A181071
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(k+1) * n/(n-k).
Original entry on oeis.org
1, 3, 7, 15, 66, 357, 1891, 20559, 257605, 3436908, 96199478, 2734569969, 96260508267, 6820892444439, 438665726703387, 43006289605790127, 7366025744010911808, 1099005822684238964181, 309398207716948885643749
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 66*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*x + 3^3*x^2 + x^3)*x^3/3
+ (1+ 4^2*x + 6^3*x^2 + 4^4*x^3 + x^5)*x^4/4
+ (1+ 5^2*x + 10^3*x^2 + 10^4*x^3 + 5^5*x^4 + x^5)*x^5/5
+ (1+ 6^2*x + 15^3*x^2 + 20^4*x^3 + 15^5*x^4 + 6^6*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181070:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 23*x^5 + 88*x^6 + 379*x^7 + 3044*x^8 + ...
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[(&+[Binomial(n-j,j)^(j+1)*(n/(n-j)): j in [0..Floor(n/2)]]): n in [1..20]]; // G. C. Greubel, Apr 04 2021
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Table[Sum[Binomial[n-k,k]^(k+1) n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* Harvey P. Dale, Sep 25 2020 *)
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a(n)=sum(k=0, n\2, binomial(n-k, k)^(k+1)*n/(n-k))
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{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(k+1)*x^k)*x^m/m)+x*O(x^n), n)}
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[sum( binomial(n-k, k)^(k+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021
A181080
Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ).
Original entry on oeis.org
1, 1, 2, 4, 14, 83, 774, 10641, 255918, 14643874, 1752083557, 320079087261, 79294841767020, 27407454296637142, 16895839815165609994, 26064121763003372842186, 82824096391548076720149081
Offset: 0
G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 +...
The logarithm of g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + 4077*x^6/6 + ... + A181081(n)*x^n/n + ...
and equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1 + 3^3*x + 3^2*x^2 + x^3)*x^3/3
+ (1 + 4^4*x + 6^3*x^2 + 4^2*x^3 + x^4)*x^4/4
+ (1 + 5^5*x + 10^4*x^2 + 10^3*x^3 + 5^2*x^4 + x^5)*x^5/5
+ (1 + 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
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m:=20;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n-k+1)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
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With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n-k+1)*x^(n+k)/n, {k,0,n}], {n,m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^(m-k+1)*x^k)*x^m/m)+x*O(x^n)),n)}
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m=20;
def A181066_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n,k)^(n-k+1)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181066_list(m) # G. C. Greubel, Apr 05 2021
A181083
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).
Original entry on oeis.org
1, 3, 13, 111, 1686, 88737, 14355265, 3583775847, 1789371713317, 4311992850152298, 23667113846872049808, 185391762466214524964649, 4305238471804328835068596175, 468653724243371951619336632177235
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
+ (1 + 3^4*x + 3^5*x^2 + x^3)*x^3/3
+ (1 + 4^5*x + 6^6*x^2 + 4^7*x^3 + x^4)*x^4/4
+ (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 + 5^9*x^4 + x^5)*x^5/5
+ (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181082:
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 + 2066401*x^7 + ...
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[(&+[Binomial(n-k,k)^n*(n/(n-k)): k in [0..Floor(n/2)]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
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A181083:= n-> add(binomial(n-k,k)^n*(n/(n-k)), k=0..floor(n/2)); seq(A181083(n), n=1..20); # G. C. Greubel, Apr 05 2021
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Table[Sum[Binomial[n-k,k]^n n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* Harvey P. Dale, Jun 24 2015 *)
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a(n)=sum(k=0, n\2, binomial(n-k, k)^n*n/(n-k))
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{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n), n)}
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[sum(binomial(n-k,k)^n*(n/(n-k)) for k in (0..n//2)) for n in (1..25)] # G. C. Greubel, Apr 05 2021
A181085
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).
Original entry on oeis.org
1, 3, 25, 327, 6336, 513657, 142074241, 52903930911, 36806786795365, 148308705637730728, 1318954828711012426638, 15279013243159345043036553, 534104982404807772659968455891, 97749134742042348389685885848315523
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
+ (1+ 3^5*x + 3^6*x^2 + x^3)*x^3/3
+ (1+ 4^6*x + 6^7*x^2 + 4^8*x^3 + x^4)*x^4/4
+ (1+ 5^7*x + 10^8*x^2 + 10^9*x^3 + 5^10*x^4 + x^5)*x^5/5
+ (1+ 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181084:
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...
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[(&+[Binomial(n-j,j)^(n+1)*(n/(n-j)): j in [0..Floor(n/2)]]): j in [1..20]]; // G. C. Greubel, Apr 04 2021
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Table[Sum[Binomial[n-k, k]^(n+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* G. C. Greubel, Apr 04 2021 *)
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a(n)=sum(k=0, n\2, binomial(n-k, k)^(n+1)*n/(n-k))
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{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k+1)*x^k)*x^m/m)+x*O(x^n), n)}
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[sum( binomial(n-k, k)^(n+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021
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