cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A160464 The Eta triangle.

Original entry on oeis.org

-1, -11, 2, -114, 29, -2, -3963, 1156, -122, 4, -104745, 32863, -4206, 222, -4, -3926745, 1287813, -184279, 12198, -366, 4, -198491580, 67029582, -10317484, 781981, -30132, 562, -4
Offset: 2

Views

Author

Johannes W. Meijer, May 24 2009

Keywords

Comments

The ES1 matrix coefficients are defined by ES1[2*m-1,n] = 2^(2*m-1) * int(y^(2*m-1)/(cosh(y))^(2*n),y=0..infinity)/(2*m-1)! for m = 1, 2, 3, .. and n = 1, 2, 3 .. .
This definition leads to ES1[2*m-1,n=1] = 2*eta(2*m-1) and the recurrence relation ES1[2*m-1,n] = ((2*n-2)/(2*n-1))*(ES1[2*m-1,n-1] - ES1[2*m-3,n-1]/(n-1)^2) which we used to extend our definition of the ES1 matrix coefficients to m = 0, -1, -2, .. . We discovered that ES1[ -1,n] = 0.5 for n = 1, 2, .. . As usual eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.
The coefficients in the columns of the ES1 matrix, for m = 1, 2, 3, .. , and n = 2, 3, 4 .. , can be generated with the polynomials GF(z,n) for which we found the following general expression GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n))/p(n).
The CFN1(z,n) polynomials depend on the central factorial numbers A008955.
The ETA(z,n) are the Eta polynomials which lead to the Eta triangle.
The zero patterns of the Eta polynomials resemble a UFO. These patterns resemble those of the Zeta, Beta and Lambda polynomials, see A160474, A160480 and A160487.
The first Maple algorithm generates the coefficients of the Eta triangle. The second Maple algorithm generates the ES1[2*m-1,n] coefficients for m= 0, -1, -2, -3, .. .
The M(n) sequence, see the second Maple algorithm, leads to Gould's sequence A001316 and a sequence that resembles the denominators of the Taylor series for tan(x), A156769(n).
Some of our results are conjectures based on numerical evidence, see especially A160466.

Examples

			The first few rows of the triangle ETA(n,m) with n=2,3,.. and m=1,2,... are
  [ -1]
  [ -11, 2]
  [ -114, 29, -2]
  [ -3963, 1156, -122, 4].
The first few ETA(z,n) polynomials are
  ETA(z,n=2) = -1;
  ETA(z,n=3) = -11+2*z^2;
  ETA(z,n=4) = -114 + 29*z^2 - 2*z^4.
The first few CFN1(z,n) polynomials are
  CFN1(z,n=2) = (z^2-1);
  CFN1(z,n=3) = (z^4 - 5*z^2 + 4);
  CFN1(z,n=4) = (z^6 - 14*z^4 + 49*z^2 - 36).
The first few generating functions GF(z;n) are:
  GF(z;n=2) = ((-1)*2*(z^2 - 1)*GF(z;n=1) + (- 1))/3;
  GF(z;n=3) = (4*(z^4 - 5*z^2+4) *GF(z;n=1) + (-11 + 2*z^2))/30;
  GF(z;n=4) = ((-1)*4*(z^6 - 14*z^4 + 49*z^2 - 36)*GF(z;n=1) + (-114 + 29*z^2 - 2*z^4))/315.
		

References

  • Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Crossrefs

The r(n) sequence equals A062383 (n>=1).
The p(n) sequence equals A160473(n) (n>=2).
The GCS(n) sequence equals the Geometric Connell sequence A049039(n).
The M(n-1) sequence equals A001316(n-1)/A156769(n) (n>=1).
The q(n) sequence leads to A081729 and the 'gossip sequence' A007456.
The first right hand column equals A053644 (n>=1).
The first left hand column equals A160465.
The row sums equal A160466.
The CFN1(z, n) and the cfn1(n, k) lead to A008955.
Cf. A094665 and A160468.
Cf. the Zeta, Beta and Lambda triangles A160474, A160480 and A160487.
Cf. A162440 (EG1 matrix).

