A024206 -id:A024206 - cp's OEIS frontend

A295380 Number of canonical forms for separation coordinates on hyperspheres S_n, ordered by increasing number of independent continuous parameters.

1, 1, 1, 2, 3, 1, 3, 8, 5, 1, 6, 20, 22, 8, 1, 11, 49, 73, 46, 11, 1, 23, 119, 233, 206, 87, 15, 1, 46, 288, 689, 807, 485, 147, 19, 1, 98, 696, 1988, 2891, 2320, 1021, 236, 24, 1, 207, 1681, 5561, 9737, 9800, 5795, 1960, 356, 29, 1, 451, 4062, 15322, 31350, 38216, 28586, 13088, 3525, 520, 35, 1, 983, 9821, 41558, 97552, 139901, 127465, 74280, 27224, 5989, 730, 41, 1

Offset: 1

Tom Copeland, Nov 21 2017

Table 1 of the Schöbel and Veselov paper with initial 1 added. Reverse of Table 2 of the Devadoss and Read paper.
Apparently A032132 contains the row sums.
From Petros Hadjicostas, Jan 28 2018: (Start)
In this triangle, which is read by rows, for 0 <= k <= n-1 and n>=1, let T(n,k) be the number of inequivalent canonical forms for separation coordinates of the hypersphere S^n with k independent continuous parameters. It is the mirror image of sequence A232206, that is, T(n, k) = A232206(n+1, n-k) for 0 <= k <= n-1 and n>=1. (Triangular array A232206(N, K) is defined for N >= 2 and 1 <= K <= N-1.)
If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))). This can be derived from the bivariate g.f. of A232206. See the comments for that sequence.
Let S(n) := Sum_{k>=0} T(n,k). The g.f. of S(n) is B(x, y=1). If we let y=1 in the above functional equation, we get x*B(x,1) = x + (1/2)*((x*B(x,1))^2/(1-x*B(x,1)) + (1 + x*B(x,1))*x^2*B(x^2,1)/(1-x^2*B(x^2,1))). After some algebra, we get 2*x*B(x,1) = x + (1/2)(x*B(x,1)/(1-x*B(x,1)) + (x*B(x,1) + x^2*B(x^2,1))/(1-x^2*B(x,1))), i.e., 2*x*B(x,1) = x + BIK(x*B(x,1)), where we have the "BIK" (reversible, indistinct, unlabeled) transform of C. G. Bower. This proves that S(n) = A032132(n+1) for n>=0, which is Copeland's claim above.
Note that for the second column we have T(n,k=2) = A048739(n-2) for 2 <= n < = 10, but T(11,2) = 4062 <> 4059 = A048739(9). In any case, they have different g.f.s (see the formula section below).
(End)

			From _Petros Hadjicostas_, Jan 27 2018: (Start)
Triangle T(n,k) begins:
n\k      0     1     2     3     4     5     6    7   8  9
----------------------------------------------------------------
(S^1)    1,
(S^2)    1,    1,
(S^3)    2,    3,    1,
(S^4)    3,    8,    5,    1,
(S^5)    6,   20,   22,    8,    1,
(S^6)   11,   49,   73,   46,   11,    1,
(S^7)   23,  119,  233,  206,   87,   15,    1,
(S^8)   46,  288,  689,  807,  485,  147,   19,   1,
(S^9)   98,  696, 1988, 2891, 2320, 1021,  236,  24,  1,
(S^10) 207, 1681, 5561, 9737, 9800, 5795, 1960, 356, 29, 1,
...
(End)

C. G. Bower, Transforms (2)
S. Devadoss and R. C. Read, Cellular structures determined by polygons and trees, arXiv/0008145 [math.CO], 2000.
S. L. Devadoss and R. C. Read, Cellular structures determined by polygons and trees, Ann. Combin., 5 (2001), 71-98.
K. Schöbel and A. Veselov, Separation coordinates, moduli spaces, and Stasheff polytopes, arXiv:1307.6132 [math.DG], 2014.
K. Schöbel and A. Veselov, Separation coordinates, moduli spaces and Stasheff polytopes, Commun. Math. Phys., 337 (2015), 1255-1274.

