A002865 -id:A002865 - cp's OEIS frontend

A182845 a(n) = A002865(2n-1)+A002865(2n).

1, 3, 6, 11, 20, 35, 58, 96, 154, 242, 375, 573, 861, 1282, 1886, 2745, 3961, 5667, 8038, 11323, 15836, 22001, 30383, 41715, 56953, 77363, 104566, 140668, 188397, 251247, 333689, 441474, 581890, 764215, 1000233, 1304815, 1696717, 2199591, 2843073, 3664312

Offset: 1

Omar E. Pol, Jan 24 2011

a(n) is also the length of the n-th "large mirror" of the "mirror" version of the shell model of partitions of A135010.

			a(1)=0+1=1. a(2)=1+2=3: a(3)=2+4=6. a(4)=4+7=11. a(5)=8+12=20. a(6)=14+21=35.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A000041, A002865, A135010. For another version see A182844.

Extended by Nathaniel Johnston, May 06 2011

A340525 Triangle read by rows: T(n,k) = A006218(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 0, 5, 0, 1, 8, 0, 3, 1, 10, 0, 5, 3, 2, 14, 0, 8, 5, 6, 2, 16, 0, 10, 8, 10, 6, 4, 20, 0, 14, 10, 16, 10, 12, 4, 23, 0, 16, 14, 20, 16, 20, 12, 7, 27, 0, 20, 16, 28, 20, 32, 20, 21, 8, 29, 0, 23, 20, 32, 28, 40, 32, 35, 24, 12, 35, 0, 27, 23, 40, 32, 56, 40, 56, 40, 36, 14

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.

			Triangle begins:
   1;
   3,  0;
   5,  0,  1;
   8,  0,  3,  1;
  10,  0,  5,  3,  2;
  14,  0,  8,  5,  6,  2;
  16,  0, 10,  8, 10,  6,  4;
  20,  0, 14, 10, 16, 10, 12,  4;
  23,  0, 16, 14, 20, 16, 20, 12,  7;
  27,  0, 20, 16, 28, 20, 32, 20, 21,  8;
  29,  0, 23, 20, 32, 28, 40, 32, 35, 24, 12;
  35,  0, 27, 23, 40, 32, 56, 40, 56, 40, 36, 14;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *  14   =  14
2      0   *  10   =   0
3      1   *   8   =   8
4      1   *   5   =   5
5      2   *   3   =   6
6      2   *   1   =   2
.           A006218
--------------------------
The sum of row 6 is 14 + 0 + 8 + 5 + 6 + 2 = 35, equaling A006128(6).

Mirror of A245095.
Row sums give A006128 (conjectured).
Columns 1, 3 and 4 are A006218.
Column 2 gives A000004.
Leading diagonal gives A002865.
Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340524, A340426.

A110142 Limit of rows of triangle A110141 after dividing respectively by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1.

Original entry on oeis.org

1, 2, 3, 8, 4, 6, 5, 48, 8, 18, 6, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 144, 40, 24, 14, 162, 18, 20, 9, 3840, 192, 144, 48, 30, 64, 16, 72, 21, 24, 50, 10, 1152, 240, 96, 56, 324, 36, 40, 18, 90, 96, 24, 28, 30, 11, 46080, 1536, 864, 288, 120, 256, 64, 144, 42, 48, 100

Offset: 1

Paul D. Hanna, Jul 13 2005

nonn

Row n of triangle A110141 lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k). A002865 equals the first differences of the partition numbers. A110144 lists terms at positions p(n)+1.

			Row 6 of A110141 is: {720,48,18,16,8,6,5,48,8,18,6};
divided respectively by: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!}
with {4!,3!,2!,1!,0!} each occurring {1,1,2,2,4} times after 6!,
yields the initial A000041(6)=11 terms: {1,2,3,8,4,6,5,48,8,18,6}.
Sum of reciprocal terms at positions p(5)+1 through p(6) =
1/48 + 1/8 + 1/18 + 1/6 = 1-1+1/2!-1/3!+1/4!-1/5!+1/6!.
Other patterns emerge when the terms are read by groups
of terms in positions p(n-1)+1 through p(n):
1;
2;
3;
8,4;
6, 5;
48,8, 18,6;
24,10, 12,7;
384,32,36,12, 15,32,8;
144,40,24,14, 162,18,20,9;
3840,192,144,48,30,64,16, 72,21,24,50,10;
1152,240,96,56,324,36,40,18, 90,96,24,28,30,11;
46080,1536,864,288,120,256,64,144,42,48,100,20, 1944,108,60,27,384,32,35,72,12;
11520,1920,576,336,1296,144,160,72,180,192,48,56,60,22, 648,126,72,150,30,160,36,40,42,13; ...

