This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144015 #33 Jun 24 2025 10:01:57
%S A144015 1,1,5,29,265,3001,42125,696149,13296145,287706481,6959431445,
%T A144015 186061833869,5448382252825,173418192216361,5961442393047965,
%U A144015 220112963745653189,8687730877758518305,365023930617143804641,16266420334783460443685,766297734521812843642109
%N A144015 Expansion of e.g.f. 1/(1 - sin(4*x))^(1/4).
%C A144015 Row sums of A186492 - _Peter Bala_, Feb 22 2011.
%F A144015 E.g.f. A(x) satisfies:
%F A144015 (1) A(x) = (cos(2*x) - sin(2*x))^(-1/2).
%F A144015 (2) A(x)^2/A(-x)^2 = 1/cos(4*x) + tan(4*x).
%F A144015 (3) A(x) = exp( Integral A(x)^2/A(-x)^2 dx).
%F A144015 (4) A'(x) = A(x)^3/A(-x)^2 with A(0) = 1.
%F A144015 (5) A(x) = 1/sqrt(1 - 2*Series_Reversion( Integral 1/sqrt(1+4*x-4*x^2) dx )).
%F A144015 G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)*(2*k+1)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 11 2013.
%F A144015 a(n) ~ 2^(3*n+5/4)*n^n/(exp(n)*Pi^(n+1/2)). - _Vaclav Kotesovec_, Jun 26 2013
%F A144015 a(n) = Sum_{k=0..n} A007696(k) * (4*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - _Seiichi Manyama_, Jun 24 2025
%e A144015 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 29*x^3/3! + 265*x^4/4! + 3001*x^5/5! +...
%e A144015 log(A(x)) = x + 4*x^2/2! + 16*x^3/3! + 128*x^4/4! + 1280*x^5/5! +...
%e A144015 A(x)^2/A(-x)^2 = 1 + 4*x + 16*x^2/2! + 128*x^3/3! +...+ 4^n*A000111(n)*x^n/n! +...
%e A144015 O.g.f.: 1/(1-x - 4*1*1*x^2/(1-5*x - 4*2*3*x^2/(1-9*x - 4*3*5*x^2/(1-13*x - 4*4*7*x^2/(1-17*x - 4*5*9*x^2/(1-...)))))) [continued fraction by Sergei Gladkovskii].
%t A144015 CoefficientList[Series[1/(1-Sin[4*x])^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)
%o A144015 (PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(2*X)-sin(2*X))^(-1/2), n)}
%o A144015 for(n=0, 20, print1(a(n), ", "))
%o A144015 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=exp(intformal(A^2/subst(A^2,x,-x))));n!*polcoeff(A,n)}
%o A144015 for(n=0, 20, print1(a(n), ", "))
%o A144015 (PARI) /* From A'(x) = A(x)^3 / A(-x)^2: */
%o A144015 {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
%o A144015 for(n=0, 20, print1(a(n), ", "))
%o A144015 (PARI) /* 1/sqrt(1-2*Series_Reversion(Integral 1/sqrt(1+4*x-4*x^2) dx)): */
%o A144015 {a(n)=local(A=1);A=1/sqrt(1-2*serreverse(intformal(1/sqrt(1+4*x-4*x^2 +x*O(x^n)))));n!*polcoeff(A, n)}
%o A144015 for(n=0,20,print1(a(n),", "))
%o A144015 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A144015 a007696(n) = prod(k=0, n-1, 4*k+1);
%o A144015 a(n) = sum(k=0, n, a007696(k)*(4*I)^(n-k)*a136630(n, k)); \\ _Seiichi Manyama_, Jun 24 2025
%Y A144015 Cf. A000111, A001586, A007788, A186492, A230134, A227544, A230114, A235329.
%Y A144015 Cf. A007696, A136630.
%K A144015 nonn
%O A144015 0,3
%A A144015 _Paul D. Hanna_, Sep 09 2008