Programs

  • Maple
    nmax:=8; 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); ETA(n, n) := 0 end do: mmax:=nmax: for m from 2 to mmax do for n from m+1 to nmax do q(n) := (1+(-1)^(n-3)*(floor(ln(n-1)/ln(2)) - floor(ln(n-2)/ln(2)))): ETA(n, m) := q(n)*((-1)*ETA(n-1, m-1)+(n-1)^2*ETA(n-1, m)) end do end do: seq(seq(ETA(n,m), m=1..n-1), n=2..nmax);
    # End first program.
    nmax1:=20; m:=1; ES1row:=1-2*m; with (combinat): cfn1 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j=-k..k) end proc: mmax1:=nmax1: for m1 from 1 to mmax1 do M(m1-1) := 2^(2*m1-2)/((2*m1-1)!); ES1[-2*m1+1,1] := 2*(1-2^(1-(1-2*m1)))*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do ES1[1-2*m1, n] := (-1)^(n-1)*M(n-1)*sum((-1)^(k+1)*cfn1(n-1,k-1)* ES1[2*k-2*n-2*m1+1, 1], k=1..n) od: od: seq(ES1[1-2*m, n], n=1..nmax1-m+1);
    # End second program.

Formula

We discovered an interesting relation between the Eta triangle coefficients ETA(n,m) = q(n)*((-1)*ETA(n-1,m-1)+(n-1)^2*ETA(n-1,m)), for n = 3, 4, ... and m = 2, 3, ... , with
q(n) = 1 + (-1)^(n-3)*(floor(log(n-1)/log(2)) - floor(log(n-2)/log(2))) for n = 3, 4, ....
See A160465 for ETA(n,m=1) and furthermore ETA(n,n) = 0 for n = 2, 3, ....
The generating functions GF(z;n) of the coefficients in the matrix columns are defined by
GF(z;n) = sum_{m>=1} ES1[2*m-1,n] * z^(2*m-2), with n = 1, 2, 3, .... This leads to
GF(z;n=1) = (2*log(2) - Psi(z) - Psi(-z) + Psi(1/2*z) + Psi(-1/2*z)); Psi(z) is the digamma-function.
GF(z;n) = ((2*n-2)/(2*n-1)-2*z^2/((n-1)*(2*n-1)))*GF(z;n-1)-1/((n-1)*(2*n-1)).
We found for GF(z;n), for n = 2, 3, ..., the following general expression:
GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n) )/p(n) with
r(n) = 2^floor(log(n-1)/log(2)+1) and
p(n) = 2^(-GCS(n))*(2*n-1)! with
GCS(n) = log(1/(2^(-(2*(n-1)-1-floor(log(n-1)/ log(2))))))/log(2).

A000457 Exponential generating function: (1+3*x)/(1-2*x)^(7/2).

Original entry on oeis.org

1, 10, 105, 1260, 17325, 270270, 4729725, 91891800, 1964187225, 45831035250, 1159525191825, 31623414322500, 924984868933125, 28887988983603750, 959493919812553125, 33774185977401870000, 1255977541034632040625
Offset: 0

Views

Author

Keywords

Examples

			G.f. = 1 + 10*x + 105*x^2 + 1260*x^3 + 17325*x^4 + 270270*x^5 + ... - _Michael Somos_, Dec 15 2023
		

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
  • F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
  • C. Jordan, Calculus of Finite Differences. Eggenberger, Budapest and Röttig-Romwalter, Sopron 1939; Chelsea, NY, 1965, p. 172.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Equals (1/2)*A000906.
Third column of triangle A001497.
Second column (m=1) of unsigned Laguerre-Sonin a=1/2 triangle |A130757|.
Diagonal k=n-1 of triangle A134991.

Programs

  • Magma
    [Factorial(2*n+3)/(6*Factorial(n)*2^n): n in [0..30]]; // G. C. Greubel, May 15 2018
  • Mathematica
    Table[(2n+3)!/(3!*n!*2^n), {n,0,30}] (* G. C. Greubel, May 15 2018 *)
  • PARI
    for(n=0, 30, print1((2*n+3)!/(3!*n!*2^n), ", ")) \\ G. C. Greubel, May 15 2018
    

Formula

a(n) = (2n+3)!/( 3!*n!*2^n ).
a(n) = (n+1)*(2*n+3)!!/3, n>=0, with (2*n+3)!! = A001147(n+2).
a(n) = Sum_{j=0..n} (j + 1) * Eulerian2(n + 2, n - j). - Peter Luschny, Feb 13 2023

Extensions

More terms from Sascha Kurz, Aug 15 2002

A049039 Geometric Connell sequence: 1 odd, 2 even, 4 odd, 8 even, ...

