A363304
Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^4 + A(x)^7).
Original entry on oeis.org
1, 2, 22, 350, 6538, 133658, 2895214, 65294502, 1516963346, 36056007602, 872615973766, 21430572885422, 532737957899290, 13379121740808266, 338941379999841758, 8651415618928816886, 222278432539991439906, 5743974149517874477922, 149192980850883703986166
Offset: 0
G.f.: A(x) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + 1516963346*x^8 + 36056007602*x^9 + ...
where A(x) = 1 + x*(A(x)^4 + A(x)^7).
RELATED SERIES.
A(x)^4 = 1 + 8*x + 112*x^2 + 1960*x^3 + 38528*x^4 + 813064*x^5 + 17998512*x^6 + 412364968*x^7 + ...
A(x)^7 = 1 + 14*x + 238*x^2 + 4578*x^3 + 95130*x^4 + 2082150*x^5 + 47295990*x^6 + 1104598378*x^7 + ...
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{a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) )}
for(n=0, 20, print1(a(n), ", "))
A363309
Expansion of g.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
Original entry on oeis.org
1, 1, 8, 67, 590, 5403, 51034, 494268, 4886794, 49153835, 501631980, 5182767291, 54115252508, 570206217940, 6055948422280, 64765311313944, 696876526961130, 7539151412082315, 81957518070961472, 894826829565106185, 9808173152466891270, 107888887505651377475
Offset: 0
G.f.: A(x) = 1 + x + 8*x^2 + 67*x^3 + 590*x^4 + 5403*x^5 + 51034*x^6 + 494268*x^7 + 4886794*x^8 + 49153835*x^9 + 501631980*x^10 + ...
such that A(x) = F(x*F(x)^5) where F(x) = 1 + x*F(x)^3 begins
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + ... + A001764(n)*x^n + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^2) which begins
B(x) = 1 + x + 10*x^2 + 120*x^3 + 1620*x^4 + 23560*x^5 + 360352*x^6 + 5714800*x^7 + 93129840*x^8 + ... + A363310(n)*x^n + ...
then
( (B(x) - 1)/x )^(1/5) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + ... + A363311(n)*x^n + ...
is an integer series.
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{a(n) = if(n==0, 1, sum(k=1, n, 5*k* binomial(3*k+1, k) * binomial(3*n+2*k, n-k) / ((3*k+1)*(3*n+2*k)) ) )}
for(n=0, 30, print1(a(n), ", "))
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/* G.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 */
{a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^3 + x*O(x^n));
polcoeff( subst(F, x, x*F^5), n)}
for(n=0, 30, print1(a(n), ", "))
A363308
Expansion of g.f. C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
Original entry on oeis.org
1, 1, 5, 26, 141, 790, 4542, 26668, 159333, 966038, 5930678, 36801660, 230491410, 1455283172, 9253674120, 59209786992, 380961295445, 2463303690790, 15998687418030, 104325569140156, 682768883525830, 4483232450501492, 29527005540912660, 195006621974036808
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 141*x^4 + 790*x^5 + 4542*x^6 + 26668*x^7 + 159333*x^8 + 966038*x^9 + 5930678*x^10 + ...
such that A(x) = C(x*C(x)^3), where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... + A000108(n)*x^n + ...
x*C(x)^3 = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + ... + A000245(n)*x^n + ...
Note that x*C(x)^3 = (C(x) - 1)*(1-x)/x - 1.
Also, the g.f. of related sequence A033296 begins
B(x) = 1 + x + 6*x^2 + 42*x^3 + 326*x^4 + 2706*x^5 + 23526*x^6 + ...
where A(x) = B(x/A(x)), B(x) = A(x*B(x)) = C(x*B(x)*C(x*B(x))^3).
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{a(n) = if(n==0,1, sum(k=1,n, 3*k* binomial(2*k+1,k) * binomial(2*n+k,n-k) / ((2*k+1)*(2*n+k)) ) )}
for(n=0, 30, print1(a(n), ", "))
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/* G.f. A(x) = C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 */
{a(n) = my(C = (1 - sqrt(1 - 4*x +x^2*O(x^n)))/(2*x)); polcoeff( subst(C, x, x*C^3), n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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