A219504
E.g.f.: Sum_{n>=0} a(n) * (cos(n*x)-sin(n*x)) * x^n/n! = 1/(1-x).
Original entry on oeis.org
1, 1, 4, 33, 512, 13005, 494976, 26383917, 1876721664, 171728626617, 19650536857600, 2749029193911033, 461590186944847872, 91611982632843733125, 21215197576393952452608, 5669317752667727770720965, 1731566894935958076783067136, 599421136964093700021081229041
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 512*x^4/4! + 13005*x^5/5! +...
By definition, the coefficients a(n) satisfy:
1/(1-x) = 1 + 1*(cos(x)-sin(x))*x + 4*(cos(2*x)-sin(2*x))*x^2/2! + 33*(cos(3*x)-sin(3*x))*x^3/3! + 512*(cos(4*x)-sin(4*x))*x^4/4! + 13005*(cos(5*x)-sin(5*x))*x^5/5! +...+ a(n)*(cos(n*x)-sin(n*x))*x^n/n! +...
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a[0] := 1; a[1] := 1; a[n_] := n! + Sum[(-1)^(Floor[(n -k-1)/2]) *Binomial[n, k]*k^(n - k)*a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 18 2017 *)
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{a(n)=local(A=[1],N);for(i=1,n,A=concat(A,0);N=#A;A[N]=(N-1)!*(1-Vec(sum(m=0,N-1,A[m+1]*x^m/m!*(cos(m*x+x*O(x^N))-sin(m*x+x*O(x^N)))))[N]));A[n+1]}
for(n=0,25,print1(a(n),", "))
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{a(n)=if(n==0|n==1, 1, n!+sum(k=1, n-1, (-1)^((n-k-1)\2)*a(k)*binomial(n, k)*k^(n-k)))}
for(n=0, 25, print1(a(n), ", "))
A221535
E.g.f.: Sum_{n>=0} a(n) * (cos(n^2*x) - sin(n^2*x)) * x^n/n! = 1 + x.
Original entry on oeis.org
1, 1, 2, 27, 1160, 113385, 21060816, 6623049027, 3256046222848, 2359986757857297, 2411094849547390720, 3350982557290104443883, 6155509617679334624756736, 14592373629282306879174535161, 43755759571493116198207431532544, 163135210694347619479784565520981395
Offset: 0
By definition, the coefficients a(n) satisfy:
1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(4*x)-sin(4*x))*x^2/2! + 27*(cos(9*x)-sin(9*x))*x^3/3! + 1160*(cos(16*x)-sin(16*x))*x^4/4! + 113385*(cos(25*x)-sin(25*x))*x^5/5! +...+ a(n)*(cos(n^2*x)-sin(n^2*x))*x^n/n! +...
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a(n)=local(A=[1, 1], N); for(i=1, n, A=concat(A, 0); N=#A; A[N]=(N-1)!*(-Vec(sum(m=0, N-1, A[m+1]*x^m/m!*(cos(m^2*x+x*O(x^N))-sin(m^2*x+x*O(x^N)))))[N])); A[n+1]
for(n=0, 25, print1(a(n), ", "))
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a(n)=if(n<2, 1, sum(k=1, n-1, (-1)^((n-k-1)\2)*a(k)*binomial(n, k)*k^(2*n-2*k)))
for(n=0, 25, print1(a(n), ", "))
A221534
E.g.f.: Sum_{n>=0} a(n) * (cos(n*x) - sin(n*x))^n * x^n/n! = 1 + x.
Original entry on oeis.org
1, 1, 2, 27, 968, 68865, 8252496, 1505652267, 390426582272, 136910626544673, 62531921536979200, 36122742294179711643, 25777397243775426776064, 22288717300246130379501921, 22978204666564567674247942144, 27861330789200983137890612877675
Offset: 0
By definition, the coefficients a(n) satisfy:
1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(2*x)-sin(2*x))^2*x^2/2! + 27*(cos(3*x)-sin(3*x))^3*x^3/3! + 968*(cos(4*x)-sin(4*x))^4*x^4/4! + 68865*(cos(5*x)-sin(5*x))^5*x^5/5! +...+ a(n)*(cos(n*x)-sin(n*x))^n*x^n/n! +...
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{a(n)=local(A=[1, 1], N); for(i=1, n, A=concat(A, 0); N=#A; A[N]=(N-1)!*(-Vec(sum(m=0, N-1, A[m+1]*x^m/m!*(cos(m*x+x*O(x^N))-sin(m*x+x*O(x^N)))^m))[N])); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
Showing 1-3 of 3 results.
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