A372870 -id:A372870 - cp's OEIS frontend

A372956 G.f. A(x) satisfies A(x)^5 = A(x^5) / (1 - 5*x)^5 with A(0)=1.

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

Seiichi Manyama, Jul 04 2024

nonn

Euler transform of 5 * A054662(n).

			A(x)^5 = 1 + 25*x + 375*x^2 + 4375*x^3 + 43750*x^4 + 393755*x^5 + ... .

Seiichi Manyama, Table of n, a(n) for n = 0..1000
OEIS Wiki, Euler transform

Cf. A372870, A372957.

Cf. A054662, A372535.

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)

G.f.: A(x) = 1 / ( Product_{k>=1} (1 - x^k)^A054662(k) )^5.

A372957 G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 - 2*x)^2 with A(0)=1.

Original entry on oeis.org

1, 2, 5, 10, 22, 44, 91, 182, 370, 740, 1490, 2980, 5979, 11958, 23950, 47900, 95865, 191730, 383580, 767160, 1534549, 3069098, 6138628, 12277256, 24555341, 49110682, 98222947, 196445894, 392894839, 785789678, 1571585230, 3143170460, 6286352290, 12572704580, 25145431172

Offset: 0

Seiichi Manyama, Jul 04 2024

nonn

Euler transform of 2 * A000048(n).

			A(x)^2 = 1 + 4*x + 14*x^2 + 40*x^3 + 109*x^4 + 276*x^5 + 678*x^6 + ... .

Seiichi Manyama, Table of n, a(n) for n = 0..1000
OEIS Wiki, Euler transform

Cf. A372870, A372956.

Cf. A000048, A123916.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1 - x^k)^b(k, 2))^2)

G.f.: A(x) = 1 / ( Product_{k>=1} (1 - x^k)^A000048(k) )^2.

cp's OEIS Frontend

A372956 G.f. A(x) satisfies A(x)^5 = A(x^5) / (1 - 5*x)^5 with A(0)=1.

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