A181076
G.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^n *x^k ] *x^n/n ).
Original entry on oeis.org
1, 1, 2, 5, 20, 168, 3659, 204644, 25503314, 7434144333, 5248999682258, 8079852389207554, 28328874782544308254, 244277149833867010587231, 4673118265932181394325207044, 198007423467261943865049734612821
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 20*x^4 + 168*x^5 + 3659*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 59*x^4/4 + 726*x^5/5 + 20832*x^6/6 +...+ A181077(n)*x^n/n +...
which equals the series:
log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
+ (1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 + 6^2*x^5 +...)*x^2/2
+ (1 + 3^3*x + 6^3*x^2 + 10^3*x^3 + 15^3*x^4 + 21^3*x^5 +...)*x^3/3
+ (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 +...)*x^4/4
+ (1 + 5^5*x + 15^5*x^2 + 35^5*x^3 + 70^5*x^4 + 126^5*x^5 +...)*x^5/5
+ (1 + 6^6*x + 21^6*x^2 + 56^6*x^3 + 126^6*x^4 + 252^6*x^5 +...)*x^6/6
+ (1 + 7^7*x + 28^7*x^2 + 84^7*x^3 + 210^7*x^4 + 462^7*x^5 +...)*x^7/7 +...
-
{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^m*x^k)*x^m/m)+x*O(x^n)), n)}
A181075
a(n) = Sum_{k=0..n-1} C(n-1,k)^(k+1) * n/(n-k).
Original entry on oeis.org
1, 3, 10, 71, 1026, 30912, 2219946, 339460991, 112986526834, 91234232847938, 161113616883239406, 619495336824891912596, 5839092706931985694730356, 124192664709851995516427897172, 5681764626723349386531457243004370
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + ...
which equals the series:
L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
+ (1 + 2^2*x + 3^3*x^2 + 4^4*x^3 + 5^5*x^4 + 6^6*x^5 + ...)*x^2/2
+ (1 + 3^2*x + 6^3*x^2 + 10^4*x^3 + 15^5*x^4 + 21^6*x^5 + ...)*x^3/3
+ (1 + 4^2*x + 10^3*x^2 + 20^4*x^3 + 35^5*x^4 + 56^6*x^5 + ...)*x^4/4
+ (1 + 5^2*x + 15^3*x^2 + 35^4*x^3 + 70^5*x^4 + 126^6*x^5 + ...)*x^5/5
+ (1 + 6^2*x + 21^3*x^2 + 56^4*x^3 + 126^5*x^4 + 252^6*x^5 + ...)*x^6/6
+ (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181074:
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 + ...
-
[(&+[Binomial(n-1, k)^(k+1)*n/(n-k): k in [0..n-1]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
-
Table[Sum[Binomial[n-1, k]^(k+1)*n/(n-k), {k,0,n-1}], {n,25}] (* G. C. Greubel, Apr 05 2021 *)
-
{a(n)=sum(k=0, n-1, binomial(n-1, k)^(k+1)*n/(n-k))}
-
{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(k+1)*x^k)*x^m/m)+x*O(x^n), n)}
-
[sum(binomial(n-1,k)^(k+1)*n/(n-k) for k in (0..n-1)) for n in (1..25)] # G. C. Greubel, Apr 05 2021
A181079
a(n) = Sum_{k=0..n-1} binomial(n-1,k)^(n-1) * n/(n-k).
Original entry on oeis.org
1, 3, 10, 95, 3126, 363132, 154742736, 238830058287, 1401973344195850, 30168336369959767298, 2525043541826640689536056, 779938173975597096091742711900, 951131113887078985926203597341181404
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + ...
which equals the series:
L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...)*x
+ (1 + 2^2*x + 3^3*x^2 + 4^4*x^3 + 5^5*x^4 + 6^6*x^5 + ...)*x^2/2
+ (1 + 3^3*x + 6^4*x^2 + 10^5*x^3 + 15^6*x^4 + 21^7*x^5 + ...)*x^3/3
+ (1 + 4^4*x + 10^5*x^2 + 20^6*x^3 + 35^7*x^4 + 56^8*x^5 + ...)*x^4/4
+ (1 + 5^5*x + 15^6*x^2 + 35^7*x^3 + 70^8*x^4 + 126^9*x^5 + ...)*x^5/5
+ (1 + 6^6*x + 21^7*x^2 + 56^8*x^3 + 126^9*x^4 + 252^10*x^5 + ...)*x^6/6
+ (1 + 7^7*x + 28^8*x^2 + 84^9*x^3 + 210^10*x^4 + 462^11*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181078:
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 + … + A181078(n)*x^n + ...
-
[(&+[Binomial(n-1,j)^(n-1)*(n/(n-j)): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Apr 04 2021
-
Table[Sum[Binomial[n-1,k]^(n-1) n/(n-k),{k,0,n-1}],{n,20}] (* Harvey P. Dale, Jun 13 2013 *)
-
{a(n)=sum(k=0, n-1, binomial(n-1, k)^(n-1)*n/(n-k))}
-
{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(m+k-1)*x^k)*x^m/m)+x*O(x^n), n)}
-
[sum( binomial(n-1, k)^(n-1)*(n/(n-k)) for k in (0..n-1)) for n in (1..20)] # G. C. Greubel, Apr 04 2021
Showing 1-3 of 3 results.
Comments