A372956
G.f. A(x) satisfies A(x)^5 = A(x^5) / (1 - 5*x)^5 with A(0)=1.
Original entry on oeis.org
1, 5, 25, 125, 625, 3126, 15630, 78150, 390750, 1953750, 9768753, 48843765, 244218825, 1221094125, 6105470625, 30527353136, 152636765680, 763183828400, 3815919142000, 19079595710000, 95397978550044, 476989892750220, 2384949463751100, 11924747318755500, 59623736593777500
Offset: 0
A(x)^5 = 1 + 25*x + 375*x^2 + 4375*x^3 + 43750*x^4 + 393755*x^5 + ... .
-
b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
my(N=30, x='x+O('x^N)); Vec(1/prod(k=1, N, (1 - x^k)^b(k, 5))^5)
A386647
G.f. A(x) satisfies: A(x)^7 = A(x^7) / (1 - 7*x).
Original entry on oeis.org
1, 1, 4, 20, 110, 638, 3828, 23515, 146968, 930797, 5957100, 38450370, 249927394, 1634140604, 10738638021, 70875009760, 469546933535, 3121106054760, 20807373517870, 139080864081230, 931841783576460, 6256651942091035, 42090203778813320, 283651372136401905, 1914646755015446620
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146968*x^9 + 930797*x^10 + 5957100*x^11 + 38450370*x^12 +...
where A(x)^7 = A(x^7) / (1 - 7*x).
Also, when expressed as the EULER transform of A373277,
A(x) = x/( (1-x) * (1-x^2)^3 * (1-x^3)^16 * (1-x^4)^84 * (1-x^5)^480 * (1-x^6)^2792 * (1-x^7)^16807 * (1-x^8)^102900 * ... * (1-x^n)^A373277(n) * ... ).
RELATED SERIES.
A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764808*x^15 + ...
-
{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^7)/(1 - 7*x +x*O(x^n)))^(1/7)); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", "))
-
/* EULER transform of A373277 */
{A373277(n) = 1/(7*n) * sumdiv(n, d, (gcd(d, 7)==1)*(moebius(d)*7^(n/d)))} \\ after Seiichi Manyama in A373277
{a(n) = my(A = x/prod(m=1, n, (1-x^m +x*O(x^n))^A373277(m))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-2 of 2 results.
Comments