A010766 -id:A010766 - cp's OEIS frontend

A212119 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the number of divisors d of n with min(d, n/d) = k.

1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0

Offset: 1

Omar E. Pol, Jul 02 2012

Column k lists the numbers A040000: 1, 2, 2, 2, 2... interleaved with k-1 zeros, starting in row k^2.
The sum of row n gives A000005(n), the number of divisors of n.
T(n,k) is also the number of divisors of n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212120.
It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253. - Omar E. Pol, Dec 03 2013

			Row 10 gives 2, 2, 0 therefore the sums of row 10 is 2+2+0 = 4, the same as A000005(10), the number of divisors of 10.
Written as an irregular triangle the sequence begins:
1;
2;
2;
2, 1;
2, 0;
2, 2;
2, 0;
2, 2;
2, 0, 1;
2, 2, 0;
2, 0, 0;
2, 2, 2;
2, 0, 0;
2, 2, 0;
2, 0, 2;
2, 2, 0, 1;
2, 0, 0, 0;
2, 2, 2, 0;
2, 0, 0, 0;
2, 2, 0, 2;
2, 0, 2, 0;
2, 2, 0, 0;
2, 0, 0, 0;
2, 2, 2, 2;
2, 0, 0, 0, 1;

Omar E. Pol, Diagram of divisors, figure 1, figure 2
Omar E. Pol, Illustration of initial terms and of row sums

Row sums give A000005. Column 1 is A040000. Column 2 gives the absolute values of A176742.

Cf. A006218, A027750, A010766, A147861, A163100, A196020, A210959, A212120, A211343, A221645, A228812-A228814.

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 39, 78, 161, 324, 658, 1316, 2657, 5314, 10668, 21347, 42777, 85554, 171290, 342580, 685498, 1371037, 2742733, 5485466, 10972351, 21944711, 43892080, 87784323, 175574004, 351148008, 702307038, 1404614076, 2809249582, 5618499824, 11237042426

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

This is a weaker condition than achirality (cf. A003238).

			The a(1) = 1 through a(6) = 18 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o)(o))   (((ooo)))
                          ((o(o)))   ((o(oo)))
                          (o((o)))   ((oo(o)))
                          ((((o))))  (o((oo)))
                                     (o(o)(o))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002541, A003238, A010766, A126656, A014668, A167865, A214577, A316782, A317099, A317100, A317712, A320230.

saue[n_]:=Sum[If[SameQ@@DeleteCases[ptn,1],If[DeleteCases[ptn,1]=={},1,saue[DeleteCases[ptn,1][[1]]]],0],{ptn,IntegerPartitions[n-1]}];
Table[saue[n],{n,15}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=2, n-1, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(n) = 1 + Sum_{k = 2..n-1} floor((n-1)/k) * a(k).

a(n) ~ c * 2^n, where c = 0.3270422384018894564479397100499014525700668391191792769625407295138546463... - Vaclav Kotesovec, Sep 07 2019

A277647 Triangle T(n,k) = floor(n/sqrt(k)) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
2, 1, 1, 1;
3, 2, 1, 1, 1, 1, 1, 1, 1;
4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
5, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10416 (the first 31 rows of the triangle)

Cf. A010766, A277646, A277648.

The 1000th row is A033432.

A277647:=func;
[A277647(n,k):k in[1..n^2],n in[1..7]];

Table[Floor[n/Sqrt@ k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

row(n) = for(k=1, n^2, print1(floor(n/sqrt(k)), ", ")); print("")
trianglerows(n) = for(k=1, n, row(k))
/* Print initial five rows of triangle as follows: */
trianglerows(5) \\ Felix Fröhlich, Nov 12 2016

T(n,k) = A000196(A277646(n,k)).

T(n,k)sqrt(k) <= n < (T(n,k)+1)sqrt(k).

A014668 a(1) = 1, a(n) = Sum_{k=1..n-1} Sum_{d|k} a(d).

