A083480 -id:A083480 - cp's OEIS frontend

A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

5, 32, 113, 299, 664, 1309, 2366, 4002, 6423, 9878, 14663, 21125, 29666, 40747, 54892, 72692, 94809, 121980, 155021, 194831, 242396, 298793, 365194, 442870, 533195, 637650, 757827, 895433, 1052294, 1230359, 1431704, 1658536, 1913197

Offset: 1

Alford Arnold, Dec 29 2003; extended May 04 2005

The diagonals are finite and sum to A047970.
Values appear to be a transformation of A006468 (rooted planar maps). Also known as well-labeled trees (cf. A000168).
First differences of the conjectured polynomial formula for A006468. [From R. J. Mathar, Jun 26 2010]

			The array begins
1
2
4
7 1
11 5
16 14 2
22 30 12
29 55 39 5
37 91 95 32 1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A105552, A006468.
Cf. A006011, A034261.
Cf. A000124 (column 1), A000330 (column 2), A086602 (column 3), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

Row sums are powers of 2.
a(n) = A000330(n) + A006011(n+1) + A034263(n-1).
a(n)= +6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). G.f.: x*(5+2*x-4*x^2+x^3)/(x-1)^6. a(n) = n*(n+1)*(4*n^3+51*n^2+159*n+86)/120. [From R. J. Mathar, Jun 26 2010]

Extended beyond a(8) by R. J. Mathar, Jun 26 2010

A089349 Triangle read by rows: counts the permutations of partitions described in A083480. Both Arrays have shape sequence A083479.

Original entry on oeis.org

1, 2, 4, 7, 1, 11, 5, 16, 14, 2, 22, 30, 12, 29, 55, 39, 5, 37, 91, 95, 32, 1, 46, 140, 195, 113, 18, 56, 204, 357, 299, 101, 7, 67, 285, 602, 664, 357, 71, 2, 79, 385, 954, 1309, 978, 350, 41, 92, 506, 1440, 2366, 2274, 1204, 292, 18

Offset: 1

Alford Arnold, Jan 03 2004

			The array begins
1
2
4
7 1
11 5
16 14 2
22 30 12
29 55 39 5
37 91 95 32 1

A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 2, 5, 3, 4, 3, 1, 6, 4, 6, 6, 6, 2, 2, 7, 5, 8, 9, 11, 9, 7, 4, 3, 1, 8, 6, 10, 12, 16, 16, 18, 12, 12, 8, 6, 2, 2, 9, 7, 12, 15, 21, 23, 29, 27, 26, 23, 21, 15, 13, 7, 4, 3, 1, 10, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40, 40, 30, 26, 18, 14, 8, 6, 2, 2

Offset: 0

Alford Arnold, Jun 19 2003

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.
In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,
the maximum order 10 defining the row length.
Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.
A083480 is a variation illustrating a relationship with numeric partitions, A000041.
The rows are formed by the nonzero entries of the columns of A049597.
The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017
If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

			When viewed as an array with A033638(r) entries per row, the table begins:
. 1 ............... : 1
. 2 ............... : 2
. 3 1 ............. : 3 + q = (1) + (1+q) + (1)
. 4 2 2 ........... : 4 + 2q + 2q^2 = 1 + (1+q+q^2) + (1+q+q^2) + 1
. 5 3 4 3 1 ....... : 5 + 3q + 4q^2 + 3q^3 + q^4
. 6 4 6 6 6 2 2
. 7 5 8 9 11 9 7 4 3 1
. 8 6 10 12 16 16 18 12 12 8 6 2 2
. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
...
The second but last row is from the sum over 7 q-polynomials coefficients:
. 1 ....... : 1 = [6,0]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q
. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
. 1 ....... : 1 = [6,6]_q

George E. Andrews, 'Theory of Partitions', 1976, page 242.

