A346061 -id:A346061 - cp's OEIS frontend

A178315 E.g.f.: A(x) = sqrt( Sum_{n>=0} 2^(n(n+1)/2) * x^n/n! ).

1, 1, 3, 23, 393, 13729, 943227, 126433847, 33245947857, 17276815511041, 17836691600303283, 36694285316980381463, 150671768689108469724633, 1235972596853128519493249569, 20265064539085026367759911941547, 664309630995695142408442512638430647

Offset: 0

Paul D. Hanna, May 24 2010

nonn

Number of equivalence classes of graphs on n vertices, where two graphs are in the same class if one can be obtained from the other by loop-switching a subset of its connected components. Here, loop-switching is a fixed-point-free involution adding a loop to every vertex that doesn't have one while simultaneously deleting the loops from all vertices that do. (see MO link)
Also, number of balanced signed graphs (without loops) on n vertices. A graph is signed if every edge has a sign, either positive or negative, and it is balanced if every cycle has an even number of negative edges. (see MO link)
Also, number of graphs on vertices {1,2,...,n}  with loops allowed, where the least vertex in each component has a loop. (see MO link)

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 393*x^4/4! +...
A(x)^2 = 1 + 2*x + 2^3*x^2/2! + 2^6*x^3/3! + 2^10*x^4/4! +...

F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (1953/54), 143-146.
F. Harary and E. M. Palmer, On the number of balanced signed graphs, Bulletin of Mathematical Biophysics 29 (1967), 759-765.

Alois P. Heinz, Table of n, a(n) for n = 0..81
R. Stanley et al., The "square root" of a graph?, MathOverflow, 2021.
F. Ardila, F. Castillo, Federico, and M. Henley, The arithmetic Tutte polynomials of the classical root systems, Int. Math. Res. Not. IMRN 12 (2015), 3830-3877.

Column k=2 of A346061.

a:= n-> n!*coeff(series(add(2^binomial(j+1, 2)
        *x^j/j!, j=0..n)^(1/2), x, n+1), x, n):
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 15 2021

{a(n)=n!*polcoeff(sqrt(sum(m=0,n,2^(m*(m+1)/2)*x^m/m!)+x*O(x^n)),n)}

Edited by Max Alekseyev, Mar 13 2021

A178319 E.g.f.: ( Sum_{n>=0} 3^(n(n + 1)/2) x^n/n! )^(1/3).

Original entry on oeis.org

1, 1, 7, 199, 17713, 4572529, 3426693463, 7575807034711, 49908659904426337, 983868034228748840161, 58130023275752925902247847, 10299771730830080877230000021479, 5474153833417147528343683843805979793, 8727821227226586439546709016484604992020049

Offset: 0

Paul D. Hanna, May 24 2010

nonn

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 199*x^3/3! + 17713*x^4/4! +...
A(x)^3 = 1 + 3*x + 3^3*x^2/2! + 3^6*x^3/3! + 3^10*x^4/4! +...
Let E(x, q) = Sum_{n>=0} q^(n*(n - 1)/2)*x^n/n!, then the coefficients of (x^n/n!) in E(qx, q)^(1/q) begin:
  1;
  1;
  q^2 - q + 1;
  q^5 - 3*q^3 + 5*q^2 - 3*q + 1;
  q^9 - 4*q^6 + q^5 + 15*q^4 - 24*q^3 + 17*q^2 - 6*q + 1;
  q^14 - 5*q^10 + 5*q^9 - 10*q^8 + 30*q^7 - 95*q^5 + 149*q^4 - 110*q^3 + 45*q^2 - 10*q + 1; ...
Setting q = 3 generates this sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..64
Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Cf. A178315 (sqrt case).

Column k=3 of A346061.

a:= n-> n!*coeff(series(add(3^binomial(j+1, 2)
        *x^j/j!, j=0..n)^(1/3), x, n+1), x, n):
seq(a(n), n=0..14);  # Alois P. Heinz, Mar 15 2021

{a(n)=n!*polcoeff(sum(m=0,n,3^(m*(m+1)/2)*x^m/m!+x*O(x^n))^(1/3),n)}

a(n) = 1 (mod 6) for n >= 0 (conjecture).

General conjecture: [x^n/n!] E(q*x, q)^(1/q) = 1 (mod q(q-1)) for n >= 0 and integer q > 1 where E(x, q) = Sum_{n>=0} q^(n*(n - 1)/2)*x^n/n!.

General conjecture restated by Paul D. Hanna, May 25 2010

A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j(j+1)/2) x^j/j!)^(1/n) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 3, 199, 249337, 6062674201, 3653786369479951, 65709007885111803731947, 40564683796482484146182142025377, 969773549559254966290998252899999751714721, 999999990999996719397362087568018696141879478712251051, 49037072510879011742983689973641327840345400616866967292640434759551

Offset: 0

Alois P. Heinz, Mar 15 2021

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..35
Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Cf. A000217, A023813, A178315, A178319.

Main diagonal of A346061.

a:= n-> `if`(n>0, coeff(series(add(n^binomial(j+1, 2)*
         x^j/j!, j=0..n)^(1/n), x, n+1), x, n)*n!, 1):
seq(a(n), n=0..12);

a(n) == 1 (mod n*(n-1)) for n >= 2 (see "general conjecture" in A178319 and link to proof by Richard Stanley above).

a(n) ~ n^((n^2 + n - 2)/2). - Vaclav Kotesovec, Jul 15 2021

cp's OEIS Frontend

A178315 E.g.f.: A(x) = sqrt( Sum_{n>=0} 2^(n(n+1)/2) * x^n/n! ).

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A178319 E.g.f.: ( Sum_{n>=0} 3^(n(n + 1)/2) x^n/n! )^(1/3).

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A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j(j+1)/2) x^j/j!)^(1/n) for n > 0, a(0) = 1.

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cp's OEIS Frontend

A178315 E.g.f.: A(x) = sqrt( Sum_{n>=0} 2^(n(n+1)/2) * x^n/n! ).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

PARI

Extensions

A178319 E.g.f.: ( Sum_{n>=0} 3^(n*(n + 1)/2) * x^n/n! )^(1/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j*(j+1)/2) * x^j/j!)^(1/n) for n > 0, a(0) = 1.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Maple

Formula

A178319 E.g.f.: ( Sum_{n>=0} 3^(n(n + 1)/2) x^n/n! )^(1/3).

A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j(j+1)/2) x^j/j!)^(1/n) for n > 0, a(0) = 1.