Original entry on oeis.org

1, 1, 3, 7, 16, 33, 71, 143, 295, 594, 1206, 2413, 4871, 9743, 19559, 39138, 78428, 156857, 314047, 628095, 1256809, 2513693, 5028594, 10057189, 20116979, 40233975, 80472823, 160945945, 321901713, 643803427, 1287627061, 2575254123, 5150547536, 10301096282

Offset: 1

Benoit Cloitre, Jun 24 2003

nonn

Equals eigensequence of triangle A010766 and starting (1, 3, 7, 16, 33, ...) = row sums of triangle A163313. - Gary W. Adamson, Jul 30 2009. Gary Adamson's comment may be restated as "This sequence shifts left by one place under the floor transform." - N. J. A. Sloane, Feb 05 2016

The Gould & Quaintance reference, published in 2007, says incorrectly that this sequence is not in the OEIS. - Olivier Gérard, Oct 20 2011

Alois P. Heinz, Table of n, a(n) for n = 1..1000
H. W. Gould and J. Quaintance, Floor and Roof function analog of the Bell Numbers, INTEGERS, 7 (2007), #A58.

Cf. A010766, A163313. - Gary W. Adamson, Jul 30 2009

with(numtheory):
a:= proc(n) option remember;
      `if`(n=1, 1, add(add(a(d), d=divisors(k)), k=1..n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 28 2011

a[1] = 1; a[n_] := a[n] = Sum[Sum[a[d], {d, Divisors[k]}], {k, 1, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 07 2015 *)

// an=vector(100); a(n)=if(n<0,0,an[n]); // an[1]=1; for(n=2,100,an[n]=sum(k=1,n-1,sumdiv(k,d,a(d))))

a(n) is asymptotic to c*2^n where c = 0.59960731361450033896934...
a(n+1) = Sum_{k=1..n} a(k)*floor(n/k). - Franklin T. Adams-Watters, Mar 21 2017
G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, Feb 25 2020

A118013 Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 9, 4, 3, 16, 8, 5, 4, 25, 12, 8, 6, 5, 36, 18, 12, 9, 7, 6, 49, 24, 16, 12, 9, 8, 7, 64, 32, 21, 16, 12, 10, 9, 8, 81, 40, 27, 20, 16, 13, 11, 10, 9, 100, 50, 33, 25, 20, 16, 14, 12, 11, 10, 121, 60, 40, 30, 24, 20, 17, 15, 13, 12, 11, 144, 72, 48, 36, 28, 24, 20, 18, 16, 14

Offset: 1

Reinhard Zumkeller, Apr 10 2006

T(n,1) = A000290(n); T(n,n) = n;
T(n,2) = A007590(n) for n>1;
T(n,3) = A000212(n) for n>2;
T(n,4) = A002620(n) for n>3;
T(n,5) = A118015(n) for n>4;
T(n,6) = A056827(n) for n>5;
central terms give A008574: T(2*k-1,k) = 4*(k-1)+0^(k-1);
row sums give A118014.

			Triangle begins:
1,
4, 2,
9, 4, 3,
16, 8, 5, 4,

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

Cf. A010766.

a118013 n k = a118013_tabl !! (n-1) !! (k-1)
a118013_row n = map (div (n^2)) [1..n]
a118013_tabl = map a118013_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

T(n,k)=n^2\k \\ Charles R Greathouse IV, Jan 15 2012

A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 3, 1, 1, 6, 1, 1, 1, 1, 12, 3, 3, 1, 1, 1, 8, 4, 1, 1, 1, 1, 1, 15, 3, 3, 3, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 18, 6, 3, 3, 3, 1, 1, 1, 1, 1, 12, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 14, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 3, 6, 3, 3, 3, 3, 1

Offset: 1

Omar E. Pol, May 01 2020

The one-part partition n = n is included in the count.
The column k is related to (k+2)-gonal numbers, assuming that 2-gonals are the nonnegative numbers, 3-gonals are the triangular numbers, 4-gonals are the squares, 5-gonals are the pentagonal numbers, and so on.
Note that the number of parts for T(n,0) = A000203(n), equaling the sum of the divisors of n.
For fixed k>0, Sum_{j=1..n} T(j,k) ~ 2^(3/2) * n^(3/2) / (3*sqrt(k)). - Vaclav Kotesovec, Oct 23 2024

