A156170
G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^n*x^k]^n/n ), a power series in x with integer coefficients.
Original entry on oeis.org
1, 1, 3, 10, 41, 219, 1602, 16635, 247171, 5242108, 157390565, 6663089873, 396778864166, 33200932308437, 3906922702271961, 646161881511137940, 150482521507292513413, 49318093291540113084965, 22790150225552744270503692, 14843990673285561887923674163, 13646527810852572644275538963207, 17710656073227095563348293151121448
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 41*x^4 + 219*x^5 + 1602*x^6 +...
log(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 +...+ A276750(n)*x^n/n +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n/n,
or,
log(A(x)) = (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 +...) +
(x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + 5^2*x^5 +...)^2/2 +
(x + 2^3*x^2 + 3^3*x^3 + 4^3*x^4 + 5^3*x^5 +...)^3/3 +
(x + 2^4*x^2 + 3^4*x^3 + 4^4*x^4 + 5^4*x^5 +...)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers like so:
log(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^2+n)/n +...
-
{a(n) = polcoeff( exp( sum(m=1,n, sum(k=1,n, k^m*x^k +x*O(x^n))^m/m ) ),n)}
for(n=0,30,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,m, A008292(m,k)*x^k/(1-x +Oxn)^(m+1) )^m / m ) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
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), ", "))
A292500
G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2*k-1)^n * x^k ]^n / n ).
Original entry on oeis.org
1, 1, 4, 18, 122, 1382, 26992, 967860, 59207134, 6539607238, 1225903048760, 407719392472476, 233686070341415140, 233030334505100451484, 407716349332865096406960, 1219594666823043463552070760, 6484753389847998264537623184230, 58288150472645787928029816422705798, 936721167715228772497787011017302901192, 25340260842241991639562678352357479545874188
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 122*x^4 + 1382*x^5 + 26992*x^6 + 967860*x^7 + 59207134*x^8 + 6539607238*x^9 + 1225903048760*x^10 + 407719392472476*x^11 + 233686070341415140*x^12 + 233030334505100451484*x^13 + 407716349332865096406960*x^14 + 1219594666823043463552070760*x^15 +...
RELATED SERIES.
log(A(x)) = x + 7*x^2/2 + 43*x^3/3 + 399*x^4/4 + 6091*x^5/5 + 151255*x^6/6 + 6550307*x^7/7 + 465127199*x^8/8 + 58293976795*x^9/9 + 12191724780647*x^10/10 + 4471204259257363*x^11/11 + 2799295142330495151*x^12/12 + 3026340345288168023883*x^13/13 + 5704756586858875194533367*x^14/14 +...+ A292502(n)*x^n/n +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 3^n*x^2 + 5^n*x^3 +...+ (2*k-1)^n*x^k +...)^n/n,
or,
log(A(x)) = (x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 +...) +
(x + 3^2*x^2 + 5^2*x^3 + 7^2*x^4 + 9^2*x^5 +...)^2/2 +
(x + 3^3*x^2 + 5^3*x^3 + 7^3*x^4 + 9^3*x^5 +...)^3/3 +
(x + 3^4*x^2 + 5^4*x^3 + 7^4*x^4 + 9^4*x^5 +...)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers of type B like so:
log(A(x)) = (x + x^2) / (1-x)^2 +
(x + 6*x^2 + x^3)^2 / (1-x)^6/2 +
(x + 23*x^2 + 23*x^3 + x^4)^3 / (1-x)^12/3 +
(x + 76*x^2 + 230*x^3 + 76*x^4 + x^5)^4 / (1-x)^20/4 +
(x + 237*x^2 + 1682*x^3 + 1682*x^4 + 237*x^5 + x^6)^5 / (1-x)^30/5 +
(x + 722*x^2 + 10543*x^3 + 23548*x^4 + 10543*x^5 + 722*x^6 + x^7)^6 / (1-x)^42/6 +
(x + 2179*x^2 + 60657*x^3 + 259723*x^4 + 259723*x^5 + 60657*x^6 + 2179*x^7 + x^8)^7 / (1-x)^56/7 +...+
[ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n / (1-x)^(n^2+n) * x^n/n +...