Seiichi Manyama, Rows n = 0..48, flattened
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Alexander Gruber, "The Egg:" Bizarre behavior of the roots of a family of polynomials Mathematics StackExchange Oct 04 2012
MathOverflow, Partition numbers and Gaussian binomial coefficient
Eric Weisstein, q-Binomial Coefficient, Mathworld.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes

Cf. A000025, A000034, A000041, A016116, A029552, A033638, A060546, A063746, A077028, A083479, A083480, A098613, A260460.

R:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n,k | Coefficient(R!((&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
[A083906(n,k): k in [0..Floor(n^2/4)], n in [0..12]]; // G. C. Greubel, Feb 13 2024

QBinomial := proc(n,m,q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ) ; expand(%) ; end:
A083906 := proc(n,k) add( QBinomial(n,m,q),m=0..n ) ; coeftayl(%,q=0,k) ; end:
for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d,",A083906(n,k)) ; od: od: # R. J. Mathar, May 28 2009
T := proc(n, k) if n < 0 or k < 0 or k > floor(n^2/4) then return 0 fi;
if n < 2 then return n + 1 fi; 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) end:
seq(print(seq(T(n, k), k = 0..floor((n/2)^2))), n = 0..8);  # Peter Luschny, Feb 16 2024

Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]],q], {n, 0, 10}] // Grid (* Geoffrey Critzer, May 14 2017 *)

{T(n, k) = polcoeff(sum(m=0, n, prod(k=0, m-1, (x^n - x^k) / (x^m - x^k))), k)}; /* Michael Somos, Jun 25 2017 */

def T(n,k): # T = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*T(n-1, k) - T(n-2, k) + T(n-2, k-n+1)
flatten([[T(n,k) for k in range(int(n^2//4)+1)] for n in range(13)]) # G. C. Greubel, Feb 13 2024

T(n, k) is the coefficient [q^k] of the Sum_{m=0..n} [n, m]_q over q-Binomial coefficients.
Row sums: Sum_{k=0..floor(n^2/4)} T(n,k) = 2^n.
For n >= k, T(n+1,k) = T(n, k) + A000041(k). - Geoffrey Critzer, Feb 12 2021
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A060546(n). - G. C. Greubel, Feb 13 2024
From Mikhail Kurkov, Feb 14 2024: (Start)
T(n, k) = 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) for n >= 2 and 0 <= k <= floor(n^2/4).
Sum_{i=0..n} T(n-i, i) = A000041(n+1). Note that upper limit of the summation can be reduced to A083479(n) = (n+2) - ceiling(sqrt(4*n)).
Both results were proved (see MathOverflow link for details). (End)
From G. C. Greubel, Feb 17 2024: (Start)
T(n, floor(n^2/4)) = A000034(n).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A016116(n+1).
Sum_{k=0..(n + 2) - ceiling(sqrt(4*n))} (-1)^k*T(n - k, k) = (-1)^n*A000025(n+1) = -A260460(n+1). (End)

Edited by R. J. Mathar, May 28 2009

New name using a comment from Geoffrey Critzer by Peter Luschny, Feb 17 2024

A083479 The natural numbers with all terms of A033638 inserted.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 57, 57

Offset: 0

Alford Arnold, Jun 08 2003

Row n of A049597 has a(n+1) nonzero values.
When considering the set of nested parabolas defined by -(x^2) + p*x for integer values of p, a(n) tells us how many parabolas are intersected by the line from (1,n) to (n,n). - Gregory R. Bryant, Apr 01 2013
Number of distinct perimeters for polyominoes with n square cells. - Wesley Prosser, Sep 06 2017

			There are three 1's, one from the natural numbers and two from A033638.
When viewed as an array the sequence begins:
   0
   1
   1  1
   2  2
   3  3  4
   5  5  6
   7  7  8  9
  10 10 11 12
  13 13 14 15 16
  17 17 18 19 20
  21 21 22 23 24 25
  26 26 27 28 29 30
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A002620, A027434, A027709, A033638, A049068, A049597.

