A276752
G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(2*n) * x^k]^n / n ), a power series in x with integer coefficients.
Original entry on oeis.org
1, 1, 5, 30, 327, 7085, 307280, 28472653, 5000661017, 1886425568702, 1331753751874235, 2008313162512681569, 5765904212733638946976, 34525801618218187545094977, 406111805399407205212602871837, 9635669704681654899673855841540822, 464496624513770925349468939192278531231, 43718131231809168093455159164707384418710045, 8598321846236415035740539472279473819390935625008
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 30*x^3 + 327*x^4 + 7085*x^5 + 307280*x^6 + 28472653*x^7 + 5000661017*x^8 + 1886425568702*x^9 + 1331753751874235*x^10 +...
log(A(x)) = x + 9*x^2/2 + 76*x^3/3 + 1157*x^4/4 + 33291*x^5/5 + 1792296*x^6/6 + 196919213*x^7/7 + 39766253741*x^8/8 + 16931726147956*x^9/9 + 13298466280839329*x^10/10 + 22076711237844558263*x^11/11 + 69166686377284889199104*x^12/12 +...+ A276754(n)*x^n/n +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^(2*n)*x^2 + 3^(2*n)*x^3 +...+ k^(2*n)*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = (x + x^2)/(1-x)^3 + (x + 11*x^2 + 11*x^3 + x^4)^2/(1-x)^10/2 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^3/(1-x)^21/3 + (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)^4/(1-x)^36/4 + (x + 1013*x^2 + 47840*x^3 + 455192*x^4 + 1310354*x^5 + 1310354*x^6 + 455192*x^7 + 47840*x^8 + 1013*x^9 + x^10)^5/(1-x)^55/5 + (x + 4083*x^2 + 478271*x^3 + 10187685*x^4 + 66318474*x^5 + 162512286*x^6 + 162512286*x^7 + 66318474*x^8 + 10187685*x^9 + 478271*x^10 + 4083*x^11 + x^12)^6/(1-x)^78/6 +...+ [ Sum_{k=1..2*n} A008292(2*n,k) * x^k ]^n / (1-x)^(2*n^2+n) /n +...
-
{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^(2*m) * x^k +x*O(x^n))^m / m ) ), n)}
for(n=0, 20, print1(a(n), ", "))
-
{A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, 2*m, A008292(2*m, k)*x^k/(1-x +Oxn)^(2*m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A276750
L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^n * x^k ]^n / n.
Original entry on oeis.org
1, 5, 22, 117, 821, 7796, 101417, 1810093, 44561794, 1515368605, 71428667861, 4677209119632, 426268582440013, 54220470799325101, 9632796180856419722, 2397253932245127919389, 835827069207839232602401, 409329501365419311969616628, 281600921299273941316256813501, 272632759803890415543364253988037, 371636574592049013061911521355729422, 713832787857018847209335427225631327093
Offset: 1
L.g.f.: A(x) = x + 5*x^2/2 + 22*x^3/3 + 117*x^4/4 + 821*x^5/5 + 7796*x^6/6 + 1810093*x^7/7 + 44561794*x^8/8 + 1515368605*x^9/9 + 71428667861*x^10/10 +...
such that A(x) equals the series:
A(x) = Sum_{n>=1} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
A(x) = x/(1-x)^2 + (x + x^2)^2/(1-x)^6/2 + (x + 4*x^2 + x^3)^3/(1-x)^12/3 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20/4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30/5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1))/n +...
where
exp(A(x)) = 1 + x + 3*x^2 + 10*x^3 + 41*x^4 + 219*x^5 + 1602*x^6 + 16635*x^7 + 247171*x^8 + 5242108*x^9 + 157390565*x^10 +...+ A156170(n)*x^n +...
-
{a(n) = n * polcoeff( sum(m=1, n, sum(k=1, n, k^m*x^k +x*O(x^n))^m/m ), n)}
for(n=1, 20, print1(a(n), ", "))
-
{A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = sum(m=1, n+1, sum(k=1, m, A008292(m, k)*x^k/(1-x +Oxn)^(m+1) )^m / m ); n*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
A276753
L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^(2*n-1) * x^k ]^n / n.
Original entry on oeis.org
1, 5, 34, 381, 8401, 334688, 27151993, 4091831133, 1251353635162, 737891198902325, 864695662715974585, 2033353960345783330704, 9255876152303901497918425, 87365856252845525476020365429, 1563265999862817889675899566032954, 59157049408983740505063226640565220029, 4200428372739940183291465697348398947046393, 634544126271277747190512917479290795324884131840
Offset: 1
L.g.f.: A(x) = x + 5*x^2/2 + 34*x^3/3 + 381*x^4/4 + 8401*x^5/5 + 334688*x^6/6 + 27151993*x^7/7 + 4091831133*x^8/8 + 1251353635162*x^9/9 + 737891198902325*x^10/10 +...
such that A(x) equals the series:
A(x) = Sum_{n>=1} (x + 2^(2*n-1)*x^2 + 3^(2*n-1)*x^3 +...+ k^(2*n-1)*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
A(x) = x/(1-x)^2 + (x + 4*x^2 + x^3)^2/(1-x)^8/2 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^3/(1-x)^18/3 + (x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7)^4/(1-x)^32/4 + (x + 502*x^2 + 14608*x^3 + 88234*x^4 + 156190*x^5 + 88234*x^6 + 14608*x^7 + 502*x^8 + x^9)^5/(1-x)^50/5 + (x + 2036*x^2 + 152637*x^3 + 2203488*x^4 + 9738114*x^5 + 15724248*x^6 + 9738114*x^7 + 2203488*x^8 + 152637*x^9 + 2036*x^10 + x^11)^6/(1-x)^72/6 +...+ [ Sum_{k=1..2*n-1} A008292(2*n-1,k) * x^k ]^n / (1-x)^(2*n^2) /n +...
where
exp(A(x)) = 1 + x + 3*x^2 + 14*x^3 + 111*x^4 + 1813*x^5 + 57846*x^6 + 3941129*x^7 + 515554887*x^8 + 139563384274*x^9 + 73929755773659*x^10 +...+ A276751(n)*x^n +...
-
{a(n) = n * polcoeff( sum(m=1, n, sum(k=1, n, k^(2*m-1)*x^k +x*O(x^n))^m/m ), n)}
for(n=1, 20, print1(a(n), ", "))
-
{A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = sum(m=1, n+1, sum(k=1, 2*m-1, A008292(2*m-1, k)*x^k/(1-x +Oxn)^(2*m) )^m / m ); n * polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
Showing 1-3 of 3 results.
Comments