A265936 -id:A265936 - 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

A303920 G.f.: A(x,y) = (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 6, 6, 0, 0, 0, 4, 56, 56, 4, 0, 0, 0, 1, 117, 722, 722, 117, 1, 0, 0, 0, 0, 126, 2982, 12012, 12012, 2982, 126, 0, 0, 0, 0, 0, 84, 6916, 79548, 246092, 246092, 79548, 6916, 84, 0, 0, 0, 0, 0, 36, 10900, 312880, 2322000, 6002824, 6002824, 2322000, 312880, 10900, 36, 0, 0, 0, 0, 0, 9, 12717, 864009, 13617765, 74916306, 170048394, 170048394, 74916306, 13617765, 864009, 12717, 9, 0, 0, 0, 0, 0, 1, 11421, 1825786, 57282026, 604000555, 2669115383, 5489377628, 5489377628, 2669115383, 604000555, 57282026, 1825786, 11421, 1, 0, 0

Offset: 0

Paul D. Hanna, May 02 2018

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^n*y^k, where T(n,k) is the term of this triangle at position k in row n.

			G.f.: A(x,y) = (1-y) * ( 1 + y*(1 + x*(1-y)^2) + y^2*(1 + x*(1-y)^2)^4 + y^3*(1 + x*(1-y)^2)^9 + y^4*(1 + x*(1-y)^2)^16 + y^5*(1 + x*(1-y)^2)^25 + ... ).
Explicitly,
A(x,y) = 1 + x*(y + y^2) + x^2*(6*y^2 + 6*y^3) + x^3*(4*y^2 + 56*y^3 + 56*y^4 + 4*y^5) + x^4*(y^2 + 117*y^3 + 722*y^4 + 722*y^5 + 117*y^6 + y^7) + x^5*(126*y^3 + 2982*y^4 + 12012*y^5 + 12012*y^6 + 2982*y^7 + 126*y^8) + x^6*(84*y^3 + 6916*y^4 + 79548*y^5 + 246092*y^6 + 246092*y^7 + 79548*y^8 + 6916*y^9 + 84*y^10) + x^7*(36*y^3 + 10900*y^4 + 312880*y^5 + 2322000*y^6 + 6002824*y^7 + 6002824*y^8 + 2322000*y^9 + 312880*y^10 + 10900*y^11 + 36*y^12) + x^8*(9*y^3 + 12717*y^4 + 864009*y^5 + 13617765*y^6 + 74916306*y^7 + 170048394*y^8 + 170048394*y^9 + 74916306*y^10 + 13617765*y^11 + 864009*y^12 + 12717*y^13 + 9*y^14) + x^9*(y^3 + 11421*y^4 + 1825786*y^5 + 57282026*y^6 + 604000555*y^7 + 2669115383*y^8 + 5489377628*y^9 + 5489377628*y^10 + 2669115383*y^11 + 604000555*y^12 + 57282026*y^13 + 1825786*y^14 + 11421*y^15 + y^16) + ...
This triangle begins:
[1];
[0, 1, 1];
[0, 0, 6, 6, 0];
[0, 0, 4, 56, 56, 4, 0];
[0, 0, 1, 117, 722, 722, 117, 1, 0];
[0, 0, 0, 126, 2982, 12012, 12012, 2982, 126, 0, 0];
[0, 0, 0, 84, 6916, 79548, 246092, 246092, 79548, 6916, 84, 0, 0];
[0, 0, 0, 36, 10900, 312880, 2322000, 6002824, 6002824, 2322000, 312880, 10900, 36, 0, 0];
[0, 0, 0, 9, 12717, 864009, 13617765, 74916306, 170048394, 170048394, 74916306, 13617765, 864009, 12717, 9, 0, 0];
[0, 0, 0, 1, 11421, 1825786, 57282026, 604000555, 2669115383, 5489377628, 5489377628, 2669115383, 604000555, 57282026, 1825786, 11421, 1, 0, 0]; ...

Paul D. Hanna, Table of n, a(n) for n = 0..2600, of rows 0..50, as a flattened triangle read by rows.

Cf. A303921 (diagonal), A303922 (column sums), A001813 (row sums), A265936 (y=2), A173217.

