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 +...
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{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),", "))
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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 = 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),", "))
A276743
G.f.: Sum_{n>=0} [ Sum_{k>=1} k^n * x^k ]^n.
Original entry on oeis.org
1, 1, 3, 12, 63, 447, 4286, 56185, 1008317, 24917676, 849963761, 40142633815, 2633061525012, 240207555735097, 30578843349537575, 5434894746337720676, 1352812180415380719387, 471689727423751377883607, 230943183470327388401886858, 158839247095790148049487792081, 153694547774391577758847456894905
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 63*x^4 + 447*x^5 + 4286*x^6 + 56185*x^7 + 1008317*x^8 + 24917676*x^9 + 849963761*x^10 +...
such that
A(x) = Sum_{n>=0} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.
Explicitly,
A(x) = 1 + (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...) +
(x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 +...)^2 +
(x + 8*x^2 + 27*x^3 + 64*x^4 + 125*x^5 + 216*x^6 +...)^3 +
(x + 16*x^2 + 81*x^3 + 256*x^4 + 625*x^5 + 1296*x^6 +...)^4 +
(x + 32*x^2 + 243*x^3 + 1024*x^4 + 3125*x^5 + 7776*x^6 +...)^5 +
...
The g.f. can be written using the Eulerian numbers like so:
A(x) = 1 + x/(1-x)^2 + (x + x^2)^2/(1-x)^6 + (x + 4*x^2 + x^3)^3/(1-x)^12 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)) +...
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{a(n) = my(A=1);
A = sum(m=0,n+1, sum(k=1,n+1, k^m * x^k +x*O(x^n))^m); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
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/* Using Eulerian numbers A008292 */
{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=0, n+1, sum(k=1, m, A008292(m, k)*x^k/(1-x +Oxn)^(m+1) )^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A292502
L.g.f.: Sum_{n>=1} [ Sum_{k>=1} (2*k-1)^n * x^k ]^n / n.
Original entry on oeis.org
1, 7, 43, 399, 6091, 151255, 6550307, 465127199, 58293976795, 12191724780647, 4471204259257363, 2799295142330495151, 3026340345288168023883, 5704756586858875194533367, 18287793731664040419412785283, 103736521111190203113027053903423, 990788254951454647260121962687606203, 16859931481746848392491523248553253264263, 481447154976629475966161111465088882379644147
Offset: 1
L.g.f: 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 + ...
The l.g.f. A(x) equals the series:
A(x) = Sum_{n>=1} (x + 3^n*x^2 + 5^n*x^3 + ... + (2*k-1)^n*x^k + ...)^n/n,
or,
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:
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 + ...
Exponentiation yields the g.f. of A292500:
exp(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 + ... + A292500(n)*x^n + ...
which is an integer series.
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{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 = 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 );
n*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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{a(n) = n*polcoeff( sum(m=1, n+1, sum(k=1, n, (2*k-1)^m * x^k +x*O(x^n))^m/m ), n)}
for(n=1, 30, print1(a(n), ", "))
A276907
L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^n * x^(2*k-1) ]^n / n.
Original entry on oeis.org
1, 1, 7, 17, 56, 199, 890, 4649, 27817, 195946, 1684398, 17397323, 208799982, 2932164012, 49785808832, 1022745137705, 24671296028079, 695270673553051, 23526126768837873, 965093874912658722, 46827415587504280547, 2655503102769481320544, 179856174616910379655073, 14761130793635395568878091, 1439881917495260610082685956, 164363140573098989525137162900, 22322323085863965044351721067969
Offset: 1
L.g.f.: 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 +...
such that A(x) equals the series:
A(x) = Sum_{n>=1} (x + 2^n*x^3 + 3^n*x^5 +...+ k^n*x^(2*k-1) +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
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 +...
where
exp(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 +...+ A276906(n)*x^n +...
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{a(n) = n * polcoeff( sum(m=1, n, sum(k=1, n, k^m * x^(2*k-1) +x*O(x^n))^m/m ), n)}
for(n=1, 30, 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, m, A008292(m, k) * x^(2*k-1)/(1-x^2 +Oxn)^(m+1) )^m / m ); n*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-4 of 4 results.
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