Cf. A110141, A110144, A000041, A002865.

a(p(n)) = n where p(n) = A000041(n) (partition numbers) for n>=1. Sum_{k=p(n-1)+1..p(n)} 1/a(k) = Sum_{k=0..n} (-1)^k/k!, for n>1.

A124222 Triangular array read by rows, with shape sequence A002865, generated from A123682.

Original entry on oeis.org

1, 1, 3, 5, 0, 9, 2, 13, 7, 1, 0, 19, 18, 4, 2, 25, 36, 12, 10, 0, 2, 0, 33, 64, 26, 32, 2, 11, 2, 1, 41, 103, 49, 78, 7, 40, 13, 6, 1, 0, 3, 0

Offset: 1

Alford Arnold, Oct 19 2006

The shape sequence for A124222 is essentially A002865 since its columns are based on source partitions (1,22,33,222,44,332,2222,333,...). The row sum sequence for A124222 is essentially A001045.

			The array begins
1
1
3
5 0
9 2
13 7 1 0
19 18 4 2
25 36 12 10 0 2 0
33 64 26 32 2 11 2 1
41 103 49 78 7 40 13 6 1 0 3 0

Cf. A001045 (row sums), A002865, A099392, A123682, A123683.

A171239 Triangle read by rows extracted from convolution product (1,2,3,...) * A002865: (1,1,2,2,4,4,7,8,...)

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 6, 4, 5, 4, 8, 6, 8, 5, 6, 7, 8, 12, 8, 10, 6, 7, 8, 14, 12, 16, 10, 12, 7, 8, 8, 14, 12, 16, 10, 12, 7, 8, 12, 16, 21, 16, 20, 12, 14, 8, 9, 14, 24, 24, 28, 20, 24, 14, 10, 9, 10

Offset: 1

Gary W. Adamson, Dec 05 2009

Row sums = A058682 starting (1, 3, 7, 13, 23, 37, 58, 87,...).

			The triangle = upward sloping diagonals of multiplication table:
.
1,..2,..3,..4,..5,..6,..7,...
1,..2,..3,..4,..5,..6,..7,...
2,..4,..6,..8,.10,.12,.14,...
2,..4,..6,..8,.10,.12,.14,...
... First few rows of the triangle =
.
1;
1, 2;
2, 2, 3;
2, 4, 3, 4;
4, 4, 6, 4, 5;
4, 8, 6, 8, 5, 6;
7, 8, 12, 8, 10, 6, 7;
8, 14, 12, 16, 10, 12, 7, 8;
12, 16, 21, 16, 20, 12, 14, 8, 9;
14, 24, 24, 28, 20, 24, 14, 10, 9, 10;
21, 28, 36, 32, 35, 24, 28, 16, 18, 10, 11;
24, 42, 42, 48, 40, 42, 28, 32, 18, 20, 11, 12;
34, 48, 63, 56, 60, 48, 49, 32, 36, 20, 22, 12, 13;
41, 68, 72, 84, 70, 72, 56, 56, 36, 40, 22, 24, 13, 14;
...

Cf. A002865, A058682

Triangle read by rows, ascending diagonals of an array formed by the product of

(1,2,3,...) and (1,1,2,2,4,4,7,8,12,...).

A345971 Irregular triangle T(n,k) read by rows in which n-th row lists numbers of series-reduced trees realized by respective degree sequences in n-th row of A345970; n >= 4, 1 <= k <= A002865(n-2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 1, 1, 2, 3, 2, 2, 4, 6, 2, 1, 1, 1, 1, 1, 2, 3, 3, 2, 2, 4, 9, 4, 8, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 2, 3, 1, 4, 9, 6, 9, 2, 8, 14, 4, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 3, 2, 2, 4, 9, 9, 9, 9, 4, 8, 25, 14, 15

