A276906 -id:A276906 - cp's OEIS frontend

A276750 L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^n * x^k ]^n / n.

1, 5, 22, 117, 821, 7796, 101417, 1810093, 44561794, 1515368605, 71428667861, 4677209119632, 426268582440013, 54220470799325101, 9632796180856419722, 2397253932245127919389, 835827069207839232602401, 409329501365419311969616628, 281600921299273941316256813501, 272632759803890415543364253988037, 371636574592049013061911521355729422, 713832787857018847209335427225631327093

Offset: 1

Paul D. Hanna, Sep 17 2016

nonn

L.g.f. equals the logarithm of the g.f. of A156170.

			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 +...

Paul D. Hanna, Table of n, a(n) for n = 1..150

Cf. A156170, A276753, A276754, A276906, A008292, A292502.

{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), ", "))

L.g.f.: Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * x^k / (1-x)^(n+1) ]^n / n, where A008292 are the Eulerian numbers.

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

Paul D. Hanna, Sep 28 2016

nonn

L.g.f. equals the logarithm of the g.f. of A276906.

			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 +...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A276906, A276750.

{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), ", "))

{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), ", "))

L.g.f.: Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * x^(2*k-1) / (1-x^2)^(n+1) ]^n / n, where A008292 are the Eulerian numbers.

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A276750 L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^n * x^k ]^n / n.

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A276907 L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^n * x^(2*k-1) ]^n / n.

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