A276751
G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(2*n-1) * x^k]^n / n ), a power series in x with integer coefficients.
Original entry on oeis.org
1, 1, 3, 14, 111, 1813, 57846, 3941129, 515554887, 139563384274, 73929755773659, 78682910542834037, 169524995438153307498, 712160156293232925362965, 6241130803695426404771763891, 104223975880844169453617144998346, 3697419824526049703366356719095712903, 247087719554207540966918934263865223952113, 35252698554630762784745670915756020139337705854, 9472029798481852471047526788494040155248502738148149
Offset: 0
G.f.: 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 +...
log(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 + 864695662715974585*x^11/11 + 2033353960345783330704*x^12/12 +...
The logarithm of g.f. A(x) equals the series:
log(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:
log(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 +...
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{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^(2*m-1) * 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-1, A008292(2*m-1, k)*x^k/(1-x +Oxn)^(2*m) )^m / m ) ); polcoeff(A, n)}
for(n=0, 20, 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), ", "))
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 +...
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{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), ", "))
A156171
G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.
Original entry on oeis.org
1, 1, 3, 11, 53, 357, 3521, 51665, 1122135, 35638903, 1639453459, 108526044099, 10298220348807, 1396920580458279, 270394562069007327, 74574294532698008703, 29276455806256470979269, 16344863466384180848085765, 12969208162308705691408055345, 14616452655308018025267503353697
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 53*x^4 + 357*x^5 + 3521*x^6 + 51665*x^7 + 1122135*x^8 + 35638903*x^9 + 1639453459*x^10 + 108526044099*x^11 +...
such that:
log(A(x)) = Sum_{n>=1} x^n/n * (1 + 2^n*x + 4^n*x^2 +...+ 2^(n*k)*x^k +...)^n
or
log(A(x)) = x*(1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...) +
x^2/2*(1 + 8*x + 48*x^2 + 256*x^3 + 1280*x^4 + 6144*x^5 +...) +
x^3/3*(1 + 24*x + 384*x^2 + 5120*x^3 + 61440*x^4 + 688128*x^5 +...) +
x^4/4*(1 + 64*x + 2560*x^2 + 81920*x^3 + 2293760*x^4 + 58720256*x^5 +...) +
x^5/5*(1 + 160*x + 15360*x^2 + 1146880*x^3 + 73400320*x^4 + 4227858432*x^5 +...) +
x^6/6*(1 + 384*x + 86016*x^2 + 14680064*x^3 + 2113929216*x^4 + 270582939648*x^5 +...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 161*x^4/4 + 1441*x^5/5 + 18305*x^6/6 + 330625*x^7/7 + 8488961*x^8/8 + 309465601*x^9/9 + 16011372545*x^10/10 + 1174870185985*x^11/11 + 122233833963521*x^12/12 +...
-
nmax = 20; CoefficientList[Series[Exp[Sum[x^k/(1 - 2^k*x)^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 17 2020 *)
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{a(n)=polcoeff(exp(sum(m=1,n,x^m/(1-2^m*x+x*O(x^n))^m/m)),n)}
A159596
G.f.: A(x) = exp( Sum_{n>=1} [ D^n x/(1-x)^2 ]^n/n ), where differential operator D = x*d/dx.
Original entry on oeis.org
1, 1, 5, 22, 121, 863, 8476, 118131, 2361313, 67467236, 2731757961, 156417295405, 12605225573076, 1432381581679361, 229016092616239411, 51628631138952017332, 16402709158903948390585, 7351149638643155728435357
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 22*x^3 + 121*x^4 + 863*x^5 +...
log(A(x)) = Sum_{n>=1} [x + 2^(n+1)*x^2 + 3^(n+1)*x^3 +...]^n/n.
D^n x/(1-x)^2 = x + 2^(n+1)*x^2 + 3^(n+1)*x^3 + 4^(n+1)*x^4 +...
-
{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,k^(m+1)*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
A276754
L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^(2*n) * x^k ]^n / n.