-
nmax = 20; CoefficientList[Series[Exp[Sum[2^(k^2) * x^k * LerchPhi[x, -k, 1/2]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 17 2020 *)
-
{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, (2*k-1)^m * x^k +x*O(x^n))^m/m ) ), n)}
for(n=0, 30, print1(a(n), ", "))
-
{A060187(n, k) = sum(j=1, k, (-1)^(k-j) * binomial(n, k-j) * (2*j-1)^(n-1))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = exp( sum(m=1,n+1, sum(k=0, m, A060187(m+1, k+1)*x^k)^m /(1-x +Oxn)^(m^2+m) * x^m/m ) );
polcoeff(A, n)}
for(n=0, 30, 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), ", "))
A292501
G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2*k-1)^n * x^k ]^n * (1-x)^n / n ).
Original entry on oeis.org
1, 1, 3, 13, 91, 1119, 23235, 879361, 55447631, 6274018595, 1192773105789, 400761393446831, 231147252957096671, 231434829013884972151, 406000810484101907916927, 1216355994930424625967455929, 6474418584620388915674215696687, 58229572245447428847208518694227279, 936163501254507409972001699357677028097, 25330794407893091120626418701416294765820223, 1224635875718403110628189182372406488768960029317
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 91*x^4 + 1119*x^5 + 23235*x^6 + 879361*x^7 + 55447631*x^8 + 6274018595*x^9 + 1192773105789*x^10 + 400761393446831*x^11 + 231147252957096671*x^12 + 231434829013884972151*x^13 + 406000810484101907916927*x^14 + 1216355994930424625967455929*x^15 +...
RELATED SERIES.
log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 305*x^4/4 + 5041*x^5/5 + 131477*x^6/6 + 5973311*x^7/7 + 436089793*x^8/8 + 55949083681*x^9/9 + 11863792842885*x^10/10 + 4395111080551775*x^11/11 + 2768928615166879025*x^12/12 + 3005637312940054635857*x^13/13 + 5680764740993004611483477*x^14/14 + 18239242940612856315412499071*x^15/15 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 3^n*x^2 + 5^n*x^3 +...+ (2*k-1)^n*x^k +...)^n * (1-x)^n/n,
or,
log(A(x)) = (x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 +...) * (1-x) +
(x + 3^2*x^2 + 5^2*x^3 + 7^2*x^4 + 9^2*x^5 +...)^2 * (1-x)^2/2 +
(x + 3^3*x^2 + 5^3*x^3 + 7^3*x^4 + 9^3*x^5 +...)^3 * (1-x)^3/3 +
(x + 3^4*x^2 + 5^4*x^3 + 7^4*x^4 + 9^4*x^5 +...)^4 * (1-x)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers of type B like so:
log(A(x)) = (x + x^2) / (1-x) +
(x + 6*x^2 + x^3)^2 / (1-x)^4/2 +
(x + 23*x^2 + 23*x^3 + x^4)^3 / (1-x)^9/3 +
(x + 76*x^2 + 230*x^3 + 76*x^4 + x^5)^4 / (1-x)^16/4 +
(x + 237*x^2 + 1682*x^3 + 1682*x^4 + 237*x^5 + x^6)^5 / (1-x)^25/5 +
(x + 722*x^2 + 10543*x^3 + 23548*x^4 + 10543*x^5 + 722*x^6 + x^7)^6 / (1-x)^36/6 +
(x + 2179*x^2 + 60657*x^3 + 259723*x^4 + 259723*x^5 + 60657*x^6 + 2179*x^7 + x^8)^7 / (1-x)^49/7 +...+
[ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n / (1-x)^(n^2) * x^n/n +...
-
{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n, (2*k-1)^m * x^k +x*O(x^n))^m*(1-x)^m/m ) ), n)}
for(n=0, 30, print1(a(n), ", "))
-
{A060187(n, k) = sum(j=1, k, (-1)^(k-j) * binomial(n, k-j) * (2*j-1)^(n-1))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = exp( sum(m=1,n+1, sum(k=0, m, A060187(m+1, k+1)*x^k)^m /(1-x +Oxn)^(m^2) * x^m/m ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-5 of 5 results.
Comments