A290261 -id:A290261 - cp's OEIS frontend

A301470 Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

1, 0, 1, 0, 1, 1, 2, 3, 5, 9, 15, 27, 47, 87, 155, 288, 524, 983, 1813, 3434, 6396, 12174, 22891, 43810, 82925, 159432, 303559, 585966, 1121446, 2171341, 4172932, 8106485, 15635332, 30445899, 58925280, 115014681, 223210718, 436603718, 849480835, 1664740873

Offset: 0

Gus Wiseman, Mar 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..3266

Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290971, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301469.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
    end:
a:= n-> `if`(n<2, 1-n, b(n-2$2)+b(n-1, n-2)):
seq(a(n), n=0..45);  # Alois P. Heinz, Jun 23 2018

a[n_]:=a[n]=(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
     If[i < 1, 0, b[n, i - 1] + a[i] b[n - i, Min[n - i, i]]]];
a[n_] := If[n < 2, 1 - n, b[n - 2, n - 2] + b[n - 1, n - 2]];
a /@ Range[0, 45] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

O.g.f.: 1/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).

a(n) = Sum_t (-1)^w(t) where the sum is over all enriched r-trees of size n and w(t) is the sum of leaves of t.

A302917 Solution to a(1) = 1 and Sum_y Product_i a(y_i) = 0 for each n > 1, where the sum is over all relatively prime or monic partitions of n.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -3, 1, 4, -5, -3, 3, 4, 2, -6, -6, 19, -8, -25, 25, 20, -12, -34, 2, 30, 38, -117, 54, 159, -173, -123, 55, 229, 32, -250, -148, 753, -365, -1022, 840, 1121, -847, -1482, -390, 2099

Offset: 1

Gus Wiseman, Apr 15 2018

sign

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

Cf. A000837, A036987, A047966, A063834, A093637, A196545, A220418, A289501, A290261, A300486, A300863-A300866, A301462, A302094, A302915, A302916.

a[n_]:=a[n]=If[n===1,1,0]-Sum[Times@@a/@y,{y,Rest[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]]}];
Array[a,20]

A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

Original entry on oeis.org

1, 1, 0, -1, -1, 1, 1, -1, -2, 0, 2, 0, -3, -1, 3, 2, -3, -3, 3, 4, -3, -6, 2, 7, -1, -8, 0, 10, 2, -11, -4, 12, 7, -13, -10, 13, 13, -13, -17, 13, 22, -11, -26, 9, 31, -6, -36, 2, 41, 3, -46, -9, 51, 17, -55, -26, 59, 36, -62, -48, 63, 61, -64, -75, 64, 92, -60, -109, 55, 127, -48, -147, 37, 167

Offset: 0

Gus Wiseman, Mar 09 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A220418, A290261, A292043, A292137, A300574.

CoefficientList[Series[QPochhammer[-x,-x^2],{x,0,100}],x]

O.g.f.: Product_{n >= 0} (1 + (-1)^n x^(2n+1)).

a(n) = Sum (-1)^k where the sum is over all strict integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.

A301469 Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

Original entry on oeis.org

2, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 3, 3, 6, 7, 11, 17, 23, 35, 53, 75, 119, 173, 264, 398, 603, 911, 1411, 2114, 3279, 4977, 7696, 11760, 18253, 27909, 43451, 66675, 103945, 160096, 249904, 385876, 603107, 933474, 1461967, 2266384, 3553167, 5521053, 8664117, 13485744

Offset: 0

Gus Wiseman, Mar 21 2018

sign

Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290973, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301470.

a[n_]:=a[n]=2(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]

O.g.f.: 2/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).

a(n) = Sum_t 2^k * (-1)^w where the sum is over all enriched r-trees of size n, k is the number of leaves, and w is the sum of leaves.

A295635 Write 2 - Zeta(s) in the form 1/Product_{n > 1}(1 + a(n)/n^s).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 6, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 12, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 10, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 6, 2, 1, 16, 2, 2

Offset: 2

Gus Wiseman, Nov 24 2017

nonn

Cf. A001055, A045778, A050376, A220418, A220420, A273866, A273873, A289501, A290261, A290262, A290971, A290973, A295279, A295632, A295636.

nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
-Solve[Table[-1==Sum[Times@@a/@f,{f,facs[n]}],{n,2,nn}],Table[a[n],{n,2,nn}]][[1,All,2]]

A295636 Write 2 - Zeta(s) in the form Product_{n > 1}(1 - a(n)/n^s).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 8, 1, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 8, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 1, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 8, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 3, 2, 1, 16, 2, 2, 2

Offset: 2

Gus Wiseman, Nov 24 2017

nonn

Cf. A001055, A045778, A050376, A220418, A220420, A273866, A273873, A289501, A290261, A290262, A290971, A290973, A295279, A295632, A295635.

nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
-Solve[Table[-1==Sum[Times@@a/@f,{f,Select[facs[n],UnsameQ@@#&]}],{n,2,nn}],Table[a[n],{n,2,nn}]][[1,All,2]]

a(n) = Sum_t (-1)^(v(t)-1) where the sum is over all strict tree-factorizations of n (see A295279 for definition) and v(t) is the number of nodes (branchings and leaves) in t.