			Square array starts:
   n\k|   0  1  2  3  4  5  6  7  8  9 10 11 12
   ---+---------------------------------------------
   1  |   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   2  |   3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   3  |   4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   4  |   7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   5  |   6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   6  |  12, 4, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, ...
   7  |   8, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, ...
   8  |  15, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, ...
   9  |  13, 6, 4, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, ...
  10  |  18, 5. 3. 1. 3. 1, 3, 1, 3, 1, 1, 1, 1, ...
  11  |  12, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, ...
  12  |  28, 4, 6, 4, 3, 1, 3, 1, 3, 1, 3, 1, 1, ...
  ...
For n = 9 we have that:
For k = 0 the partitions of 9 into consecutive parts that differ by 0 (or simply: the partitions of 9 into equal parts) are [9], [3,3,3], [1,1,1,1,1,1,1,1,1]. In total there are 13 parts, so T(9,0) = 13.
For k = 1 the partitions of 9 into consecutive parts that differ by 1 (or simply: the partitions of 9 into consecutive parts) are [9], [5,4], [4,3,2]. In total there are six parts, so T(9,1) = 6.
For k = 2 the partitions of 9 into consecutive parts that differ by 2 are [9], [5, 3, 1]. In total there are four parts, so T(9,2) = 4.

Columns k: A000203 (k=0), A204217 (k=1), A066839 (k=2), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6), A377300 (k=7), A377301 (k=8).
Triangles whose row sums give the column k: A127093 (k=0), A285914 (k=1), A330466 (k=2) (conjectured), A330888 (k=3), A334462 (k=4), A334540 (k=5), A339947 (k=6).
Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).
Tables of partitions related to column k: A010766 (k=0), A286001 (k=1), A332266 (k=2), A334945 (k=3), A334618 (k=4).
Polygonal numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).
Cf. A051735, A237048, A303300, A330887, A334460, A334465, A334946.
Cf. A038040, A245579, A060872, A334467, A327262, A334733, A334953.
Cf. A057145, A323345.

nmax = 14;
col[k_] := col[k] = CoefficientList[Sum[n x^(n(k n - k + 2)/2)/(1 - x^n), {n, 1, nmax}] + O[x]^(nmax+1), x];
T[n_, k_] := col[k][[n+1]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

The g.f. for column k is Sum_{n>=1} n*x^(n*(k*n-k+2)/2)/(1-x^n). (For proof, see A330889. - N. J. A. Sloane, Nov 21 2020)

A003988 Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is [ i/j ].

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 1, 0, 0, 5, 2, 1, 0, 0, 6, 2, 1, 0, 0, 0, 7, 3, 1, 1, 0, 0, 0, 8, 3, 2, 1, 0, 0, 0, 0, 9, 4, 2, 1, 1, 0, 0, 0, 0, 10, 4, 2, 1, 1, 0, 0, 0, 0, 0, 11, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 12, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 13, 6, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 14, 6, 4, 2, 2, 1, 1, 0, 0, 0, 0

Offset: 1

Marc LeBrun

Another version of A010766.

Reinhard Zumkeller, Table of n, a(n) for n = 1..5050

Cf. A010766, A003056, A049581, A003991, A004247, A077049.

Row sums are in A006218. Antidiagonal sums are in A002541.

a003988 n k = (n + 1 - k) `div` k
a003988_row n = zipWith div [n,n-1..1] [1..n]
a003988_tabl = map a003988_row [1..]
-- Reinhard Zumkeller, Apr 13 2012

t[n_, k_] := Quotient[n, k]; Table[t[n-k+1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 21 2013 *)

From Franklin T. Adams-Watters, Jan 28 2006: (Start)
T(n,k) = Sum_{i=1..k} A077049(n,i).
G.f.: (1/(1-x))*Sum_{k>0} x^k*y^k/(1-x^k) = (1/(1-x))*Sum_{k>0} x^k * y / (1 - x^k y) = (1/(1-x)) * Sum_{k>0} x^k * Sum_{d|k} y^d = A(x,y)/(1-x) where A(x,y) is the g.f. of A077049. (End)
T(n,k) = floor((n + 1 - k) / k). - Reinhard Zumkeller, Apr 13 2012

More terms from James Sellers

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).

Original entry on oeis.org

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

N. J. A. Sloane

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

A064649 gives the row sums.
Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).
See also A247917, A047918, A047919.