Original entry on oeis.org
1, 9, 76, 1157, 33291, 1792296, 196919213, 39766253741, 16931726147956, 13298466280839329, 22076711237844558263, 69166686377284889199104, 448760359479425463648647769, 5685081590883001302122022078913, 144528951819771627855280850227089996, 7431791795502279858136165452572662669213, 743200333842768450767851829731370148558347843, 154769006272445896954868694741314742556915451805336
Offset: 1
L.g.f.: 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 +...
such that A(x) equals the series:
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:
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 +...
where
exp(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 +...+ A276752(n)*x^n +...
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{a(n) = n * polcoeff( sum(m=1, n, sum(k=1, n, k^(2*m)*x^k +x*O(x^n))^m/m ), n)}
for(n=1, 20, print1(a(n), ", "))
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{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, A008292(2*m, k)*x^k/(1-x +Oxn)^(2*m+1) )^m / m ); n * polcoeff(A, n)}
for(n=1, 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 +...
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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 *)
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{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), ", "))
A276906
G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^n * x^(2*k-1) ]^n / n ), a power series in x with integer coefficients.
Original entry on oeis.org
1, 1, 1, 3, 7, 18, 53, 188, 799, 4001, 24050, 179248, 1639637, 17764040, 227653634, 3550628492, 67513114323, 1519274903363, 40153164845377, 1278514703044023, 49536414234360980, 2279497269454146657, 122986833567853232448, 7942922462379370617039, 622994706862172074402587, 58218522316121110190816538, 6379893924028925326363565894
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 18*x^5 + 53*x^6 + 188*x^7 + 799*x^8 + 4001*x^9 + 24050*x^10 + 179248*x^11 + 1639637*x^12 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^3 + 3^n*x^5 +...+ k^n*x^(2*k-1) +...)^n/n.
Explicitly,
log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 56*x^5/5 + 199*x^6/6 + 890*x^7/7 + 4649*x^8/8 + 27817*x^9/9 + 195946*x^10/10 + 1684398*x^11/11 + 17397323*x^12/12 +...+ A276907(n)*x^n/n +...
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x^2)^2 + (x + x^3)^2/(1-x^2)^6/2 + (x + 4*x^3 + x^5)^3/(1-x^2)^12/3 + (x + 11*x^3 + 11*x^5 + x^7)^4/(1-x^2)^20/4 + (x + 26*x^3 + 66*x^5 + 26*x^7 + x^9)^5/(1-x^2)^30/5 + (x + 57*x^3 + 302*x^5 + 302*x^7 + 57*x^9 + x^11)^6/(1-x^2)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * x^(2*k-1) ]^n / (1-x^2)^(n*(n+1))/n +...
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{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^m*x^(2*k-1) +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+1, A008292(m, k)*x^(2*k-1)/(1-x^2 +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A159597
G.f.: A(x) = exp( Sum_{n>=1} [ D^n x/(1-x)^3 ]^n/n ), where differential operator D = x*d/dx.
Original entry on oeis.org
1, 1, 7, 37, 245, 2094, 24661, 410376, 9809637, 334520167, 16192227784, 1107914634442, 106788033119369, 14525652771018918, 2780328926392863928, 751651711717655433750, 286240041470280077141769
Offset: 0
G.f.: A(x) = 1 + x + 7*x^2 + 37*x^3 + 245*x^4 + 2094*x^5 +...
log(A(x)) = Sum_{n>=1} [x + 2^n*3*x^2 + 3^n*6*x^3 +...]^n/n.
D^n x/(1-x)^3 = x + 2^n*3*x^2 + 3^n*6*x^3 + 4^n*10*x^4 +...
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,k^m*k*(k+1)/2*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
A159598
G.f.: A(x) = exp( Sum_{n>=1} [ D^n x(1+x)/(1-x)^3 ]^n/n ), where differential operator D = x*d/dx.
Original entry on oeis.org
1, 1, 9, 52, 389, 3741, 49908, 938799, 25477165, 984680146, 54180019253, 4211350678751, 462028240134476, 71561459522839253, 15611478225943599423, 4816139618587302209166, 2092942812095475521879845
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 52*x^3 + 389*x^4 + 3741*x^5 +...
log(A(x)) = Sum_{n>=1} [x + 2^(n+2)*x^2 + 3^(n+2)*x^3 +...]^n/n.
D^n x(1+x)/(1-x)^2 = x + 2^(n+2)*x^2 + 3^(n+2)*x^3 + 4^(n+2)*x^4 +...
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,k^(m+2)*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
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