Cf. A054243, A060510, A080037, A083480, A083906.

a083479 n = a083479_list !! n
a083479_list = m [0..] a033638_list where
   m xs'@(x:xs) ys'@(y:ys) | x <= y    = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Apr 06 2012

[n eq 0 select 0 else (n+2)-Ceiling(Sqrt(4*n)): n in [0..100]]; // G. C. Greubel, Feb 17 2024

Table[(n + 2) - Ceiling@ Sqrt[4 n] - 2 Boole[n == 0], {n, 0, 73}] (* Michael De Vlieger, Sep 05 2017 *)

a(n):=((n+2)-ceiling(sqrt(4*n))); /* Gregory R. Bryant, Apr 01 2013 */

from math import isqrt
def A083479(n): return n+1-isqrt((n<<2)-1) if n else 0 # Chai Wah Wu, Jul 28 2022

[(n+2)-ceil(sqrt(4*n)) -2*int(n==0) for n in range(101)] # G. C. Greubel, Feb 17 2024

a(n) = (n+2) - ceiling(sqrt(4*n)), for n > 0. - Gregory R. Bryant, Apr 01 2013
From Wesley Prosser, Sep 06 2017: (Start)
a(n) = (n+2) - A027709(n)/2.
a(n) = (n+2) - A027434(n).
a(n) = (2n+2) - A049068(n).
a(n) = (2n+3) - A080037(n).
(End)

Edited and extended by David Wasserman, Nov 16 2004

A105552 Irregular triangle T(n,k) read down columns: the number of compositions c of n with largest_part(c)+length(c)=k+1 in row n, column k.

Original entry on oeis.org

1, 2, 4, 1, 7, 5, 2, 11, 14, 12, 5, 1, 16, 30, 39, 32, 18, 7, 2, 22, 55, 95, 113, 101, 71, 41, 18, 6, 1, 29, 91, 195, 299, 357, 350, 292, 207, 126, 64, 27, 9, 2, 37, 140, 357, 664, 978, 1204, 1283, 1198, 992, 731, 482, 284, 148, 66, 25, 7, 1, 46, 204, 602, 1309, 2274, 3329, 4253

Offset: 1

Alford Arnold, May 03 2005

For each of the A000041(n) partitions of n, one can assign a weight to the partition which counts the permutations of that partition, given by the multinomial coefficient derived from the frequency representation of the parts.
An equivalent representation is given by writing down all compositions of n.
The entries count those partitions multiplied by their weights (=compositions) of n where the sum of the largest addend plus number of parts equals k+1. Only nonzero counts are entered into the sequence.
Each entry can also be interpreted as counting a subset of numbers in A055932, because there is a 1-to-1 correspondence between their prime signature and ordered partitions.
Each diagonal of T(n,k) can be decomposed into p(n) sequences. For example,
A086602 = 2 12 39 95 195 ... is the sum of
A000330 = 1 5 14 30 55 ... plus
A001296 = 1 7 25 65 140 ...
The main diagonal and subdiagonals in order of appearance are A000124, A000330, A086602, A089574, A107600, A107601, A109125, ...

			The row n=7 starts from the partitions (weights in parentheses) 7 (1), 6+1 (2), 5+2 (2), 4+3 (2), 5+1+1 (3), 4+2+1 (6=3!/1!/1!/1!), 3+3+1 (3), 3+2+2 (3), 4+1+1+1 (4=4!/1!/3!), 3+2+1+1 (12 = 4!/1!/1!/2!), 2+2+2+1 (4), 3+1+1+1+1+1 (5), 2+2+1+1+1 (10=5!/2!/3!), 2+1+1+1+1 (6), 1+1+1+1+1+1 (1).
Then T(7,7) = 1+2+3+4+5+6+1 = 22 is the sum of the weights of partitions with largest part 7 and length 1, largest part 6 and length 2,... largest part 1 and length 7.
T(7,6) = 2+6+12+10 = 30 is the sum of the weights of the partitions with largest part 6 and length 1, largest part 5 and length 2, ..., largest part 1 and length 6.
T(7,5) = 2+3+3+4 = 12 collects all the partitions with largest part 5 and length 1 down to largest part 1 and length 5.
The array has A033638(k) nonzero entries per column, starting at n=1 as :
1
..2
....4
....1..7
.......5..11
.......2..14..16
..........12..30..22
...........5..39..55..29
...........1..32..95..91..37
..............18.113.195.140
...............7.101.299.357
...............2

Alois P. Heinz, Columns k = 1..40, flattened

Cf. A047969, A047970, A055932, A057335, A083480, A083906, A089349, A033638, A086602 (subdiagonal), A089574 (subdiagonal).

