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.

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A364232 Expansion of Sum_{k>=0} x^(3*k+1) / (1 + x^(3*k+1))^2.

Original entry on oeis.org

1, -2, 3, -3, 5, -6, 8, -10, 9, -9, 11, -9, 14, -16, 15, -19, 17, -18, 20, -17, 24, -21, 23, -30, 26, -28, 27, -24, 29, -27, 32, -42, 33, -33, 40, -27, 38, -40, 42, -53, 41, -48, 44, -35, 45, -45, 47, -57, 57, -47, 51, -42, 53, -54, 56, -80, 60, -57, 59, -51, 62, -64, 72, -83, 70, -63, 68, -53, 69, -72
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 14 2023

Keywords

Links

Crossrefs

Cf. A002129, A186690, A326399, A364012, A364204, A364235.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[x^(3 k + 1)/(1 + x^(3 k + 1))^2, {k, 0, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, MemberQ[{1}, Mod[n/#, 3]] &], {n, 1, 70}]

Formula

a(n) = Sum_{d|n, n/d==1 (mod 3)} (-1)^(d+1) * d.

A364235 Expansion of Sum_{k>=0} x^(3*k+2) / (1 + x^(3*k+2))^2.

Original entry on oeis.org

0, 1, 0, -2, 1, 3, 0, -3, 0, 3, 1, -6, 0, 8, 3, -10, 1, 9, 0, -13, 0, 9, 1, -9, 5, 14, 0, -16, 1, 9, 0, -19, 3, 15, 8, -18, 0, 20, 0, -25, 1, 24, 0, -25, 9, 21, 1, -30, 0, 16, 3, -28, 1, 27, 16, -24, 0, 27, 1, -39, 0, 32, 0, -42, 14, 27, 0, -37, 3, 24, 1, -27, 0, 38, 15, -40, 8, 42, 0, -69
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 14 2023

Keywords

Links

Crossrefs

Cf. A002129, A186690, A326400, A364015, A364205, A364232.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[x^(3 k + 2)/(1 + x^(3 k + 2))^2, {k, 0, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 80}]

Formula

a(n) = Sum_{d|n, n/d==2 (mod 3)} (-1)^(d+1) * d.

A015664 Expansion of e.g.f. theta_3^(1/2).

Original entry on oeis.org

1, 1, -1, 3, 9, -15, 135, -2205, 21105, 76545, 694575, -6392925, -56600775, 66891825, -19964169225, 741313447875, 5375639894625, 44667168170625, -2328500019470625, 5663134786183875, -466442955127524375, 11513119609487120625
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The sequence shows the coefficients of sqrt(theta_3) regarded as an exponential generating function.

Examples

			sqrt(theta_3) = 1 + q - (1/2)*q^2 + (1/2)*q^3 + (3/8)*q^4 - (1/8)*q^5 + (3/16)*q^6 - (7/16)*q^7 + (67/128)*q^8 + (27/128)*q^9 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Links

Crossrefs

Cf. A000122 (theta_3), A015680, A186690.
Cf. A015665, A015666, A015667, A015669, A015671, A015672, A015673, A015675, A015676, A015677, A015678, A015679.

Programs

  • Maple
    # get basic theta series in maple
maxd:=201:
# get th2, th3, th4 = Jacobi theta constants out to degree maxd
temp0:=trunc(evalf(sqrt(maxd)))+2:
a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
th2:=series(a,q,maxd); # A098108
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
th3:=series(a,q,maxd); # A000122
th4:=series(subs(q=-q,th3),q,maxd); # A002448
series(sqrt(th3),q,maxd); # this sequence
  • Mathematica
    nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 23 2018 *)

Formula

E.g.f. appears to equal exp( Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - Peter Bala, Dec 23 2021
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A186690(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Jul 07 2023

Extensions

Entry revised by N. J. A. Sloane, Oct 22 2018

A193538 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^2/2 * x^n/n ).

Original entry on oeis.org

1, 2, 6, 20, 46, 116, 284, 632, 1414, 3102, 6536, 13636, 28020, 56300, 111888, 219608, 424694, 813104, 1540818, 2888060, 5366072, 9884616, 18050428, 32713048, 58851972, 105113942, 186505864, 328821408, 576153008, 1003687444, 1738735728, 2995837872
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Comments

Here sigma(n) = A000203(n) is the sum of divisors of n. Compare g.f. to the formula for Jacobi theta_4(x) given by
theta_4(x) = exp( Sum_{n>=1} (sigma(n)-sigma(2*n))*x^n/n )
where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 46*x^4 + 116*x^5 + 284*x^6 +...
log(A(x)) = 2^2*x/2 + 4^2*x^2/4 + 8^2*x^3/6 + 8^2*x^4/8 + 12^2*x^5/10 + 16^2*x^6/12 + 16^2*x^7/14 + 16^2*x^8/16 + 26^2*x^9/18 +...+ A054785(n)^2/2*x^n/n +...

Crossrefs

Cf. A177398, A054785, A186690.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^2/2*x^m/m)+x*O(x^n)), n)}

Formula

Self-convolution yields A177398.

A193539 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^3 * x^n/n ).

Original entry on oeis.org

1, 8, 64, 512, 3200, 19392, 112128, 598016, 3088896, 15362408, 73331264, 340653056, 1538392064, 6762336448, 29072665600, 122299068416, 504128374784, 2040557142592, 8116582974656, 31760991869952, 122408808197120, 464983163273216, 1742277357389312
Offset: 0

Views

Author

Paul D. Hanna, Jul 30 2011

Keywords

Comments

Here sigma(n) = A000203(n) is the sum of divisors of n. Compare g.f. to the formula for Jacobi theta_4(x) given by:
theta_4(x) = exp( Sum{n>=1} (sigma(n)-sigma(2*n))*x^n/n )
where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).

Examples

			G.f.: A(x) = 1 + 8*x + 64*x^2 + 512*x^3 + 3200*x^4 + 19392*x^5 +...
log(A(x)) = 2^3*x + 4^3*x^2/2 + 8^3*x^3/3 + 8^3*x^4/4 + 12^3*x^5/5 + 16^3*x^6/6 + 16^3*x^7/7 + 16^3*x^8/8 + 26^3*x^9/9 +...+ A054785(n)^3*x^n/n +...

Crossrefs

Cf. A177398, A054785, A186690.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^3*x^m/m)+x*O(x^n)), n)}
Previous Showing 21-25 of 25 results.

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

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