A218798
E.g.f.: Sum_{n>=0} a(n) * (cos(n*x)-sin(n*x)) * x^n/n! = 1 + x.
Original entry on oeis.org
1, 1, 2, 15, 224, 5665, 215136, 11466903, 815542272, 74626924833, 8539305740800, 1194616544819423, 200588161750990848, 39810821495648979009, 9219256372209251966976, 2463653490073311266825895, 752468103154551980520636416, 260483894045203867466646001345
Offset: 0
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 224*x^4/4! + 5665*x^5/5! +...
By definition, the coefficients a(n) satisfy:
1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(2*x)-sin(2*x))*x^2/2! + 15*(cos(3*x)-sin(3*x))*x^3/3! + 224*(cos(4*x)-sin(4*x))*x^4/4! + 5665*(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(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)))))[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^(n-k)))
for(n=0,25,print1(a(n),", "))
A209316
E.g.f.: Sum_{n>=0} a(n) * (cos(n^2*x) - sin(n^2*x)) * x^n/n! = 1/(1-x).
Original entry on oeis.org
1, 1, 4, 57, 2456, 240205, 44616096, 14030856525, 6897867308800, 4999592004999705, 5107861266649227520, 7098997630368216900833, 13040338287878632604362752, 30913685990004537377333201253, 92695803952674372198927320920064, 345599063527286969179932122231749365
Offset: 0
By definition, the coefficients a(n) satisfy:
1/(1-x) = 1 + 1*(cos(x)-sin(x))*x + 4*(cos(4*x)-sin(4*x))*x^2/2! + 57*(cos(9*x)-sin(9*x))*x^3/3! + 2456*(cos(16*x)-sin(16*x))*x^4/4! + 240205*(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)!*(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==0 || n==1, 1, n!+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), ", "))
A209317
E.g.f.: Sum_{n>=0} a(n) * (cos(n*x) - sin(n*x))^n * x^n/n! = 1/(1-x).
Original entry on oeis.org
1, 1, 4, 57, 2072, 147925, 17749536, 3240106485, 840395708928, 294739255397385, 134627422799345920, 77773271544276025553, 55500837134575871643648, 47990173549409999557055133, 49475217831781002832374386688, 59989657372751900405803761497805, 84553864714598468031554754299887616
Offset: 0
By definition, the coefficients a(n) satisfy:
1/(1-x) = 1 + 1*(cos(x)-sin(x))*x + 4*(cos(2*x)-sin(2*x))^2*x^2/2! + 57*(cos(3*x)-sin(3*x))^3*x^3/3! + 2072*(cos(4*x)-sin(4*x))^4*x^4/4! + 147925*(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)!*(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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