A033638 := proc(n) ( (7+(-1)^n)/2 + n^2 )/4 ; end proc:
freq := proc(L,n) local a,p; a := 0 ; for p in L do if p = n then a := a+1 ; end if; end do: a ; end proc:
M3 := proc(L) local a,i; a := factorial(nops(L)) ; for i in convert(L,set) do a := a/factorial(freq(L,i)) ; end do: a ; end proc:
A105552 := proc(n,k) local p,a,l ; a := 0 ; for p in combinat[partition](n) do if max(op(p)) + nops(p) = k+1 then a := a+ M3(p); end if; end do ; a ; end proc:
for k from 1 to 15 do for n from k to k+A033638(k)+1 do T := A105552(n,k) ; if T >0 then printf("%d,", A105552(n,k)) ; end if; end do: printf("\n") ; end do: # R. J. Mathar, Jun 26 2010
# second Maple program:
b:= proc(n, k, p) option remember; `if`(n=0 and k=0, 1,
     `if`(k<1, 0, add(b(n-j, k-1-max(p, j)+p, max(p, j)), j=1..n)))
    end:
T:= k-> seq(b(n, k+1, 0), n=k..k+floor((k-1)^2/4)):
seq(T(k), k=1..10);  # Alois P. Heinz, Jul 24 2013

b[n_, k_, p_] := b[n, k, p] = If[n == 0 && k == 0, 1, If[k < 1, 0, Sum[b[n - j, k - 1 - Max[p, j] + p, Max[p, j]], {j, 1, n}]]]; T[k_] := Table[b[n, k + 1, 0], {n, k, k + Floor[(k - 1)^2/4]}]; Table[T[k], {k, 1, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

Row sums: Sum_{k=0..n} T(n,k) = 2^(n-1).

Column sums: Sum_{n>=k} T(n,k) = A047970(n).

Definition clarified by R. J. Mathar, Jun 26 2010

A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 1, 0, 1, 4, 2, 0, 0, 2, 5, 3, 2, 0, 2, 3, 6, 4, 4, 0, 0, 4, 4, 7, 5, 6, 3, 0, 3, 6, 5, 8, 7, 8, 7, 0, 1, 6, 8, 6, 9, 9, 10, 11, 4, 0, 6, 9, 10, 7, 10, 13, 12, 15, 10, 0, 2, 11, 12, 12, 8, 11

Offset: 1

Omar E. Pol, Dec 12 2019

This triangle has the property that it contains the triangle A049597, since if we replace with zeros the positive terms before the first zero in the row n of this triangle, we get the triangle A049597.
Hence the sum of the terms after the last zero in row n equals A000041(n), the number of partitions of n (see the Example section).
Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).

