A120787 -id:A120787 - cp's OEIS frontend

A120777 a(n) = 2^(2*n - valuation(CatalanNumber(n), 2)).

1, 4, 8, 64, 128, 512, 1024, 16384, 32768, 131072, 262144, 2097152, 4194304, 16777216, 33554432, 1073741824, 2147483648, 8589934592, 17179869184, 137438953472, 274877906944, 1099511627776, 2199023255552, 35184372088832, 70368744177664, 281474976710656, 562949953421312

Offset: 0

Wolfdieter Lang, Jul 20 2006

Previous name: One half of denominators of partial sums of a series for sqrt(2).
Also denominators of partial sums Sum_{k=0..n} (C(k)/(-4)^k) = A120788(n)/A120777(n).
One half of denominators of partial sums which involve Catalan numbers A000108(k) divided by 4^k with alternating signs.
The listed numbers coincide with the denominators of sum(C(k)/4^k, k=0..n). See numerators A120778. In general these denominators may be different. See e.g. A120783 versus A120793 and A120787 versus A120796.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

Appears in A162446.

Cf. A048881, A120777, A120783, A120787, A120788, A120793, A120796, A283208.

a := n -> denom(binomial(2*n+2, n+1) / 2^(2*n+1)):
seq(a(n), n=0..22); # Johannes W. Meijer, Sep 23 2012
Conjecture: The following Maple program appears to generate this sequence! Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(denom(coeff(gser, z, n))/2, n=0..22); # Zerinvary Lajos, May 21 2008
a := proc(n) option remember: if n = 0 then b(0):=0 else b(n) := b(n-1) + A001511(n+1) fi: a(n) := 2^b(n) end proc: A001511 := proc(n) option remember: if n = 1 then 1 else procname(n-1) + (-1)^n * procname(floor(n/2)) fi: end proc:
seq(a(n), n=0..22); # Johannes W. Meijer, Jul 06 2009, revised Sep 23 2012

Table[Denominator[CatalanNumber[k]/(-4)^k], {k, 0, 22}] (* Jean-François Alcover, Jun 21 2013 *)
(* Alternative: *)
A120777[n_] := 2^(2*n - IntegerExponent[CatalanNumber[n], 2]);
Table[A120777[n], {n, 0, 26}]  (* Peter Luschny, Apr 16 2024 *)

a(n) = denominator(r(n)), with the rationals r(n) defined under A120088.
From Johannes W. Meijer, Jul 06 2009: (Start)
a(n) = denominator(C(2*n+2,n+1)/2^(2*n+1)).
If b(n) = log(a(n))/log(2) then c(n) = b(n+1)-b(n) = A001511(n+1) i.e. the ruler function. (End)
a(n) = 2^(2*n- A048881(n)) = 2^A283208(n). - Amiram Eldar, Apr 18 2024

New name by Peter Luschny, Apr 16 2024

A120785 Denominators of partial sums of Catalan numbers scaled by powers of 1/16.

Original entry on oeis.org

1, 16, 128, 4096, 32768, 524288, 4194304, 268435456, 2147483648, 34359738368, 274877906944, 8796093022208, 70368744177664, 1125899906842624, 9007199254740992, 1152921504606846976

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are given under A120784.
The listed numbers coincide with the denominators of sum(C(k)/(-16)^k,k=0..n). For the numerators see A120794. In general these denominators may be different. See e.g. A120783 versus A120793 and A120787 versus A120796.
Also one fourth of denominators of partial sums for sqrt(2)+sqrt(3). See A121503 for the numerators.

a(n)=denominator(r(n)), with the rationals r(n):=sum(C(k)/16^k,k=0..n) with C(k):=A000108(k) (Catalan numbers).

A120786 Numerators of partial sums of Catalan numbers scaled by powers of 1/20.