Cf. A001190, A024206, A032132, A232206.

From Petros Hadjicostas, Jan 28 2018: (Start)
G.f.: If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))).
If c(N,K) = A232206(N,K) and C(x,y) = Sum_{N,K>=0} c(N,K)*x^N*y^K (with c(1,0) = 1 and c(N,K) = 0 for 0 <= N <= K), then C(x,y) = x*B(x*y, 1/y) and B(x,y) = C(x*y, 1/y)/(x*y).
Setting y=0 in the above functional equation, we get x*B(x,0) = x + (1/2)*((x*B(x,0))^2 + x^2*B(x^2,0)), which is the functional equation for the g.f. of the first column. This proves that T(n,k=0) = A001190(n+1) for n>=0 (assuming T(0,0) = 1).
The g.f. of the second column is B_1(x,0) = Sum_{n>=0} T(n,2)*x^n = lim_{y->0} (B(x,y)-B(x,0))/y, where B(x,0) = 1 + x + x^2 + ... is the g.f. of the first column. We get B_1(x,0) = x*B(x,0)*(B(x,0) - 1)/(1 - x*B(x,0)).
(End)

Typo for T(11,3)=15322 corrected by Petros Hadjicostas, Jan 28 2018

A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 8, 5, 9, 3, 15, 3, 9, 9, 11, 3, 15, 3, 15, 9, 9, 3, 24, 5, 9, 8, 15, 3, 27, 3, 15, 9, 9, 9, 25, 3, 9, 9, 24, 3, 27, 3, 15, 15, 9, 3, 33, 5, 15, 9, 15, 3, 24, 9, 24, 9, 9, 3, 45, 3, 9, 15, 19, 9, 27, 3, 15, 9, 27, 3, 40, 3, 9, 15

Offset: 1

Ilya Gutkovskiy, Feb 21 2020

Inverse Moebius transform of A322483.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A007425, A024206, A048691, A124315, A124316, A295316, A322483, A332619, A332713.

Table[Sum[DivisorSigma[0, d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e+3)/2]; A322483[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Table[Sum[A322483[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e + 1)*(e + 5)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n) = Sum_{d|n} A322483(d).
a(n) = Sum_{d|n} tau(n/d) * A295316(d).
Multiplicative with a(p^e) = floor((e+1)*(e+5)/4) = A024206(e+2). - Amiram Eldar, Dec 05 2022

A126696 Tenth-squares: floor(n/10)*ceiling(n/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 12, 12, 12, 12, 16, 20, 20, 20, 20, 20, 20, 20, 20, 20, 25, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 42, 42, 42, 42, 42, 42, 42, 42, 42, 49, 56, 56, 56, 56, 56, 56, 56, 56, 56

Offset: 0

Jonathan Vos Post, May 27 2007

Cf. A002620, A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013.

[ Floor(n/10)*Ceiling(n/10) : n in [0..100]];

f[n_]:=Module[{c=n/10},Floor[c]Ceiling[c]];f[Range[0,90]] (* Harvey P. Dale, Apr 04 2011 *)

Equivalently, floor(n^2/100).

A271647 Irregular triangle read by rows: the natural numbers from right to left.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 8, 7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, 17, 25, 24, 23, 22, 21, 30, 29, 28, 27, 26, 36, 35, 34, 33, 32, 31, 42, 41, 40, 39, 38, 37, 49, 48, 47, 46, 45, 44, 43, 56, 55, 54, 53, 52, 51, 50, 64, 63, 62, 61, 60, 59, 58, 57

Offset: 1

Paul Curtz, Apr 11 2016

A permutation of the natural numbers. Mentioned as d(n) in A269837.
Difference table:
1,  2,  4,  3,  6,  5,  9,  8,  7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, ...
1,  2, -1,  3, -1,  4, -1, -1,  5, -1, -1,  6, -1, -1, -1,  7, -1, -1, -1, ...
1, -3,  4, -4,  5, -5,  0,  6, -6,  0,  7, -7,  0,  0,  8, -8,  0,  0,  9, ...
etc.

