A199813
G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n) * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).
Original entry on oeis.org
1, 4, 38, 504, 8249, 154036, 3149326, 68741880, 1576163328, 37548785408, 922252542128, 23222906277952, 596981991939677, 15616173859832740, 414621835401615110, 11150969618415168280, 303278916800906999191, 8330190277527648516572, 230814933905555392525290
Offset: 0
G.f.: A(x) = 1 + 4*x + 38*x^2 + 504*x^3 + 8249*x^4 + 154036*x^5 +...
where
log(A(x)) = 2*2*x + 6*10*x^2/2 + 20*56*x^3/3 + 70*346*x^4/4 + 252*2252*x^5/5 + 924*15184*x^6/6 +...+ A000984(n)*A000172(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m, m)*sum(k=0, m, binomial(m, k)^3)*x^m/m)+x*O(x^n)),n)}
A199816
G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n)/4 * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).
Original entry on oeis.org
1, 1, 8, 101, 1639, 30665, 630225, 13836981, 319062453, 7640441894, 188534274850, 4767113222750, 122998902095908, 3228067183537455, 85960229675478804, 2317956019913480326, 63193008693741620771, 1739473925024629613227, 48292271242981605779173
Offset: 0
G.f.: A(x) = 1 + x + 8*x^2 + 101*x^3 + 1639*x^4 + 30665*x^5 +...
where
log(A(x)) = 1*1*x + 3*5*x^2/2 + 10*28*x^3/3 + 35*173*x^4/4 + 126*1126*x^5/5 + 462*7592*x^6/6 +...+ A000984(n)/2*A000172(n)/2*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m, m)/2*sum(k=0, m, binomial(m, k)^3)/2*x^m/m)+x*O(x^n)), n)}
A208890
a(n) = A000984(n)*A004981(n), the term-wise product of the coefficients in (1-4*x)^(-1/2) and (1-8*x)^(-1/4).
Original entry on oeis.org
1, 4, 60, 1200, 27300, 668304, 17153136, 455083200, 12372574500, 342766138000, 9638583800560, 274341178587840, 7887308884400400, 228685287180840000, 6678543795015960000, 196260140322869011200, 5798873833602270315300, 172160337343624495866000
Offset: 0
G.f.: A(x) = 1 + 4*x + 60*x^2 + 1200*x^3 + 27300*x^4 + 668304*x^5 +...
The terms are the term-wise products of the sequences:
A000984 = [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...],
A004981 = [1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, ...].
Related sequences:
A^2: [1, 8, 136, 2880, 67800, 1699008, 44368704, 1193107968, ...],
A^4: [1, 16, 336, 7936, 200176, 5266176, 142657536, 3948773376, ...],
A^8: [1, 32, 928, 26624, 767200, 22270976, 651331072, 19178651648, ...].
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{A000984(n)=polcoeff((1-4*x +x*O(x^n))^(-1/2),n)}
{A004981(n)=polcoeff((1-8*x +x*O(x^n))^(-1/4),n)}
{a(n)=A000984(n)*A004981(n)}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.
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