(-1/4)^k*CatalanNumber[k]/2, {k, 0, n}]; Numerator[Table[ r[n], {n, 0, 50}]] (* G. C. Greubel, Mar 27 2018 *)

{r(n) = 1 + sum(k=0,n, (-1/4)^k*binomial(2*k,k)/(2*(k+1)))};
for(n=0,30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 27 2018

a(n) = numerator(r(n)), with the rationals defined above.

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A160466 Row sums of the Eta triangle A160464.

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A120069 Denominators of partial sums of a convergent series involving scaled Catalan numbers A000108.

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A120088 Numerators of partial sums of a series for sqrt(2).

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