			Triangle begins:
   1;
   0,  2;
   0,  0,  3;
   1,  0,  1,  4;
   2,  0,  0,  2,  5;
   3,  2,  0,  2,  3,  6;
   4,  4,  0,  0,  4,  4,  7;
   5,  6,  3,  0,  3,  6,  5,  8;
   7,  8,  7,  0,  1,  6,  8,  6,  9;
   9, 10, 11,  4,  0,  6,  9, 10,  7, 10;
  13, 12, 15, 10,  0,  2, 11, 12, 12,  8, 11;
Figure 1 below shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.
.
.                                     Right-angles   Right
Part   Ferrers diagram         Part   diagram        angle
                                      _ _ _ _ _ _ _
  7    * * * * * * *             7   |  _ _ _ _ _ _|  14
  6    * * * * * *               6   | |  _ _ _ _|     8
  3    * * *                     3   | | | |           2
  3    * * *                     3   | | |_|
  2    * *                       2   | |_|
  1    *                         1   | |
  1    *                         1   | |
  1    *                         1   |_|
.
       Figure 1.                      Figure 2.
.
For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:
.
    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _
  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8
  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|_|
    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _
  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8
  1| |              1|_|
  1|_|
.
    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _
  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7
  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1
  1| |      1| |        1| |          1|_|
  1| |      1| |        1|_|
  1| |      1|_|
  1|_|
.
    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _
  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6
  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2
  2| |_|    2| |_|      1| |        2|_|_|        1|_|
  1| |      1|_|        1|_|
  1|_|
.
    _ _       _ _ _        _ _ _ _
  2|  _|5   3|  _ _|5    4|  _ _ _|5
  2| | |3   3| |  _|3    4|_|_ _ _|3
  2| | |    2|_|_|
  2|_|_|
.
There are  5 right angles of size 1, so T(8,1) = 5.
There are  6 right angles of size 2, so T(8,2) = 6.
There are  3 right angles of size 3, so T(8,3) = 3.
There are no right angle  of size 4, so T(8,4) = 0.
There are  3 right angles of size 5, so T(8,5) = 3.
There are  6 right angles of size 6, so T(8,6) = 6.
There are  5 right angles of size 7, so T(8,7) = 5.
There are  8 right angles of size 8, so T(8,8) = 8.
Hence the 8th row of triangle is [5, 6, 3, 0, 3, 6, 5, 8].
Note that the sum of the terms after the last zero is 3 + 6 + 5 + 8 = 22, equaling A000041(8) = 22, the number of partitions of 8.

G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143 [Defines the right angles in the Ferrers graph of a partition. - N. J. A. Sloane, Nov 20 2020]

Row sums give A115995.
Right border gives A000027.
Cf. A000041, A049597, A083480, A083906, A188674, A259481, A325458, A330375, A330378, A330379.

A161162 An irregular array distributing the values of A053445 having powers of two as column sums. (cf. A011782).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 4, 2, 2, 5, 3, 4, 6, 3, 4, 1, 6, 7, 6, 5, 6, 8, 8, 2, 9, 6, 2, 11, 10, 9, 9, 12, 7, 7, 16, 12, 10, 4, 16, 15, 8, 3, 18, 21, 14, 11, 1, 12, 23, 18, 9, 12, 29, 26, 16, 12, 8, 27, 30, 21, 10, 6, 26, 40, 31, 18, 13, 2, 23, 42, 37, 24, 11, 2, 21, 48, 51, 36, 20, 14, 15, 44

Offset: 1

Alford Arnold, Jun 08 2009

Diagonals match the reverse of the rows in sequence A083480.
Contribution from Alford Arnold, Jun 09 2009: (Start)
Sequence A161162 is also closely related to sequence A161161 as illustrated.
consider the two vectors in A161161 summing to 32 and 16 respectively:
1 1 2 3 5 7 5 4 3 1
1 1 2 3 5 2 2
they differ by
0 0 0 0 0 5 3 4 3 1 which is the vector in A161162 summing to 16.
(End)

			Column six is 5 3 4 3 1 and sums to 16.
The Tabf begins 1
.0
.1
..0
..2
...0
...3
...1
....4
....2
....2.5

A161161 [From Alford Arnold, Jun 09 2009]

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A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

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A089349 Triangle read by rows: counts the permutations of partitions described in A083480. Both Arrays have shape sequence A083479.

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A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

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A083479 The natural numbers with all terms of A033638 inserted.

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A105552 Irregular triangle T(n,k) read down columns: the number of compositions c of n with largest_part(c)+length(c)=k+1 in row n, column k.

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A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.

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A161162 An irregular array distributing the values of A053445 having powers of two as column sums. (cf. A011782).

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