A338661
a(n) = Sum_{d|n} d^n * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 5, 28, 289, 3126, 49036, 823544, 17040385, 387538588, 10048833246, 285311670612, 8929334253419, 302875106592254, 11116754387182648, 437894348359764856, 18448995959423107073, 827240261886336764178, 39347761059781438793815, 1978419655660313589123980
Offset: 1
-
a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 2, #-1] &]; Array[a, 20] (* Amiram Eldar, Apr 22 2021 *)
-
a(n) = sumdiv(n, d, d^n*binomial(d+n/d-2, d-1));
-
my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-(k*x)^k))^k))
A343573
a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 5, 28, 265, 3126, 46750, 823544, 16778257, 387420652, 10000015646, 285311670612, 8916100731047, 302875106592254, 11112006831322846, 437893890380906656, 18446744073843774497, 827240261886336764178, 39346408075300025340205
Offset: 1
Cf.
A023887,
A157019,
A157020,
A324158,
A324159,
A338661,
A339481,
A339482,
A339712,
A343567,
A343574.
-
a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)
-
a(n) = sumdiv(n, d, d^d*binomial(d+n/d-2, d-1));
-
my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-x^k))^k))
A339712
a(n) = Sum_{d|n} d^(d+n/d-1) * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 5, 28, 273, 3126, 46948, 823544, 16781441, 387421948, 10000078446, 285311670612, 8916102176891, 302875106592254, 11112006865913416, 437893890382064056, 18446744074783625217, 827240261886336764178, 39346408075327954053967, 1978419655660313589123980
Offset: 1
-
a[n_] := DivisorSum[n, #^(# + n/# - 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)
-
a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d+n/d-2, d-1));
-
my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-k*x^k))^k))
A339482
a(n) = Sum_{d|n} d^(n-d+1) * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 3, 4, 21, 6, 346, 8, 4617, 13132, 80696, 12, 4903847, 14, 40410966, 756336736, 2416181265, 18, 306560794753, 20, 6941876836216, 132964265599502, 34522735212626, 24, 116720277621236637, 33378601074218776, 51185893450298400, 60788365423272068968
Offset: 1
-
a[n_] := DivisorSum[n, #^(n - # + 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
-
a(n) = sumdiv(n, d, d^(n-d+1)*binomial(d+n/d-2, d-1));
-
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*(x/(1-(k*x)^k))^k))
A360795
Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).
Original entry on oeis.org
1, 3, 4, 17, 6, 211, 8, 1929, 7300, 22601, 12, 1724809, 14, 6703047, 223678576, 738787345, 18, 65630598229, 20, 2119646503661, 24448573943662, 3423809253371, 24, 21453113652593665, 12016296386718776, 4240253019018225, 8255251542208471048, 67251293544533119589, 30
Offset: 1
-
a[n_] := DivisorSum[n, #^(n-#) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)
-
my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)^(k+1)))
-
a(n) = sumdiv(n, d, d^(n-d)*binomial(d+n/d-1, d));
A360802
Expansion of Sum_{k>0} (x / (1 - (2 * x)^k))^k.
Original entry on oeis.org
1, 3, 5, 17, 17, 105, 65, 449, 641, 1953, 1025, 16257, 4097, 37761, 93185, 247809, 65537, 1499649, 262145, 6596609, 8847361, 13654017, 4194305, 210026497, 90177537, 251764737, 833880065, 2659418113, 268435457, 18345328641, 1073741825, 53553922049, 75438751745
Offset: 1
-
a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)
-
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(2*x)^k))^k))
-
a(n) = sumdiv(n, d, 2^(n-d)*binomial(d+n/d-2, d-1));
A360812
Expansion of Sum_{k>=0} ( x / (1 - (k * x)^2) )^k.
Original entry on oeis.org
1, 1, 1, 2, 9, 29, 113, 613, 3033, 17010, 110929, 713249, 5061097, 38762873, 302389553, 2544613578, 22404995001, 203762678941, 1960880744337, 19509713674397, 201306862742217, 2166901479447194, 24018963506471921, 275731857268608673, 3271769647891351705
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-(k*x)^2))^k))
-
a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n-k-1, k));
A360813
Expansion of Sum_{k>=0} ( x / (1 - (k * x)^3) )^k.
Original entry on oeis.org
1, 1, 1, 1, 2, 17, 82, 258, 818, 5671, 43363, 240520, 1183168, 8547054, 77831681, 596258173, 4031934111, 33313129161, 338733239446, 3187239159511, 27197807726066, 260179611473044, 2918973182685904, 31820249821418229, 324099587971865989
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-(k*x)^3))^k))
-
a(n) = sum(k=0, n\3, (n-3*k)^(3*k)*binomial(n-2*k-1, k));
Showing 1-8 of 8 results.