A135746
E.g.f.: A(x) = Sum_{n>=0} exp(n^2*x) * x^n/n!.
Original entry on oeis.org
1, 1, 3, 16, 137, 1536, 22417, 407884, 8920641, 230576320, 6928080641, 238375169484, 9288784476193, 406150114297552, 19761959813464065, 1062437048084297596, 62727815353861478273, 4045278841893314992896
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1536*x^5/5! + ...
where A(x) = 1 + exp(x)*x + exp(4*x)*x^2/2! + exp(9*x)*x^3/3! + exp(16*x)*x^4/4! + exp(25*x)*x^5/5! + ...
O.g.f.: F(x) = 1 + x + 3*x^2 + 16*x^3 + 137*x^4 + 1536*x^5 + 22417*x^6 + ...
where F(x) = 1 + x/(1-x)^2 + x^2/(1-4*x)^3 + x^3/(1-9*x)^4 + x^4/(1-16*x)^5 + x^5/(1-25*x)^6 + ...
-
Flatten[{1, Table[Sum[Binomial[n, k]*k^(2*(n - k)), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)
-
{a(n)=sum(k=0,n,binomial(n,k)*(k^2)^(n-k))}
-
{a(n)=n!*polcoeff(sum(k=0,n,exp(k^2*x +x*O(x^n))*x^k/k!),n)}
-
{a(n)=polcoeff(sum(k=0, n, x^k/(1-k^2*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Aug 08 2009
A135742
E.g.f.: A(x) = Sum_{n>=0} exp( n*(n-1)/2 * x ) * x^n / n!.
Original entry on oeis.org
1, 1, 1, 4, 19, 131, 1156, 12622, 166825, 2600677, 47038456, 974165336, 22829939089, 599668759483, 17512623094240, 564613124026876, 19972670155565761, 771019774737952313, 32326390781950804048
Offset: 0
-
Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[k, 2]^(n - k), {k, 0, n}], {n,1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)
-
{a(n)=sum(k=0,n,binomial(n,k)*(k*(k-1)/2)^(n-k))}
for(n=0,25,print1(a(n),", "))
-
{a(n)=n!*polcoeff(sum(k=0,n,exp(k*(k-1)/2*x +x*O(x^n))*x^k/k!),n)}
for(n=0,25,print1(a(n),", "))
-
/* From Sum_{n>=0} x^n/(1 - n*(n-1)/2*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k-1)/2*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))
A135744
E.g.f.: A(x) = Sum_{n>=0} exp( n*(n+1)*x ) * x^n/n!.
Original entry on oeis.org
1, 1, 5, 31, 297, 3781, 60373, 1188699, 28003825, 772499593, 24613769061, 894386029879, 36653691106585, 1679283513161229, 85360981759418485, 4781873479864452211, 293487961919565420897, 19633825959679051986961
Offset: 0
-
Flatten[{1, Table[Sum[Binomial[n, k]*(2*Binomial[k + 1, 2])^(n - k), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)
-
{a(n)=sum(k=0,n,binomial(n,k)*(k*(k+1))^(n-k))}
for(n=0,25,print1(a(n),", "))
-
{a(n)=n!*polcoeff(sum(k=0,n,exp(k*(k+1)*x +x*O(x^n))*x^k/k!),n)}
for(n=0,25,print1(a(n),", "))
-
/* From Sum_{n>=0} x^n/(1 - n*(n+1)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k+1)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))
A135745
E.g.f.: A(x) = Sum_{n>=0} exp((n-1)*x)^n * x^n/n!.
Original entry on oeis.org
1, 1, 1, 7, 49, 501, 6841, 115123, 2362305, 57768553, 1646192881, 53952383871, 2010872281969, 84330050952733, 3945169959883881, 204416253047774251, 11655594262050124801, 727189793270478477777, 49395902623624761264865
Offset: 0
-
Flatten[{1, Table[Sum[Binomial[n, k]*(k*(k - 1))^(n - k), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)
-
{a(n)=sum(k=0,n,binomial(n,k)*(k*(k-1))^(n-k))}
for(n=0,25,print1(a(n),", "))
-
{a(n)=n!*polcoeff(sum(k=0,n,exp(k*(k-1)*x +x*O(x^n))*x^k/k!),n)}
for(n=0,25,print1(a(n),", "))
-
/* From Sum_{n>=0} x^n/(1 - n*(n-1)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k-1)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))
A135747
E.g.f.: A(x) = Sum_{n>=0} exp( (n^2-1)*x ) * x^n/n!.
Original entry on oeis.org
1, 0, 2, 9, 88, 985, 14976, 278929, 6208000, 163268865, 4979147680, 173500986241, 6838921208736, 302161792811905, 14840867887070512, 804732692174218305, 47888731015720316416, 3110871265807567331329, 219546952410733092279360
Offset: 0
-
Flatten[{1, Table[Sum[Binomial[n, k]*(k^2 - 1)^(n - k), {k, 0, n}], {n,1,25}]}] (* G. C. Greubel, Nov 05 2016 *)
-
{a(n)=sum(k=0,n,binomial(n,k)*(k^2-1)^(n-k))}
for(n=0,25,print1(a(n),", "))
-
{a(n)=n!*polcoeff(sum(k=0,n,exp((k^2-1)*x +x*O(x^n))*x^k/k!),n)}
for(n=0,25,print1(a(n),", "))
-
/* From Sum_{n>=0} x^n/(1 - (n^2-1)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(k^2-1)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))
Showing 1-5 of 5 results.
Comments