A373509 -id:A373509 - cp's OEIS frontend

A371927 Expansion of 1/(1 - x/(1 - 8*x^2)^(1/4)).

1, 1, 1, 3, 5, 17, 33, 113, 237, 803, 1769, 5915, 13493, 44547, 104337, 340527, 814397, 2630857, 6399865, 20486905, 50548997, 160507953, 400834465, 1263577141, 3188428301, 9985916077, 25426685961, 79168607025, 203193847381, 629311885861, 1626634117809

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A373509, A004981.

A371927 := proc(n)
    add(8^k*binomial((n+2*k)/4-1,k),k=0..floor(n/2)) ;
end proc:
seq(A371927(n),n=0..70) ; # R. J. Mathar, Jun 07 2024

CoefficientList[Series[1/(1-x/(1-8x^2)^(1/4)),{x,0,30}],x] (* Harvey P. Dale, Dec 20 2024 *)

a(n) = sum(k=0, n\2, 8^k*binomial((n+2*k)/4-1, k));

a(n) = Sum_{k=0..floor(n/2)} 8^k * binomial((n+2*k)/4-1,k).

A373512 Expansion of 1/(1 - x/(1 - 16*x^8)^(1/8)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 37, 61, 89, 121, 157, 197, 241, 289, 545, 877, 1293, 1801, 2409, 3125, 3957, 4913, 8551, 13469, 19891, 28057, 38223, 50661, 65659, 83521, 138227, 213997, 315575, 448297, 618123, 831669, 1096239, 1419857

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A098615, A373509.

a(n) = sum(k=0, n\8, 16^k*binomial(n/8-1, k));

a(8*n) = 17^(n-1) for n > 0.

a(n) = Sum_{k=0..floor(n/8)} 16^k * binomial(n/8-1,k).

A373627 Expansion of 1 / ( (1 - 8x^4) (1 - x/(1 - 8*x^4)^(1/4)) ).

Original entry on oeis.org

1, 1, 1, 1, 9, 11, 13, 15, 81, 109, 141, 177, 729, 1041, 1429, 1901, 6561, 9759, 13981, 19419, 59049, 90483, 133893, 192327, 531441, 832911, 1264173, 1865539, 4782969, 7628799, 11816853, 17828163, 43046721, 69620541, 109646397, 168500385, 387420489, 633634769

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A373509, A373583.

a(n) = sum(k=0, n\4, 8^k*binomial(n/4, k));

a(4*n) = 9^n for n >= 0.
a(n) = Sum_{k=0..floor(n/4)} 8^k * binomial(n/4,k).
a(n) == 1 (mod 2).

A371490 Expansion of 1/(1 - x/(1 - 8*x)^(1/4)).

Original entry on oeis.org

1, 1, 3, 15, 91, 601, 4155, 29553, 214303, 1575931, 11712599, 87776507, 662224819, 5023611579, 38284084575, 292892970967, 2248271735299, 17307950940833, 133580448494227, 1033263820897777, 8008342899292167, 62179343789159945, 483553052098053915

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A373509.

a(n) = sum(k=0, n, 8^k*binomial((n+3*k)/4-1, k));

a(n) = Sum_{k=0..n} 8^k * binomial((n+3*k)/4-1,k).

A373292 Expansion of 1/(1 - x/(1 - 8*x^3)^(1/4)).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 7, 19, 35, 55, 139, 267, 447, 1077, 2115, 3689, 8595, 17101, 30703, 69797, 140027, 256873, 573167, 1156221, 2156555, 4742759, 9603287, 18149083, 39457727, 80104735, 153007747, 329580959, 670338231, 1291649283, 2761199459, 5623490391

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A373509.

a(n) = sum(k=0, n\3, 8^k*binomial((n+k)/4-1, k));

a(n) = Sum_{k=0..floor(n/3)} 8^k * binomial((n+k)/4-1,k).

cp's OEIS Frontend

A371927 Expansion of 1/(1 - x/(1 - 8*x^2)^(1/4)).

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A373512 Expansion of 1/(1 - x/(1 - 16*x^8)^(1/8)).

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A373627 Expansion of 1 / ( (1 - 8*x^4) * (1 - x/(1 - 8*x^4)^(1/4)) ).

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A371490 Expansion of 1/(1 - x/(1 - 8*x)^(1/4)).

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A373292 Expansion of 1/(1 - x/(1 - 8*x^3)^(1/4)).

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A373627 Expansion of 1 / ( (1 - 8x^4) (1 - x/(1 - 8*x^4)^(1/4)) ).