A215125
E.g.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / (2*n)!.
Original entry on oeis.org
1, 1, 7, 33, 223, 1753, 14391, 137137, 1382383, 14981673, 174494983, 2135204161, 27643067007, 375548195833, 5326762882903, 78889684038993, 1213984929832591, 19377034523034697, 320293617185965863, 5468629894127442913, 96328047496084810783, 1747805367475759936281
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 33*x^3/3! + 223*x^4/4! + 1753*x^5/5! +...
such that, by definition:
A(x) = 1 + d/dx (x+x^2)^2/2! + d^2/dx^2 (x+x^2)^4/4! + d^3/dx^3 (x+x^2)^6/6! + d^4/dx^4 (x+x^2)^8/8! + d^5/dx^5 (x+x^2)^10/10! +...
Compare to the trivial identity:
exp(x) = 1 + d/dx x^2/2! + d^2/dx^2 x^4/4! + d^3/dx^3 x^6/6! + d^4/dx^4 x^8/8! + d^5/dx^5 x^10/10! +...
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{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, (x+x^2+x*O(x^n))^(2*m)/(2*m)!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
A215129
G.f.: Sum_{n>=0} d^n/dx^n (2*x + x^2)^(2*n) / n!.
Original entry on oeis.org
1, 8, 108, 1604, 25000, 400680, 6539932, 108124704, 1804745052, 30345973020, 513246523504, 8722051475280, 148808353519644, 2547317293116512, 43729600850054640, 752553083643333008, 12978731348733064680, 224258766666165861192
Offset: 0
G.f.: A(x) = 1 + 2*x + 12*x^2 + 64*x^3 + 370*x^4 + 2184*x^5 + 13132*x^6 +...
such that, by definition:
A(x) = 1 + d/dx (2*x+x^2)^2 + d^2/dx^2 (2*x+x^2)^4/2! + d^3/dx^3 (2*x+x^2)^6/3! + d^4/dx^4 (2*x+x^2)^8/4! + d^5/dx^5 (2*x+x^2)^10/5! +...
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{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, (2*x+x^2+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
A361303
Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(3*n) / n!.
Original entry on oeis.org
1, 2, 15, 92, 615, 4200, 29190, 205416, 1458909, 10436030, 75079719, 542669244, 3937604853, 28664996080, 209261546580, 1531373181120, 11230365782130, 82512324300222, 607246350958449, 4475646134515360, 33031356134381220, 244073892799489500, 1805479496422561740
Offset: 0
G.f.: A(x) = 1 + 2*x + 15*x^2 + 92*x^3 + 615*x^4 + 4200*x^5 + 29190*x^6 + 205416*x^7 + 1458909*x^8 + 10436030*x^9 + ...
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{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); A = sum(m=0, n, Dx(m, x^(2*m)*(1+x +O(x^(n+1)))^(3*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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/* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^3 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-3 of 3 results.
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