A273866 -id:A273866 - cp's OEIS frontend

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.

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]]

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

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

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