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 1-4 of 4 results.

A002039 Convolution inverse of A143348.

Original entry on oeis.org

1, 3, 5, 10, 25, 64, 160, 390, 940, 2270, 5515, 13440, 32735, 79610, 193480, 470306, 1143585, 2781070, 6762990, 16445100, 39987325, 97232450, 236432060, 574915770, 1397981470, 3399360474, 8265943685, 20099618590, 48874630750
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Gandhi denotes f(-x) by Phi(x) and a(n) by G(n).

Examples

			1 + 3*x + 5*x^2 + 10*x^3 + 25*x^4 + 64*x^5 + 160*x^6 + 390*x^7 + 940*x^8 + ...

References

  • 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).

Links

Crossrefs

Cf. A002040, A143348.

Programs

  • Mathematica
    max = 28; f[x_] := -x / Sum[ k*(-x)^k/(1-(-x)^k), {k, 1, max+1}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 07 2011, after Michael Somos *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / log( eta( -x + x^2 * O(x^n)))', n))} /* Michael Somos, Apr 05 2003 */

Formula

G.f.: -x / (Sum_{k>0} k * (-x)^k / (1 - (-x)^k)) = 1 / (log( f(x) )') where f(-x) = Product_{k>0} (1 - x^k) is one of Ramanujan's theta functions. - Michael Somos, Apr 08 2003
a(n) ~ c * d^n, where d = -1/A143441 = 2.43161993449532399475429572773256778... and c = 0.765603960074106532799232452562411022387973764575133091283490410339311... - Vaclav Kotesovec, Jun 02 2018
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma(k+1) * a(n-k). - Ilya Gutkovskiy, May 27 2020

A242168 Decimal expansion of the integral of the q-Pochhammer symbol (reciprocal of the partition function) over the real interval -1 to 1.

Original entry on oeis.org

1, 2, 8, 8, 3, 0, 0, 8, 8, 8, 6, 7, 3, 9, 2, 1, 2, 3, 0, 1, 8, 0, 9, 0, 1, 4, 9, 3, 9, 3, 0, 9, 6, 3, 4, 4, 4, 2, 2, 5, 8, 7, 3, 8, 0, 7, 1, 3, 8, 7, 9, 6, 1, 9, 5, 0, 3, 2, 0, 1, 4, 9, 4, 2, 6, 9, 8, 6, 4, 4, 2, 4, 1, 8, 5, 2, 0, 4, 9, 7, 8, 8, 7, 6, 8, 2, 0, 9, 3, 4, 4, 4, 4, 1, 1, 1, 3, 3, 9, 8, 1, 3, 6, 3, 3
Offset: 1

Views

Author

William J. Keith, May 05 2014

Keywords

Comments

As a function, the q-Pochhammer symbol is an irregularly left-skewed bell curve. It has limiting value 0 at -1 and 1, and its maximum is at -0.411248... (decimal value given by A143441).

Examples

			1.2883008886739212301809014939309634442258738...

Links

Crossrefs

Cf. A010815, A143441, A258232.

Programs

  • Maple
    evalf(4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1), 120); # Vaclav Kotesovec, Jun 02 2015
  • Mathematica
    NIntegrate[QPochhammer[q, q], {q, -1, 1}, WorkingPrecision -> 45]
RealDigits[4*Sqrt[3/23]*Pi*(2*Sinh[Sqrt[23]*Pi/6] + Sqrt[2]*Sinh[Sqrt[23]*Pi/4]) / (2*Cosh[Sqrt[23]*Pi/3]-1), 10, 105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)
  • PARI
    eta2(q)=if(q==0,1,my(p=log(10^-38)/log(abs(q)),N=floor(sqrt(2*p/3)));sum(n=-N,N,(-1)^n*q^((3*n^2-n)/2),0.))
intnum(q=-.99999,.99999,eta2(q)) \\ Bill Allombert, May 06 2014

Formula

Equals 4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1). - Vaclav Kotesovec, Jun 02 2015

Extensions

More digits from Vaclav Kotesovec, Jun 02 2015

A002040 Related to partitions.

Original entry on oeis.org

1, 2, 4, 8, 21, 52, 131, 316, 765, 1846, 4494, 10944, 26654, 64798, 157502, 382868, 931028, 2264106, 5505777, 13387880, 32553601, 79156974, 192479838, 468039888, 1138098210, 2767421826, 6729311459, 16363118556, 39788886610, 96751470494
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 21*x^4 + 52*x^5 + 131*x^6 + 316*x^7 + ...

References

  • 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).

Links

Crossrefs

Cf. A002039, A000203, A010815.

Programs

  • Mathematica
    max = 29; f[q_] := Product[1 - (-q)^k, {k, 1, max + 1}]; CoefficientList[ Series[1/f'[q], {q, 0, max}], q] (* Jean-François Alcover, Jun 18 2012, after Michael Somos *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / D[ Normal @ Series[ QPochhammer[ -x], {x, 0, n + 1}], x], {x, 0, n}]]; (* Michael Somos, May 31 2016 *)
  • PARI
    {a(n) = polcoeff( 1 / eta( -x + x^2 * O(x^n))', n)};

Formula

G.f.: 1/(f(q)') where f(-q)=Product_{k>0} (1-q^k) is one of Ramanujan's theta functions. - Michael Somos, Apr 08 2003
a(n) = sum_{k=0..n} (-1)^k*A000041(k)*A002039(n-k). - Mircea Merca, Feb 27 2014
a(n) ~ c * d^n, where d = -1/A143441 = 2.431619934495323994754... and c = 0.623278923942755977756856780504941340332933121682037117752100... - Vaclav Kotesovec, Jun 02 2018

Extensions

Formula corrected and sequence extended by Michael Somos

A143440 Decimal expansion of the maximum value of the q-Pochhammer symbol along [ -1, 1].

Original entry on oeis.org

1, 2, 2, 8, 3, 4, 8, 8, 6, 7, 0, 3, 8, 5, 7, 5, 1, 1, 2, 5, 8, 6, 8, 7, 8, 3, 8, 9, 8, 6, 0, 0, 9, 6, 8, 2, 4, 9, 9, 0, 3, 2, 7, 9, 1, 5, 7, 6, 9, 5, 2, 6, 7, 4, 6, 7, 9, 6, 2, 9, 8, 3, 1, 5, 1, 0, 3, 0, 8, 2, 5, 4, 5, 1, 7, 7, 8, 3, 2, 8, 8, 6, 4, 8, 0, 1, 9, 9, 9, 3, 6, 3, 2, 4, 2, 5, 6, 1, 1, 9, 6, 4, 0, 7, 7
Offset: 1

Views

Author

Eric W. Weisstein, Aug 14 2008

Keywords

Examples

			1.2283488670385751125...

Links

Crossrefs

Cf. A143441, A242168, A258232.

Programs

  • Mathematica
    q0 = q /. FindRoot[QPochhammer'[q] == 0, {q, -1/2}, WorkingPrecision -> 300]; RealDigits[QPochhammer[q0], 10, 105] // First (* Jean-François Alcover, Dec 05 2013 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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