Offset: 4

Washington Bomfim, Jul 05 2021

The first floor((n-2)/2) terms of n-th row are all 1 thus the first floor((n-2)/2) degree sequences of n-th row of A345970 have only one realization.
Let a "unigraph" be a graph which is the only realization of its degree sequence. Among all series-reduced trees on n vertices we have floor((n-2)/2) + [n>=8] * [(n-8) == 0 (mod 3)] unigraphs.
Let T be a series-reduced tree of diameter dT, with h nodes of degree >= 3, and degree sequence D. If h <= 2, dT <= 3, and T is a unigraph [R. H. Johnson Corollary 2.3]. For each degree sequence the value of h is equal to the number of parts of a unique partition [Myerson], thus the number of unigraphs would be equal to the number of partitions (without parts 1) of n-2 with at most 2 parts, which is floor((n-2)/2). Degree sequences of the form [d,d,d,1,..,1] give an additional unigraph when n >= 8 and (n-8) == 0 (mod 3). These unigraphic sequences can be depicted as:
||_| , |_|| , ...
           | | |
A250308(n) = Sum_{ k= 1 .. A002865(2*n-2) } ( T(2*n,k) * odd( Decode( A345970(2*n, k) ) ), where odd(D) is 1 if all d in D are odd, and 0 otherwize.

			Row number 9 is {1, 1, 1, 2} because the 9th row of A345970 is {4864, 8320, 9856, 11200} which can be decoded (using Decode() of A345970) to the 4-degree sequences [8,1,1,1,1,1,1,1,1], which obviously has just 1 realization, [6,3,1,1,1,1,1,1,1], [5,4,1,1,1,1,1,1,1], that also have one, and [4,3,3,1,1,1,1,1,1] which realizes the 2 trees:
.
       *  *  *            *  *  *
       |  |  |            |  |  |
    *--0--*--*         *--*--0--*
       |     |               |  |
       *     *               *  *
.
Triangle begins
   n \ k 1  2  3  4  5  6  7  8  9 10 11 12
    4    1;
    5    1;
    6    1, 1;
    7    1, 1;
    8    1, 1, 1, 1;
    9    1, 1, 1, 2;
   10    1, 1, 1, 1, 2, 2, 2;
   11    1, 1, 1, 1, 2, 3, 1, 4;
   12    1, 1, 1, 1, 1, 2, 3, 2, 2, 4, 6, 2;
   ...

Nima Amini, Not That Good, Will Hunting
R. H. Johnson, Properties of unique realizations-a survey, Discrete Mathematics, Volume 31 January, 1980 pp 185-192.
B. D. McKay, Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.
Gerry Myerson, Problems2004
Index entries for sequences related to trees

Cf. A000014 (row sums), A002865 (row widths), A345970 (encoded degree sequences), A250308.

D_Generator(n) = { my(D = vectorsmall(n), j);
M = Map();                                \\ For each partition of n-2, "P",
forpart( P = n-2,                         \\ P without parts 1, make D =
   for(i = 1, n-#P, D[i] = 1); j = n-#P;  \\ [1..1 0..0], n-#P terms 1, and
   for(i = 1,   #P, D[j++] = P[i] + 1);   \\ #p terms 0. Complete D.
   mapput(M, D, 0) , [2, n-2] )        \\ store D.
};
EdgesList2D(n, Tr) = {my(D = vectorsmall(n), E = strsplit(Tr, "  "), u_v);
for(j = 1, n-1, u_v = strsplit(E[j], " "); u_v = eval(u_v);
   D[ u_v[1]+1 ]++; D[ u_v[2]+1 ]++); vecsort(D) };
                                      \\ Using files hitree4.txt etc from McKay.
Rows(r1, r2) = {my(Trees, D, j, C); for(n = r1, r2,
Trees = readstr(Str("hitree", n, ".txt")); D_Generator(n);
for(i = 1, #Trees, D = EdgesList2D(n, Trees[i]); j = mapget(M, D); mapput(~M, D, j+1));
C = Mat(M)[, 2]; print1(n" "); for(i = 1, #C, print1(C[i]", ")); print() ) };

A128627 Triangle read by rows. Convolution triangle based on A002865.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 2, 5, 3, 4, 0, 1, 4, 6, 9, 4, 5, 0, 1, 4, 13, 12, 14, 5, 6, 0, 1, 7, 16, 28, 20, 20, 6, 7, 0, 1, 8, 30, 39, 50, 30, 27, 7, 8, 0, 1, 12, 40, 78, 76, 80, 42, 35, 8, 9, 0, 1, 14, 66, 115, 161, 130, 119, 56, 44, 9, 10, 0, 1

Offset: 1

Alford Arnold, Mar 22 2007

Triangular array illustrating the application of cyclic partitions to the computation of partitions of an integer into parts of k kinds (cf. A060850).
The array is constructed by summing sequences associated with each cyclic partition as indicated below: (n' here denotes the sum of preceding sequences).
    1     2     3
   1     3     6
  4'     2     5     9
    1     2     3     4
   1     4     9    16
  5'     2     6    12    20
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     3     6    10    15    21    28    36    45
  1     4    10    20    35    56    84   120   165
  6'     4    13    28    50    80   119   168   228   300
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     4     9    16    25    36    49    64    81
  1     6    18    40    75   126   196   288   405
  7'     4    16    39    76   130   204   301   424   576
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     4     9    16    25    36    49    64    81
   1     3     6    10    15    21    28    36    45
  1     6    18    40    75   126   196   288   405
  1     6    18    40    75   126   196   288   405
 1     5    15    35    70   126   210   330   495
  8'     7    30    78   161   290   477   735  1078  1521

			The diagonal 9th diagonal of A060850 is 22 185 810 2580 6765 ... and can be computed from a(n) and A007318 as illustrated:
   1
   0    1
   1    0    1
   1    2    0    1
   2    2    3    0
   2    5    3    4
   4    6    9    4
   4   13   12   14
   7   16   28   20
       30   39   50
            78   76
                161
times
   1
   1    9
   1    8   45
   1    7   36  165
   1    6   28  120
   1    5   21   84
   1    4   15   56
   1    3   10   35
   1    2    6   20
        1    3   10
             1    4
                  1
yields
   1
   0    9
   1    0   45
   1   14    0  165
   2   12   84    0
   2   25   63  336
   4   24  135  224
   4   39  120  490
   7   32  168  400
       30  117  500
            78  304
                161
summing to
  22  185  810 2580 ...
Triangle T(n, k) starts:
  [ 1] 1;
  [ 2] 0,  1;
  [ 3] 1,  0,  1;
  [ 4] 1,  2,  0,  1;
  [ 5] 2,  2,  3,  0,  1;
  [ 6] 2,  5,  3,  4,  0,  1;
  [ 7] 4,  6,  9,  4,  5,  0,  1;
  [ 8] 4, 13, 12, 14,  5,  6,  0,  1;
  [ 9] 7, 16, 28, 20, 20,  6,  7,  0,  1;
  [10] 8, 30, 39, 50, 30, 27,  7,  8,  0,  1;

Cf. A002865, A060850, A007318.

# Using function A002865 and function PMatrix from A357368.
A128627Triangle := proc(dim) local M, Row, r;
M := PMatrix(dim, n -> A002865(n-1));
Row := r -> convert(linalg:-row(M, r), list)[2..r];
for r from 2 to dim do lprint(Row(r)) od end:
A128627Triangle(11); # Peter Luschny, Oct 03 2022

New name by Peter Luschny, Oct 03 2022

A133734 A000012 * A002865 as a diagonalized matrix.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 2, 4, 1, 0, 1, 1, 2, 2, 4, 4, 1, 0, 1, 1, 2, 2, 4, 4, 7, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8

Offset: 0

Gary W. Adamson, Sep 22 2007

Row sums = the partition numbers, A000041: (1, 1, 2, 3, 5, 7, 11, 15, ...).

			First few rows of the triangle:
  1;
  1, 0;
  1, 0, 1;
  1, 0, 1, 1;
  1, 0, 1, 1, 2;
  1, 0, 1, 1, 2, 2;
  1, 0, 1, 1, 2, 2, 4;
  1, 0, 1, 1, 2, 2, 4, 4;
  1, 0, 1, 1, 2, 2, 4, 4, 7;
  ...

Cf. A000041, A002865.

A000012 * A002865 as a diagonalized matrix, the latter = an infinite lower triangular matrix with A002865: (1, 0, 1, 1, 2, 2, 4, 4, 7, 8, ...) as the main diagonal and the rest zeros. Triangle read by rows, n-th row = first (n+1) terms of A002865.

A182748 Triangle T(n,k) read by rows in which row n lists the first n terms of A002865, except the first term, in reverse order together with 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 4, 2, 2, 1, 1, 0, 0, 4, 4, 2, 2, 1, 1, 0, 0, 7, 4, 4, 2, 2, 1, 1, 0, 0, 8, 7, 4, 4, 2, 2, 1, 1, 0, 0, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 0

Offset: 0

Omar E. Pol, Dec 01 2010

			Triangle (0<=k<=n) begins:
0,
0, 0,
1, 0, 0,
1, 1, 0, 0,
2, 1, 1, 0, 0,
2, 2, 1, 1, 0, 0,
4, 2, 2, 1, 1, 0, 0,
4, 4, 2, 2, 1, 1, 0, 0,
7, 4, 4, 2, 2, 1, 1, 0, 0,
8, 7, 4, 4, 2, 2, 1, 1, 0, 0,

Cf. A002865, A182746, A182747.

A345970 Irregular triangle T(n,k) read by rows in which n-th row lists in colex order all series-reduced tree degree sequences D of n nodes encoded as t = Product_{d in D} prime(d); n >= 4, 1 <= k <= A002865(n-2).

Original entry on oeis.org

40, 112, 352, 400, 832, 1120, 2176, 3520, 3136, 4000, 4864, 8320, 9856, 11200, 11776, 21760, 23296, 30976, 35200, 31360, 40000, 29696, 48640, 60928, 73216, 83200, 98560, 87808, 112000, 63488, 117760, 136192, 191488, 173056, 217600, 232960, 309760, 275968, 352000, 313600, 400000

Offset: 4

Washington Bomfim, Jul 01 2021

Tree degree sequences of n nodes are in one-to-one correspondence with the partitions of n-2, as for instance set out in Myerson's collection of problems [Myerson]. For series-reduced trees, these partitions have no part 1.
Given a term t, the respective degree sequence D is determined by Decode(t). See second (PARI) entry.
A250308(n) = Sum_{k= 1 .. A002865(2*n-2) } ( A345971(2*n,k) * odd( Decode( T(2*n,k) ) ), where odd(D) is 1 if all d in D are odd, and 0 otherwize.

			Triangle begins:
  n \ k|  1    2 ...           n \ k| 1                2            ...
  -----+-------------          -----+-----------------------------------
  4    |   40;                 4    |       [3,1,1,1];
  5    |  112;                 5    |     [4,1,1,1,1];
  6    |  352,  400;    <=>    6    |   [5,1,1,1,1,1],   [3,3,1,1,1,1];
  7    |  832, 1120;           7    | [6,1,1,1,1,1,1], [4,3,1,1,1,1,1];
  ...                          ...
Row n = 7 follows from table
                                                                         .
  +---------------------+------------------+---------------------------+
  | Partitions of n-2 = |                  |                           |
  | 5 without parts 1   | Degree sequences |       Encoding            |
  +---------------------+------------------+---------------------------+
  |                 [5] |    6,1,1,1,1,1,1 |            prime(6) * 2^6 |
  |              [2, 3] |    4,3,1,1,1,1,1 | prime(4) * prime(3) * 2^5 |
  +---------------------+------------------+---------------------------+

Gerry Myerson, Problems 2004
Index entries for sequences computed from indices in prime factorization

Cf. A002865 (row widths), A265127 (column k=1), A345971 (number of trees by degree sequence), A344122 (free tree degree sequences), A250308.

Row(n) = {my(j=0, V = vector(numbpart(n-2) - numbpart(n-3)));
forpart(P=n-2, V[j++] = prod(k=1,#P, prime(P[k]+1)) << (n-#P),[2, n-2]); V};

Decode(t) = {my(V = [], i = 1, p); while(t > 1, p = prime(i); while(t % p == 0, t /= p; V = concat(V, Vec(i)) ); i++); vecsort(V, (x,y)->y-x) };

cp's OEIS Frontend

A182845 a(n) = A002865(2*n-1)+A002865(2*n).

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A340525 Triangle read by rows: T(n,k) = A006218(n-k+1)*A002865(k-1), 1 <= k <= n.

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A110142 Limit of rows of triangle A110141 after dividing respectively by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1.

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A124222 Triangular array read by rows, with shape sequence A002865, generated from A123682.

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A171239 Triangle read by rows extracted from convolution product (1,2,3,...) * A002865: (1,1,2,2,4,4,7,8,...)

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A345971 Irregular triangle T(n,k) read by rows in which n-th row lists numbers of series-reduced trees realized by respective degree sequences in n-th row of A345970; n >= 4, 1 <= k <= A002865(n-2).

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PARI

A128627 Triangle read by rows. Convolution triangle based on A002865.

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Maple

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A133734 A000012 * A002865 as a diagonalized matrix.

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A182748 Triangle T(n,k) read by rows in which row n lists the first n terms of A002865, except the first term, in reverse order together with 0.

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A345970 Irregular triangle T(n,k) read by rows in which n-th row lists in colex order all series-reduced tree degree sequences D of n nodes encoded as t = Product_{d in D} prime(d); n >= 4, 1 <= k <= A002865(n-2).

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A182845 a(n) = A002865(2n-1)+A002865(2n).