Original entry on oeis.org

1, 21, 211, 1689, 84457, 1689161, 16891643, 1351331869, 2702663881, 270266390531, 2702663909509, 108106556409753, 1081065564149533, 4324262256635277, 43242622566419631, 6918819610629079929

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are given under A120787.
From the expansion of 2*sqrt(5)/5 = sqrt(1-1/5) = 1-(1/10)*Sum_{k>=0} C(k)/20^k one has r := lim_{n->oo} r(n) = 2*(5 - 2*sqrt(5)) = 2*(7 - 4*phi) = 1.055728090..., where phi := (1+sqrt(5))/2 (golden section) and the partial sums r(n) are defined below.
This is the second member (p=1) in the second p-family of partial sums of the normalized scaled Catalan series CsnII(p) := Sum_{k>=0} C(k)/((5^k)*F(2*p+1)^(2*k)) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi), with C(n) = A000108(n) (Catalan), F(n) = A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi := (1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned second p-family are rII(p;n) := Sum_{k=0..n} C(k)/((5^k)*F(2*p+1)^(2*k)), n>=0, for p=0,1,...
For more details about this p-family and the other three ones see the W. Lang link under A120996.

			Rationals r(n): [1, 21/20, 211/200, 1689/1600, 84457/80000, 1689161/1600000, 16891643/16000000, 1351331869/1280000000,...].

Wolfdieter Lang, Rationals r(n) and limit.

a(n) = numerator(r(n)), with the rationals r(n) := Sum_{k=0..n} C(k)/20^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

A120781 Denominators of partial sums of Catalan numbers scaled by powers of 1/8.

Original entry on oeis.org

1, 8, 32, 512, 2048, 16384, 65536, 2097152, 8388608, 67108864, 268435456, 4294967296, 17179869184, 137438953472, 549755813888, 35184372088832, 140737488355328, 1125899906842624, 4503599627370496, 72057594037927936

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are given under A120780.

The listed numbers coincide with denominators of sum(C(k)/(-8)^k,k=0..n). For the numerators see A120789. In general these denominators may be different. See e.g. A120783 versus A120793 and A120787 versus A120796.

a(n)=denominator(r(n)), with the rationals r(n):=sum(C(k)/8^k,k=0..n) with C(k):=A000108(k) (Catalan numbers).

A121004 Numerators of partial sums of Catalan numbers scaled by powers of 1/(5*5^2)=1/125.

Original entry on oeis.org

1, 126, 15752, 393801, 246125639, 30765704917, 3845713114757, 480714139345054, 12017853483626636, 7511158427266652362, 938894803408331562046, 117361850426041445314536, 14670231303255180664525012

Offset: 0

Wolfdieter Lang, Aug 16 2006

Denominators are given under A121005.
This is the third member (p=2) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned second p-family are rII(p;n):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..n), n>=0, for p=0,1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.

			Rationals r(n): [1, 126/125, 15752/15625, 393801/390625,
246125639/244140625, 30765704917/30517578125,...].

W. Lang: Rationals r(n), limit.

The second member (p=2) is A120786/A120787.

The value of the series is lim_{n->infinity}(r(n) := rII(2;n)) = 5*(18 - 11*phi) = 5*sqrt(5)/phi^5 = 1.0081306187560 (maple10, 15 digits).

a(n)=numerator(r(n)) with r(n) := rII(p=2,n) = sum(C(k)/5^(3*k),k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121003 Denominators of partial sums of Catalan numbers scaled by powers of 1/5.

Original entry on oeis.org

1, 5, 25, 25, 625, 3125, 15625, 78125, 3125, 1953125, 9765625, 48828125, 244140625, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 152587890625, 3814697265625, 19073486328125

Offset: 0

Wolfdieter Lang, Aug 16 2006

Numerators are given under A121002.
This is the first member (p=0) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
For more details on this p-family and the other three ones see the W. Lang links under A120996 and A121002.

			Rationals r(n): [1, 6/5, 32/25, 33/25, 839/625, 4237/3125,
21317/15625, 107014/78125, 4292/3125, 2687362/1953125,...].
A120787 (denominators, second member p=1).

a(n)=denominator(r(n)) with r(n) := rII(p=0,n) = sum(C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

cp's OEIS Frontend

A120777 a(n) = 2^(2*n - valuation(CatalanNumber(n), 2)).

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A120785 Denominators of partial sums of Catalan numbers scaled by powers of 1/16.

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A120786 Numerators of partial sums of Catalan numbers scaled by powers of 1/20.

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A120781 Denominators of partial sums of Catalan numbers scaled by powers of 1/8.

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A121004 Numerators of partial sums of Catalan numbers scaled by powers of 1/(5*5^2)=1/125.

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A121003 Denominators of partial sums of Catalan numbers scaled by powers of 1/5.

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