A173218 -id:A173218 - cp's OEIS frontend

A173217 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n^2) / 2^(n+1).

1, 3, 36, 744, 21606, 807912, 36948912, 1997801520, 124666314300, 8817945612300, 697162848757056, 60925366551278592, 5831682410241684192, 606763511537812563648, 68184018356901256320192, 8229830886505821175612416, 1061871008421711265790015880, 145851902823090076435152800208, 21247730059665104564252809209792

Offset: 0

Paul D. Hanna, Mar 05 2010

nonn

Variant of A104209, which enumerates labeled directed multigraphs.

Number of labeled digraphs with n edges and no vertices of degree zero, in which loops are permitted but not duplicate edges. - David Bevan, Apr 22 2013

Paul D. Hanna, Table of n, a(n) for n = 0..100
StackExchange, Combinatorial puzzle (2013)

Cf. A104209, A173218, A301466, A301468, A265936.

Table[Sum[Binomial[k^2, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 21 2018 *)
Table[Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -2*j, 0]/2, {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Mar 21 2018 *)

{a(n)=local(A=sum(m=0,n^2+100,(1+x +O(x^(n+2)))^(m^2)/2^(m+1)));round(polcoeff(A,n))}

/* Continued fraction expression: */
{a(n) = my(CF=1, q = 1+x +x*O(x^n)); for(k=0, n, CF = 1/(2 - q^(4*n-4*k+1)/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*CF)) ); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 18 2018

G.f.: 1/(2 - q/(1 - q*(q^2-1)/(2 - q^5/(1 - q^3*(q^4-1)/(2 - q^9/(1 - q^5*(q^6-1)/(2 - q^13/(1 - q^7*(q^8-1)/(2 - ...))))))))) where q = (1+x), a continued fraction due to a partial elliptic theta function identity. - Paul D. Hanna, Mar 18 2018
G.f.: Sum_{n>=0} 1/2^(n+1) * (1+x)^n * Product_{k=1..n} (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)), due to a q-series identity. - Paul D. Hanna, Mar 18 2018
a(n) ~ 2^(2*n - 1/2 - log(2)/8) * n^n / (exp(n) * log(2)^(2*n + 1)). - Vaclav Kotesovec, Mar 21 2018

A173219 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).

Original entry on oeis.org

1, 2, 12, 124, 1800, 33648, 769336, 20796960, 648841680, 22945907520, 907036108432, 39631833652320, 1896696894062880, 98669609894805600, 5543804125505195040, 334563594743197602272, 21583554094995765302592

Offset: 0

Paul D. Hanna, Mar 05 2010

nonn

a(n) is the number of nonnegative integer matrices with n distinct columns and any number of nonzero rows with 2 ones in every column and columns in decreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=2 of A331278.

Cf. A121251, A173217, A173218.

Table[Sum[StirlingS1[n, j] * Sum[Binomial[j, s]*HurwitzLerchPhi[1/2, -j - s, 0], {s, 0, j}] / 2^(j+1), {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2019 *)

{a(n)=local(A=sum(m=0,n^2+100,(1+x +O(x^(n+2)))^(m*(m+1)/2)/2^(m+1)));round(polcoeff(A,n))}

a(n) = A265937(n)/2. - Vaclav Kotesovec, Oct 08 2019
a(n) ~ 2^n * n^n / (2^(log(2)/4) * log(2)^(2*n+1) * exp(n)). - Vaclav Kotesovec, Oct 08 2019
a(n) = 2*A121251(n) for n > 0. - Andrew Howroyd, Jan 15 2020

cp's OEIS Frontend

A173217 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n^2) / 2^(n+1).

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A173219 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula