A184730
G.f.: exp( Sum_{n>=1} A184731(n)*x^n/n ) where A184731(n) = Sum_{k=0..n} C(n,k)^(k+1).
Original entry on oeis.org
1, 2, 5, 20, 159, 3152, 168036, 20428850, 5796209814, 4052041564524, 6210335115944263, 21470958882165989854, 183818137919395949397148, 3517964195874870876682733562, 147909303669340763210833833705995, 15391220509661795085065182391703575606
Offset: 0
G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 159*x^4 + 3152*x^5 +...
log(A(x)) = 2*x + 6*x^2/2 + 38*x^3/3 + 490*x^4/4 + 14152*x^5/5 + 969444*x^6/6 +...+ A184731(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^(k+1))*x^m/m)+x*O(x^n)), n)}
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
A181074
Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(k+1) *x^k ] *x^n/n ).
Original entry on oeis.org
1, 1, 2, 5, 23, 231, 5405, 322799, 42761356, 12597156231, 9136063939651, 14655841196011960, 51639276405198967750, 449212631407010945983244, 8871353886432410987179493370, 378793180251425841753491012596531
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + 30912*x^6/6 +...+ A181075(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 + 3^3*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 + ...
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m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
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With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1,k]^(k+1)*x^(n+k)/n, {k,0,m+2}], {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, n, binomial(m+k-1,k)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}
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m=30;
def A181066_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n+k-1,k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181066_list(m) # G. C. Greubel, Apr 05 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
A181078
Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1) *x^k ] *x^n/n ).
Original entry on oeis.org
1, 1, 2, 5, 29, 657, 61207, 22168009, 29875987984, 155804714312491, 3016989471632014921, 229552430038667549657248, 64995077386747098368845127628, 73163996832774559516266954450479682
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + 363132*x^6/6 + ... + A181079(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 + 3^3*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 + ...
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m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^(n+k-1)*x^(n+k)/n : k in [0..m+2]]): 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-1,k]^(n+k-1)*x^(n+k)/n, {k,0,m+2}], {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, n, binomial(m+k-1,k)^(m+k-1)*x^k)*x^m/m)+x*O(x^n)), n)}
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m=30;
def A181078_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n+k-1,k)^(n+k-1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181078_list(m) # G. C. Greubel, Apr 05 2021
A228899
Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1
Offset: 0
This triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 12, 1;
1, 10, 71, 76, 1;
1, 15, 281, 2153, 701, 1;
1, 21, 861, 29166, 129509, 8477, 1;
1, 28, 2212, 244725, 7664343, 12391414, 126126, 1;
1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1;
1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ...
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x
+ (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3
+ (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4
+ (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5
+ (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +...
in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1).
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{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
Showing 1-6 of 6 results.
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