			Irregular triangle:
1,
2,
4,   3,
6,   5,
9,   8,  7,
12, 11, 10,
16, 15, 14, 13,
20, 19, 18, 17,
25, 24, 23, 22, 21,
30, 29, 28, 27, 26,
etc.

Cf. A000027, A002620, A024206, A033638, A075356, A235355, A269837, A271584

count:= 0:
for r from 1 to 20 do
  d:= ceil(r/2);
  for i from 0 to d-1 do A[r,i]:= count+ d-i od;
  count:= count+d;
od:
seq(seq(A[r,i],i=0..ceil(r/2)-1),r=1..20); # Robert Israel, Apr 11 2016

Table[Reverse@ Range[Floor[n/2]] + Floor[(n - 1)^2/4], {n, 16}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

With offset=0, a(n) = A271584(n) + A269837(n)

Empirical g.f. as triangle: (1-y*x^3+y^2*x^4-2*y*x^4-y^2*x^5+y*x^5+y^2*x^7)*x/((1+x)*(1-x)^3*(1-y*x^2)^3). - Robert Israel, Apr 11 2016

A302800 Irregular triangle read by rows: T(n,k) is the area of the k-th region of the diagram with n rows described in A237591.

Original entry on oeis.org

1, 3, 5, 1, 8, 2, 11, 4, 15, 5, 1, 19, 7, 2, 24, 9, 3, 29, 11, 5, 35, 13, 6, 1, 41, 16, 7, 2, 48, 18, 9, 3, 55, 21, 11, 4, 63, 24, 12, 6, 71, 27, 14, 7, 1, 80, 30, 16, 8, 2, 89, 34, 18, 9, 3, 99, 37, 20, 11, 4, 109, 41, 22, 13, 5, 120, 45, 24, 14, 7, 131, 49, 27, 15, 8, 1, 143, 53, 29, 17, 9, 2

Offset: 1

Omar E. Pol, Apr 13 2018

Column k lists the partial sums of the k-th column of triangle A237591.

We can see this sequence in the front view of the pyramid described in A245092.

			Triangle begins:
    1;
    3;
    5,  1;
    8,  2;
   11,  4;
   15,  5,  1;
   19,  7,  2;
   24,  9,  3;
   29, 11,  5;
   35, 13,  6,  1;
   41, 16,  7,  2;
   48, 18,  9,  3;
   55, 21, 11,  4;
   63, 24, 12,  6;
   71, 27, 14,  7,  1;
   80, 30, 16,  8,  2;
   89, 34, 18,  9,  3;
   99, 37, 20, 11,  4;
  109, 41, 22, 13,  5;
  120, 45, 24, 14,  7;
  131, 49, 27, 15,  8,  1;
...
Illustration for n = 10:
We draw the first 10 rows of the infinite diagram described in A237591 as shown below:
Row                           _
1                           _| |
2                         _|  _|
3                       _|   | |
4                     _|    _| |
5                   _|     |  _|
6                 _|      _| | |
7               _|       |   | |
8             _|        _|  _| |
9           _|         |   |  _|
10         |_ _ _ _ _ _|_ _|_|_|
Area             35     13  6 1
.
The diagram contains four regions and the areas of the successives regions from left to right are respectively [35, 13, 6, 1], so the 10th row of this triangle is [35, 13, 6, 1].
Note that this infinite diagram gives a correspondence between the number of partitions into k consecutive parts and the symmetric representation of A000203, A024916, A004125 and many other integer sequences. For more information see A196020, A236104, A235791, A237048, A237593, A262626, A286000 and A286001.