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017

A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 4, 4, 6, 3, 8, 3, 6, 7, 4, 3, 8, 3, 8, 7, 6, 3, 8, 4, 6, 4, 8, 3, 11, 3, 4, 7, 6, 7, 8, 3, 6, 7, 8, 3, 11, 3, 8, 8, 6, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 6, 3, 16, 3, 6, 8, 4, 7, 12, 3, 8, 7, 14, 3, 8, 3, 6, 8, 8

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k]. a(n) = 4 when n is in A123710. a(n) = 8 when n is in A123711. a(n) = 16 when n is in A123712.

			a(n) = 3 when n is an odd prime.
a(n) = 7 when n is the product of two different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]
a(n) = 15 when n is the product of three different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]

M. F. Hasler, Table of n, a(n) for n = 1..500
Peter Luschny, Re: Is the A123706 triangle an extension of the Moebius function?, seqcomp list, Feb 12 2012

Cf. A123706, A123707, A123708, A123710, A123711, A123712; A010766.

Moebius[i_,j_]:=If[Divisible[i,j], MoebiusMu[i/j],0];
A123709[n_]:=Length[Select[Table[Moebius[n,j]-Moebius[n,j+1],{j,1,n}],#!=0&]];
Array[A123709, 500] (* Enrique Pérez Herrero, Feb 13 2012 *)

{a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1); sum(k=1,n,if(M[n,k]==0,0,1))}

A123709(n)=#select((matrix(n, n, r, c, r\c)^-1)[n,],x->x)  \\ M. F. Hasler, Feb 12 2012

A123709(n)={ my(t=moebius(n)); sum(k=2,n, t+0 != t=if(n%k,0,moebius(n\k)))+1}  /* the "t+0 != ..." is required because of a bug in PARI versions <= 2.4.2, maybe beyond, which seems to be fixed in v. 2.5.1 */ \\ M. F. Hasler, Feb 13 2012

a(n) = 2^(m+1) - 1 when n is the product of m distinct odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
For any k>1, a(n)=2^k if, and only if, n is a nonsquarefree number with A001221(n) = k-1 (= omega(n), number of distinct prime factors), with the only exception of a(n=6)=2^2. - M. F. Hasler, Feb 12 2012
A123709(n) = 1 + #{ k in 1..n-1 | Moebius(n,k+1) <> Moebius(n,k) }, where Moebius(n,k)={moebius(n/k) if n=0 (mod k), 0 else}, cf. link to message by P. Luschny. - M. F. Hasler, Feb 13 2012

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 44, 70, 119, 189, 314, 506, 830, 1336, 2186, 3522, 5737, 9266, 15047, 24313, 39444, 63759, 103322, 167098, 270616, 437714, 708676, 1146390, 1855582, 3002017, 4858429, 7860454, 12720310, 20580764, 33303260, 53884144, 87190964

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

This is a weaker condition than achirality (cf. A167865).

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 1 through a(8) = 9 rooted trees:
  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
               (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                        (oo(oo))  (oo(ooo))   (oo(oooo))
                                  (ooo(oo))   (ooo(ooo))
                                  ((oo)(oo))  (oooo(oo))
                                  (o(o(oo)))  (o(o(ooo)))
                                              (o(oo)(oo))
                                              (o(oo(oo)))
                                              (oo(o(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269.

saum[n_]:=Sum[If[DeleteCases[ptn,1]=={},1,saum[DeleteCases[ptn,1][[1]]]],{ptn,Select[IntegerPartitions[n-1],And[Length[#]!=1,SameQ@@DeleteCases[#,1]]&]}];
Array[saum,20]

seq(n)={my(v=vector(n)); v[1]=1; for(n=3, n, v[n] = 1 + sum(k=2, n-2, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2..n-2} floor((n-1)/k) * a(k).

cp's OEIS Frontend

A212119 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the number of divisors d of n with min(d, n/d) = k.

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A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

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Mathematica

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A277647 Triangle T(n,k) = floor(n/sqrt(k)) for 1 <= k <= n^2, read by rows.

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Magma

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A014668 a(1) = 1, a(n) = Sum_{k=1..n-1} Sum_{d|k} a(d).

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Maple

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A118013 Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.

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Haskell

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A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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A003988 Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is [ i/j ].

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A047916 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).