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Expansion of 1/(1 - x/(1 - 8*x^2)^(1/4)).
+10
1
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
FORMULA
a(n) = Sum_{k=0..floor(n/2)} 8^k * binomial((n+2*k)/4-1,k).
MAPLE
add(8^k*binomial((n+2*k)/4-1, k), k=0..floor(n/2)) ;
end proc:
PROG
(PARI) a(n) = sum(k=0, n\2, 8^k*binomial((n+2*k)/4-1, k));
Expansion of 1/(1 - x/(1 - 16*x^8)^(1/8)).
+10
1
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
FORMULA
a(8*n) = 17^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/8)} 16^k * binomial(n/8-1,k).
PROG
(PARI) a(n) = sum(k=0, n\8, 16^k*binomial(n/8-1, k));
Expansion of 1 / ( (1 - 8*x^4) * (1 - x/(1 - 8*x^4)^(1/4)) ).
+10
1
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
FORMULA
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).
PROG
(PARI) a(n) = sum(k=0, n\4, 8^k*binomial(n/4, k));
Expansion of 1/(1 - x/(1 - 8*x)^(1/4)).
+10
0
1, 1, 3, 15, 91, 601, 4155, 29553, 214303, 1575931, 11712599, 87776507, 662224819, 5023611579, 38284084575, 292892970967, 2248271735299, 17307950940833, 133580448494227, 1033263820897777, 8008342899292167, 62179343789159945, 483553052098053915
FORMULA
a(n) = Sum_{k=0..n} 8^k * binomial((n+3*k)/4-1,k).
PROG
(PARI) a(n) = sum(k=0, n, 8^k*binomial((n+3*k)/4-1, k));
Expansion of 1/(1 - x/(1 - 8*x^3)^(1/4)).
+10
0
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
FORMULA
a(n) = Sum_{k=0..floor(n/3)} 8^k * binomial((n+k)/4-1,k).
PROG
(PARI) a(n) = sum(k=0, n\3, 8^k*binomial((n+k)/4-1, k));
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