A290320 Write 1 - t * x/(1-x) as an inverse power product 1/(1+c(1)x) * 1/(1+c(2)x^2) * 1/(1+c(3)x^3) * ... The sequence is a regular triangle where T(n,k) is the coefficient of t^k in c(n), 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 2, 1, 0, 1, 3, 4, 2, 0, 0, 1, 3, 5, 5, 3, 1, 0, 1, 4, 9, 13, 13, 9, 4, 1, 1, 4, 9, 13, 13, 9, 4, 1, 0, 1, 5, 14, 25, 30, 24, 12, 3, 0, 0, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 0, 1, 6, 21, 48, 75, 81, 60, 30, 10, 2, 0, 0

Offset: 1

Gus Wiseman, Jul 27 2017

An irregular triangle with only the nonzero coefficients is given by A290262.

			Triangle begins:
  1;
  1,  1;
  1,  1,  0;
  1,  2,  2,  1;
  1,  2,  2,  1,  0;
  1,  3,  4,  2,  0,  0;
  1,  3,  5,  5,  3,  1,  0;
  1,  4,  9, 13, 13,  9,  4,  1;
  1,  4,  9, 13, 13,  9,  4,  1,  0;
  1,  5, 14, 25, 30, 24, 12,  3,  0,  0;
  1,  5, 15, 30, 42, 42, 30, 15,  5,  1,  0;
  1,  6, 21, 48, 75, 81, 60, 30, 10,  2,  0,  0;

Cf. A220418, A273866, A289501, A290261, A290262.

nn=12;Solve[Table[Expand[SeriesCoefficient[Product[1/(1+c[k]x^k),{k,n}],{x,0,n}]]==-t,{n,nn}],Table[c[n],{n,nn}]][[1,All,2]]

A305572 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} a(n/d)^d.

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 2, 0, 10, 4, 2, 32, 2, 4, 42, 0, 2, 228, 2, 32, 138, 4, 2, 1536, 34, 4, 1514, 32, 2, 3940, 2, 0, 2058, 4, 162, 102944, 2, 4, 8202, 1536, 2, 51940, 2, 32, 207370, 4, 2, 3538944, 130, 3204, 131082, 32, 2, 15668836, 2082, 1536, 524298, 4, 2, 54327840

Offset: 1

Gus Wiseman, Jun 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6911

Cf. A196545, A273873, A281145, A289078, A289079, A289501, A290261, A290971, A291441, A300862-A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[a[n/y]^y,{y,Divisors[n]//Rest}];
Array[a,40]

A305572(n) = ((-1)^(n-1) + sumdiv(n,d,if(d==1,0,A305572(n/d)^d))); \\ Antti Karttunen, Dec 05 2021

a(n) = Sum_t (-1)^(n-k) where the sum is over all same-trees of weight n (see A281145 for definition) and k is the number of leaves.

A305610 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} binomial(a(n/d) + d - 1, d).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 2, 0, 6, 3, 2, 11, 2, 3, 12, 0, 2, 38, 2, 11, 14, 3, 2, 90, 8, 3, 68, 11, 2, 127, 2, 0, 18, 3, 16, 1194, 2, 3, 20, 90, 2, 173, 2, 11, 644, 3, 2, 5158, 10, 68, 24, 11, 2, 12762, 20, 90, 26, 3, 2, 12910, 2, 3, 1386, 0, 22, 289, 2, 11, 30, 219, 2

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A063834, A196545, A220418, A273866, A273873, A275870, A281145, A281146, A289078, A289079, A289501, A290261, A290971, A291441, A300862-A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[Binomial[a[n/d]+d-1,d],{d,Divisors[n]//Rest}];
Array[a,40]

A305610(n) = ((-1)^(n-1) + sumdiv(n,d,if(d==1,0,binomial(A305610(n/d)+d-1, d)))); \\ Antti Karttunen, Dec 05 2021

cp's OEIS Frontend

A301470 Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

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A302917 Solution to a(1) = 1 and Sum_y Product_i a(y_i) = 0 for each n > 1, where the sum is over all relatively prime or monic partitions of n.

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A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

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A301469 Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

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A295635 Write 2 - Zeta(s) in the form 1/Product_{n > 1}(1 + a(n)/n^s).

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A295636 Write 2 - Zeta(s) in the form Product_{n > 1}(1 - a(n)/n^s).

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A290320 Write 1 - t * x/(1-x) as an inverse power product 1/(1+c(1)x) * 1/(1+c(2)x^2) * 1/(1+c(3)x^3) * ... The sequence is a regular triangle where T(n,k) is the coefficient of t^k in c(n), 1 <= k <= n.

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A305572 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} a(n/d)^d.

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A305610 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} binomial(a(n/d) + d - 1, d).

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