A198954 -id:A198954 - cp's OEIS frontend

A245552 G.f.: Sum_{n>=0} (2n+1)x^(n^2+n+1).

0, 1, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19

Offset: 0

N. J. A. Sloane, Aug 02 2014

nonn

Related to g.f. for A053187.

Apart from signs and a factor of 2, this is the classical Jacobi theta-function theta'_1(q), see A002483.

Antti Karttunen, Table of n, a(n) for n = 0..10101

Cf. A002483, A053187, A198954.

Join[{0},Flatten[Table[Join[{n,PadRight[{},n,0]}],{n,1,19,2}]]] (* Harvey P. Dale, Dec 14 2014 *)

A198954(n) = { my(m); if(issquare(8*n + 1, &m), m, 0) }; \\ This function from Michael Somos
A245552(n) = if(!(n%2),0,A198954((n-1)/2)); \\ After Robert Israel's formula - Antti Karttunen, Jul 24 2017

a(2*n+1) = A198954(n), a(2*n) = 0.- Robert Israel, Aug 05 2014

A168038 Squares closest to 2*n.

Original entry on oeis.org

0, 1, 4, 4, 9, 9, 9, 16, 16, 16, 16, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 0

Zak Seidov, Nov 17 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002024, A093995.

Essentially partial sums of A198954.

Mathematica
```
Round[Sqrt[2*Range[0,120]]]^2
```

from math import isqrt
def A168038(n): return (isqrt(n<<3)+1>>1)**2 # Chai Wah Wu, Jun 19 2024

a(n) = A002024(n)^2. - Chai Wah Wu, Jun 19 2024

A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule.

Original entry on oeis.org

1, 1, 2, 8, 72, 1440, 55008, 3507840, 342679680, 48401625600, 9472057781760, 2484361405532160, 850218223244544000, 371335242657899520000, 203148791342840318976000, 137006974339300359770112000

Offset: 0

Michael Somos, May 15 2005

nonn

			1 + 3*exp(-2*x) + 5*exp(-6*x) + 7*exp(-10*x) + ... ~ 1/x + 1/3 + (1/15)*x + (4/315)*x^2 + ...

G. Herzberg, Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules, D. Van Nostrand, 1945. see page 505
D. A. McQuarrie, Statistical Mechanics, University Science Books, 2000, see page 100 equ. (6-35)
G. H. Wannier, Statistical Physics, Dover Publications, 1987. see page 216 equ. (11.21)

Cf. A198954.

a[ n_] := If[ n < 0, 0, Sum[ BernoulliB[n + j] / (j! (n - j)!), {j, 0, n }] (2 n + 1)! / (-2)^n]; (* Michael Somos, Dec 04 2013 *)

{a(n) = if( n<0, 0, sum( j=0, n, bernfrac(n+j) / ((n-j)! * j!)) * (2*n + 1)! / (-2)^n)};

Sum_{k>=0} (2*k + 1) * exp(-x*(k^2 + k)) ~ (1/x) * Sum_{k>=0} a(k) * (2*x)^k / (2*k + 1)!.

a(n) ~ 2^(n + 7/2) * n^(3*n + 3/2) / (exp(3*n) * Pi^(2*n - 1/2)). - Vaclav Kotesovec, Jun 08 2019

cp's OEIS Frontend

A245552 G.f.: Sum_{n>=0} (2n+1)x^(n^2+n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A168038 Squares closest to 2*n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A245552 G.f.: Sum_{n>=0} (2*n+1)*x^(n^2+n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A168038 Squares closest to 2*n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

A245552 G.f.: Sum_{n>=0} (2n+1)x^(n^2+n+1).