Row n has length A003056(n) hence column k starts in row A000217(k).
Row sums give A000217, n >= 1.
Column 1 gives A024206 without its initial zero.
Column 2 gives the partial sums of the A261348.
Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A286000, A286001.

A320657 a(n) is the number of non-unimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 7, 12, 16, 24, 30, 41, 50, 65, 77, 96, 112, 136, 156, 185, 210, 245, 275, 316, 352, 400, 442, 497, 546, 609, 665, 736, 800, 880, 952, 1041, 1122, 1221, 1311, 1420, 1520, 1640, 1750, 1881, 2002, 2145, 2277, 2432, 2576, 2744, 2900, 3081, 3250, 3445, 3627, 3836, 4032

Offset: 1

Tricia Muldoon Brown, Oct 17 2018

nonn

For integers x,y,p,q >= 0, set (s_i){i>=1} to be the sequence of p ones followed by the binomial coefficients C(x,j) for 0 <= j <= x followed by an infinite string of zeros, and set (t_i){i>=1} to be the sequence of q ones followed by the binomial coefficients C(y,j) for 0 <= j <= y followed by an infinite string of zeros. Then a(n) is the number of non-unimodal sequences (r_i){i>=1} where r_i = Sum{j=1..i} s_j*t_{i-j} for some(s_i) and (t_i) such that x + y + p + q + 1 = n.

Let T be a rooted tree created by identifying the root vertices of two broom graphs. a(n) is the number of trees T on n vertices whose poset of connected, vertex-induced subgraphs is not rank unimodal.

T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See table 1 on page 17.
M. Jacobson, A. E. Kézdy, and S. Seif, The poset on connected induced subgraphs of a graph need not be Sperner, Order, 12 (1995) 315-318.

Cf. A005993, A024206. Equals A005581 for n even.

Table[If[EvenQ[n], 2*(Sum[Floor[i(i+4)/4], {i,0,(n/2)}]) - Floor[n^2/16], 2*(Sum[Floor[i(i+4)/4], {i,0,(n-1)/2}]) - Floor[(n-1)^2/16] + Floor[(n+1)(n+9)/16]], {n,0,40}]

a(n+10) = 2*(Sum_{i=1..n/2} floor(i*(i+4)/4)) - floor(n^2/16) for n even.

a(n+10) = 2*(Sum_{i=1..(n-1)/2} floor(i(i+4)/4)) - floor((n-1)^2/16) + floor((n+1)*(n+9)/16) for n odd.

A362894 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having Hadwiger number k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 3, 12, 5, 1, 0, 6, 50, 47, 8, 1, 0, 11, 230, 448, 152, 11, 1

Offset: 1

Peter Kagey, May 08 2023

All planar graphs have Hadwiger number <= 4. The converse is not true since planar graphs also disallow a minor of K_{3,3}. - Andrew Howroyd, Jun 18 2025

			Triangle begins:
1;
0,  1;
0,  1,   1;
0,  2,   3,   1;
0,  3,  12,   5,   1;
0,  6,  50,  47,   8,  1;
0, 11, 230, 448, 152, 11, 1;

Eric Weisstein's World of Mathematics, Hadwiger Number
Wikipedia, Hadwiger number

Row sums are A001349.
Column 2 is A000055 for n > 1.
Subdiagonal is A024206.
Cf. A084269.
Cf. A032766 (n-cocktail party graph). A353212 (n-path complement graph).

cp's OEIS Frontend

A295380 Number of canonical forms for separation coordinates on hyperspheres S_n, ordered by increasing number of independent continuous parameters.

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A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

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A126696 Tenth-squares: floor(n/10)*ceiling(n/10).

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Magma

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A271647 Irregular triangle read by rows: the natural numbers from right to left.

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Maple

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A302800 Irregular triangle read by rows: T(n,k) is the area of the k-th region of the diagram with n rows described in A237591.

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A320657 a(n) is the number of non-unimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.

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A362894 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having Hadwiger number k, 1 <= k <= n.

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