/* G.f. by Definition: */
{T(n,k) = my(A = (1-y) * sum(m=0,2*n, y^m * (1 + x*(1-y)^2  +x*O(x^(2*n)) )^(m^2))); polcoeff(polcoeff(A, n,x),k,y)}
for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", ")); print(""))

/* Continued fraction expression: */
{T(n,k) = my(CF=1, q = 1 + x*(1-y)^2 +x*O(x^(2*n))); for(k=0, n, CF = 1/(1 - q^(4*n-4*k+1)*y/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*y*CF)) ); polcoeff(polcoeff((1-y)*CF, n,x),k,y)}
for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", ")); print(""))

/* G.f. by q-series identity: */
{T(n,k) = my(A =1, q = 1 + x*(1-y)^2 +x*O(x^(2*n))); A = (1-y) * sum(m=0,2*n, y^m*q^m * prod(k=1,m, (1 - y*q^(4*k-3)) / (1 - y*q^(4*k-1) +x*O(x^(2*n))) )); polcoeff(polcoeff(A, n,x),k,y)}
for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", ")); print(""))

GENERATING FUNCTIONS.
(1) A(x,y) = (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2).
(2) A(x,y) = (1-y) * Sum_{n>=0} y^n * q^n * Product_{k=1..n} (1 - q^(4*k-3)*y) / (1 - q^(4*k-1)*y), where q = 1 + x*(1-y)^2, due to a q-series identity.
(3) A(x,y) = (1-y)/(1 - q*y/(1 - q*(q^2-1)*y/(1 - q^5*y/(1 - q^3*(q^4-1)*y/(1 - q^9*y/(1- q^5*(q^6-1)*y/(1 - q^13*y/(1 - q^7*(q^8-1)*y/(1 - ...))))))))), where q = 1 + x*(1-y)^2, a continued fraction due to an identity of a partial elliptic theta function.
FORMULAS INVOLVING TERMS.
Sum_{k=0..2*n} T(n,k)  =  (2*n)!/n!, for n>=0 (row sums = A001813).
Sum_{k=0..2*n} T(n,k) * (-1)^k  =  0, for n>=1 (symmetric rows).
Sum_{k=0..2*n} T(n,k) * 2^k  =  A265936(n), for n>=1.
Sum_{k=0..2*n} T(n,k) / 2^k  =  A173217(n) / 4^n, for n>=0.
Sum_{j=0..k^2} T(j,k) = A303922(k), for k>=0 (column sums).
T(n,n) = A303921(n), for n>=0 (diagonal).

A265937 G.f.: Sum_{n>=0} (1 + x)^(n*(n+1)/2) / 2^n.

Original entry on oeis.org

2, 4, 24, 248, 3600, 67296, 1538672, 41593920, 1297683360, 45891815040, 1814072216864, 79263667304640, 3793393788125760, 197339219789611200, 11087608251010390080, 669127189486395204544, 43167108189991530605184, 2964541208087967215725440, 215934869210274766223069440, 16627513858173093851116296960, 1349582577808759197056647917696, 115158206188199564942934814336896, 10305721256666828267464573643658240

Offset: 0

Paul D. Hanna, Dec 23 2015

nonn

			G.f.: A(x) = 2 + 4*x + 24*x^2 + 248*x^3 + 3600*x^4 + 67296*x^5 + 1538672*x^6 + 41593920*x^7 + 1297683360*x^8 + 45891815040*x^9 + 1814072216864*x^10 +...
where
A(x) = 1 + (1+x)/2 + (1+x)^3/2^2 + (1+x)^6/2^3 + (1+x)^10/2^4 + (1+x)^15/2^5 + (1+x)^21/2^6 + (1+x)^28/2^7 + (1+x)^36/2^8 +...+ (1+x)^(n*(n+1)/2)/2^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A265936.

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

/* Informal listing of terms: */
{Vec( round( sum(n=0, 600, (1+x +O(x^31))^(n*(n+1)/2)/2^n * 1.) ) )}
{Vec( round( sum(n=0, 200, (1.+x)^n/2^n * prod(k=1, n, (2 - (1+x)^(2*k-1)) / (2 - (1+x)^(2*k)) +O(x^21) ) ) ) )}

G.f.: Sum_{n>=0} (1+x)^n/2^n * Product_{k=1..n} (2 - (1+x)^(2*k-1))/(2 - (1+x)^(2*k)) due to a q-series identity.
G.f.: 1/(1 - (1+x)/2 /(1 - (1+x)*((1+x)-1)/2 /(1 - (1+x)^3/2 /(1 - (1+x)^2*((1+x)^2-1)/2 /(1 - (1+x)^5/2 /(1 - (1+x)^3*((1+x)^3-1)/2 /(1 - (1+x)^7/2 /(1 - (1+x)^4*((1+x)^4-1)/2 /(1 - ...))))))))), a continued fraction due to a partial elliptic theta function identity.
a(n) = Sum_{k>=(sqrt(8*n+1)-1)/2} binomial(k*(k+1)/2,n) / 2^k.
a(n) = 2*A173219(n). - Vaclav Kotesovec, Oct 08 2019
a(n) ~ 2^(n+1) * n^n / (2^(log(2)/4) * log(2)^(2*n+1) * exp(n)). - Vaclav Kotesovec, Oct 08 2019

cp's OEIS Frontend

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

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A303920 G.f.: A(x,y) = (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2).

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A265937 G.f.: Sum_{n>=0} (1 + x)^(n*(n+1)/2) / 2^n.

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Mathematica

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