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 26, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 121, 123, 125
Offset: 1

Views

Author

Keywords

Crossrefs

Cf. A337300 (partial sums), A043529 (first differences).
Cf. A160464, A160465 and A160473. - Johannes W. Meijer, May 24 2009

Programs

  • Haskell
    a049039 n k = a049039_tabl !! (n-1) !! (k-1)
    a049039_row n = a049039_tabl !! (n-1)
    a049039_tabl = f 1 1 [1..] where
       f k p xs = ys : f (2 * k) (1 - p) (dropWhile (<= last ys) xs) where
         ys  = take k $ filter ((== p) . (`mod` 2)) xs
    -- Reinhard Zumkeller, Jan 18 2012, Jul 08 2011
    
  • Maple
    Digits := 100: [seq(2*n-1-floor(evalf(log(n)/log(2))), n=1..100)];
  • Mathematica
    a[0] = 0; a[n_?EvenQ] := a[n] = a[n/2]+n-1; a[n_?OddQ] := a[n] = a[(n-1)/2]+n; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 27 2011, after Ralf Stephan *)
  • PARI
    a(n) = n<<1 - 1 - logint(n,2); \\ Kevin Ryde, Feb 12 2022
    
  • Python
    def A049039(n): return (n<<1)-n.bit_length() # Chai Wah Wu, Aug 01 2022

Formula

a(n) = 2n - 1 - floor(log_2(n)).
a(2^n-1) = 2^(n+1) - (n+2) = A000295(n+1), the Eulerian numbers.
a(0)=0, a(2n) = a(n) + 2n - 1, a(2n+1) = a(n) + 2n + 1. - Ralf Stephan, Oct 11 2003

Extensions

Keyword tabf added by Reinhard Zumkeller, Jan 22 2012

A054243 Number of partitions of n into distinct positive parts <= n, where parts are combined by XOR.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 8, 16, 16, 32, 64, 128, 256, 512, 1024, 2048, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 67108864, 134217728, 268435456, 536870912
Offset: 0

Views

Author

Marc LeBrun, Feb 08 2000

Keywords

Comments

Usually successive powers of 2, but "stutters" when n is power of 2. (G.f. must satisfy some interesting functional equations!). Empty partition of 0 defined as 1.
This is an instance of what I like to call "numbral theory": whenever you have a set of indexed objects that you can do some kind of arithmetic on, then the indices act as "shadows" of the objects and you can generally talk about lots of analogs, such as partitions, primes, even generating functions, etc. It would be worthwhile to systematically "fill out" the entries for as many of these systems as possible in the OEIS.
The "AND" version is just the all-ones sequence. - Christian G. Bower, Jun 07 2005
a(n) is the number of orbits of the FlipAfter1 map on integers with n+1 binary digits. The FlipAfter1 map on an integer in binary form is: flip each bit that is immediately preceded by a "1". For example, the orbits on 4-bit numbers are 1000 -> 1100 -> 1010 -> 1111 and 1001 -> 1101 -> 1011 -> 1110. The orbits on n-bit numbers are all of length 2^floor(log_2(n-1)+1) (for n >= 2), A062383. There is precisely one member of each orbit in the following set: integers in binary form such that each bit at distance a power of two from the leading "1" is 0. This set of orbit representatives begins 1, 10, 100, 1000, 1001, 10000, 10010, 100000, 100001, 100100, 100101. - David Callan, Oct 13 2012

Examples

			a(5)=4 thus: 5 4+1 5+3+2+1 4+3+2 (where "+" = XOR).
		

Crossrefs

Inclusive-OR (or IOR) version: A054244.
Cf. A160473.

Programs

Formula

a(n) = 2^floor(n - log_2(n) - 1) = A000079(n)/A062383(n). - Henry Bottomley, Nov 22 2001

A162440 The pg(n) sequence that is associated with the Eta triangle A160464.

