login
a(n) = [x^n] Product_{k>=0} (1 + n*x^(2^k)).
1

%I #6 Apr 09 2018 22:33:35

%S 1,1,2,9,4,25,36,343,8,81,100,1331,144,2197,2744,50625,16,289,324,

%T 6859,400,9261,10648,279841,576,15625,17576,531441,21952,707281,

%U 810000,28629151,32,1089,1156,42875,1296,50653,54872,2313441,1600,68921,74088,3418801,85184,4100625,4477456,229345007,2304

%N a(n) = [x^n] Product_{k>=0} (1 + n*x^(2^k)).

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(n) = n^A000120(n).

%F a(n) = A256140(n,n).

%F a(2^k) = 2^k.

%F a(2^k-1) = (2^k - 1)^k.

%e +---+-----+---+----------+

%e | n | bin.|1's| a(n) |

%e +---+-----+---+----------+

%e | 0 | 0 | 0 | 0^0 = 1 |

%e | 1 | 1 | 1 | 1^1 = 1 |

%e | 2 | 10 | 1 | 2^1 = 2 |

%e | 3 | 11 | 2 | 3^2 = 9 |

%e | 4 | 100 | 1 | 4^1 = 4 |

%e | 5 | 101 | 2 | 5^2 = 25 |

%e | 6 | 110 | 2 | 6^2 = 36 |

%e +---+-----+---+----------+

%e bin. - n written in base 2;

%e 1's - number of 1's in binary expansion of n.

%t Table[SeriesCoefficient[Product[(1 + n x^(2^k)), {k, 0, n}], {x, 0, n}], {n, 0, 48}]

%t Join[{1}, Table[n^DigitCount[n, 2, 1], {n, 48}]]

%o (PARI) a(n) = n^hammingweight(n); \\ _Altug Alkan_, Apr 08 2018

%Y Main diagonal of A256140.

%Y Cf. A000120, A001316, A048883, A102376, A245788.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 08 2018