A281429
E.g.f.: C(x) + S(x) = exp( Integral C(x)^4 dx ) where C(x) and S(x) is described by A281428 and A281427, respectively.
Original entry on oeis.org
1, 1, 1, 5, 17, 145, 865, 10325, 88865, 1357825, 15335425, 284963525, 3993275825, 87274812625, 1462392957025, 36716097543125, 716611617346625, 20309401097610625, 452780458211706625, 14290053364475013125, 358439197464543820625, 12462411363013047060625
Offset: 0
E.g.f: C(x) + S(x) = 1 + x + x^2/2! + 5*x^3/3! + 17*x^4/4! + 145*x^5/5! + 865*x^6/6! + 10325*x^7/7! + 88865*x^8/8! + 1357825*x^9/9! + 15335425*x^10/10! + 284963525*x^11/11! + 3993275825*x^12/12! + 87274812625*x^13/13! + 1462392957025*x^14/14! + 36716097543125*x^15/15! + 716611617346625*x^16/16! + 20309401097610625*x^17/17! + 452780458211706625*x^18/18! + 14290053364475013125*x^19/19! + 358439197464543820625*x^20/20! +...
where log( C(x) + S(x) ) = Integral C(x)^4 dx, and
C(x)^4 = 1 + 4*x^2/2! + 104*x^4/4! + 6880*x^6/6! + 855680*x^8/8! + 171673600*x^10/10! + 50628300800*x^12/12! + 20616410214400*x^14/14! + 11081874771968000*x^16/16! + 7600553402810368000*x^18/18! + 6477130108444835840000*x^20/20! +...
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{a(n) = my(S=x, C=1); for(i=0, n, S = intformal( C^5 +x*O(x^n)); C = 1 + intformal( S*C^4 ) ); n!*polcoeff(C+S, n)}
for(n=0, 30, print1(a(n), ", "))
A281430
E.g.f.: C(x)^4 where C(x) is described by A281428.
Original entry on oeis.org
1, 4, 104, 6880, 855680, 171673600, 50628300800, 20616410214400, 11081874771968000, 7600553402810368000, 6477130108444835840000, 6713789344917138964480000, 8317650472128427128258560000, 12137529532422860667092992000000
Offset: 0
C(x)^4 = 1 + 4*x^2/2! + 104*x^4/4! + 6880*x^6/6! + 855680*x^8/8! + 171673600*x^10/10! + 50628300800*x^12/12! + 20616410214400*x^14/14! + 11081874771968000*x^16/16! + 7600553402810368000*x^18/18! + 6477130108444835840000*x^20/20! + 6713789344917138964480000*x^22/22! + 8317650472128427128258560000*x^24/24! +...
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{a(n) = my(S=x, C=1); for(i=0, n, S = intformal( C^5 +x*O(x^(2*n))); C = 1 + intformal( S*C^4 ) ); (2*n)!*polcoeff(C^4, 2*n)}
for(n=0, 30, print1(a(n), ", "))
A281427
E.g.f. S(x) satisfies: S(x) = Integral (1 + S(x)^2)^(5/2) dx.
Original entry on oeis.org
1, 5, 145, 10325, 1357825, 284963525, 87274812625, 36716097543125, 20309401097610625, 14290053364475013125, 12462411363013047060625, 13192751210140624100103125, 16663953549286540157926890625, 24756557919279291667433199453125, 42733906625427778939437818074140625, 84814104213422372894487597292655703125
Offset: 1
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a[n_] := Module[{S = x, C = 1, C5, SC4}, For[i = 1, i <= n, i++, C5 = C^5 + x*O[x]^(2n) // Normal; S = Integrate[C5, x]; SC4 = S*C^4 + O[x]^(2n-1) // Normal; C = 1 + Integrate[SC4, x] ]; (2n-1)!*Coefficient[S, x, 2n-1]]; Array[a, 16] (* Jean-François Alcover, Mar 01 2017, translated from Pari *)
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{a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^5 +x*O(x^(2*n))); C = 1 + intformal( S*C^4 ) ); (2*n-1)!*polcoeff(S, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.