Original entry on oeis.org

2, 16, 144, 4608, 115200, 4147200, 203212800, 26011238400, 2106910310400, 210691031040000, 25493614755840000, 3671080524840960000, 620412608698122240000, 121600871304831959040000
Offset: 2

Views

Author

Johannes W. Meijer, Jul 06 2009

Keywords

Comments

The EG1 matrix coefficients are defined by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 with m = .., -2, -1, 0, 1, 2, ... and n = 1, 2, 3, ... . As usual, eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2 matrix, the even counterpart of the EG1 matrix, see A008955.
The coefficients in the columns of the EG1 matrix, for m >= 1 and n >= 2, can be generated with GFE(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GFE(z;n=1) + ETA(z,n))/pg(n) for n >= 2.
The CFN1(z,n) polynomials depend on the central factorial numbers A008955 and the ETA(z,n) are the Eta polynomials which led to the Eta triangle, see for both A160464.
The pg(n) sequence can be generated with the first Maple program and the EG1[2m-1,n] matrix coefficients can be generated with the second Maple program.
The EG1 matrix is related to the ES1 matrix, see A160464 and the formulas below.

Examples

			The first few generating functions GFE(z;n) are:
GFE(z;n=2) = ((-1)*2*(z^2 - 1)*GFE(z;n=1) + (-1))/2,
GFE(z;n=3) = ((+1)*4*(z^4 - 5*z^2 + 4)*GFE(z;n=1) + (-11 + 2*z^2))/16,
GFE(z;n=4) = ((-1)*4*(z^6-14*z^4+49*z^2-36)*GFE(z;n=1) + (-114+29*z^2-2*z^4))/144.
		

References

  • Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Crossrefs

The ETA(z, n) polynomials and the ES1 matrix lead to the Eta triangle A160464.
The CFN1(z, n), the t1(n, m) and the EG2 matrix lead to A008955.
The EG1[ -1, n] equal (1/2)*A001803(n-1)/A046161(n-1).
The r(n) sequence equals A062383(n) (n>=1).
The e(n) sequence equals A029837(n) (n>=1).
Cf. A160473 (p(n) sequence).
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

Programs

  • Maple
    nmax := 16; seq((n-1)!^2*2^floor(ln(n-1)/ln(2)+1), n=2..nmax);
    # End program 1
    nmax1 := 5; coln := 4; mmax1 := nmax1: for n from 0 to nmax1 do t1(n, 0) := 1 end do: for n from 0 to nmax1 do t1(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do t1(n, m) := t1(n-1, m-1)*n^2 + t1(n-1, m) end do: end do: for m from 1 to mmax1 do EG1[1-2*m, 1] := evalf((2^(2*m)-1)* bernoulli(2*m)/(m)) od: EG1[1, 1] := evalf(2*ln(2)): for m from 2 to mmax1 do EG1[2*m-1, 1] := evalf(2*(1-2^(1-(2*m-1))) * Zeta(2*m-1)) od: for m from -mmax1+coln to mmax1 do EG1[2*m-1, coln]:= (-1)^(coln+1)*sum((-1)^k*t1(coln-1, k) * EG1[1-2*coln+2*m+2*k, 1], k=0..coln-1)/(coln-1)!^2 od;
    # End program 2 (Edited by Johannes W. Meijer, Sep 21 2012)

Formula

pg(n) = (n-1)!^2*2^floor(log(n-1)/log(2)+1) for n >= 2.
r(n) = 2^e(n) = 2^floor(log(n-1)/log(2)+1) for n >= 2.
EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2) for n >= 1.
GFE(z;n) = sum (EG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFE(z;n) = (1-z^2/(n-1)^2)*GFE(z;n-1)-EG1[ -1,n-1]/(n-1)^2 for n = >2. with GFE(z;n=1) = 2*log(2)-Psi(z)-Psi(-z)+Psi(z/2)+Psi(-z/2) and Psi(z) is the digamma function.
EG1[2m-1,n] = (2*2^(1-2*n)*(2*n-1)!/((n-1)!^2)) * ES1[2m-1,n].

A160465 First left hand column of the Eta triangle A160464.

Original entry on oeis.org

-1, -11, -114, -3963, -104745, -3926745, -198491580, -26045435115, -2153099119815, -219022225836750, -26891482281048000, -3921682257253270125, -670160622793156369875, -132649536458654226136125
Offset: 2

Views

Author

Johannes W. Meijer, May 24 2009

Keywords

Crossrefs

A160464 is the Eta triangle.
The GCS(n) sequence equals the geometric Connell sequence A049039(n).
The p(n) sequence is given by A160473.

Programs

  • Maple
    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: seq(ETA(n, 1), n=2..nmax);
Showing 1-6 of 6 results.