A005744 -id:A005744 - cp's OEIS frontend

A024206 Expansion of x^2(1+x-x^2)/((1-x^2)(1-x)^2).

0, 1, 3, 5, 8, 11, 15, 19, 24, 29, 35, 41, 48, 55, 63, 71, 80, 89, 99, 109, 120, 131, 143, 155, 168, 181, 195, 209, 224, 239, 255, 271, 288, 305, 323, 341, 360, 379, 399, 419, 440, 461, 483, 505, 528, 551, 575, 599, 624, 649, 675, 701, 728, 755, 783, 811, 840

Offset: 1

Clark Kimberling

a(n+1) is the number of 2 X n binary matrices with no zero rows or columns, up to row and column permutation.
[ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.
First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .
Let M_n denotes the n X n matrix m(i,j) = 1 if i =j; m(i,j) = 1 if (i+j) is odd; m(i,j) = 0 if i+j is even, then a(n) = -det M_(n+1) - Benoit Cloitre, Jun 19 2002
a(n) is the number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.
Starting (1, 3, 5, 8, 11, ...), = row sums of triangle A135841. - Gary W. Adamson, Dec 01 2007
Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec, Feb 22 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-4)=-coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010
Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). - Gary W. Adamson, May 21 2010
Conjecture: if a(n) = p#(primorial)-1 for some prime number p, then q=(n+1) is also a prime number where p#=floor(q^2/4). Tested up to n=10^100000 no counterexamples are found. It seems that the subsequence is very scattered. So far the triples (p,q,a(q-1)) are {(2,3,1), (3,5,5), (5,11,29), (7,29,209), (17,1429,510509)}. - David Morales Marciel, Oct 02 2015
Numbers of an Ulam spiral starting at 0 in which the shape of the spiral is exactly a rectangle. E.g., a(4)=5 the Ulam spiral is including at that moment only the elements 0,1,2,3,4,5 and the shape is a rectangle. The area is always a(n)+1. E.g., for a(4) the area of the rectangle is 2(rows) X 3(columns) = 6 = a(4) + 1. - David Morales Marciel, Apr 05 2016
Numbers of different quadratic forms (quadrics) in the real projective space P^n(R). - Serkan Sonel, Aug 26 2020
a(n+1) is the number of one-dimensional subspaces of (F_3)^n, counted up to coordinate permutation. E.g.: For n=4, there are five one-dimensional subspaces in (F_3)^3 up to coordinate permutation: [1 2 2]  [0 2 2]  [1 0 2]  [0 0 2]  [1 1 1]. This example suggests a bijection (which has to be adjusted for the all-ones matrix) with the binary matrices of the first comment. - Álvar Ibeas, Sep 21 2021

			There are five 2 X 3 binary matrices with no zero rows or columns up to row and column permutation:
   [1 0 0]  [1 0 0]  [1 1 0]  [1 1 0]  [1 1 1]
   [0 1 1]  [1 1 1]  [0 1 1]  [1 1 1]  [1 1 1].

O. Giering, Vorlesungen über höhere Geometrie, Vieweg, Braunschweig, 1982. See p. 59.

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Tricia Muldoon Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See p. 15.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
William F. Lunnon, A postage stamp problem, Comput. J. 12 (1969), 377-380.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, Vol. 8 (2008), pp. 45-52.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A014616, A135841, A034856, A005744 (partial sums), A008619 (1st differences).
A row or column of the array A196416 (possibly with 1 subtracted from it).
Cf. A008619.
Second column of A232206.

a:=[0,1,3,5];; for n in [5..65] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; a; # Muniru A Asiru, Oct 23 2018

a024206 n = (n - 1) * (n + 3) `div` 4
a024206_list = scanl (+) 0 $ tail a008619_list
-- Reinhard Zumkeller, Dec 18 2013

[(2*n^2+4*n-7-(-1)^n)/8 : n in [1..100]]; // Wesley Ivan Hurt, Jul 22 2014

A024206:=n->(2*n^2+4*n-7-(-1)^n)/8: seq(A024206(n), n=1..100);

f[x_, y_] := Floor[ Abs[ y/x - x/y]]; Table[ Floor[ f[2, n^2 + 2 n - 2] /2], {n, 57}] (* Robert G. Wilson v, Aug 11 2010 *)
LinearRecurrence[{2,0,-2,1},{0,1,3,5},60] (* Harvey P. Dale, Jun 14 2013 *)
Rest[CoefficientList[Series[x^2 (1 + x - x^2)/((1 - x^2) (1 - x)^2), {x, 0, 70}], x]] (* Vincenzo Librandi, Oct 02 2015 *)

a(n)=(n-1)*(n+3)\4 \\ Charles R Greathouse IV, Jun 26 2013

x='x+O('x^99); concat(0, Vec(x^2*(1+x-x^2)/ ((1-x^2)*(1-x)^2))) \\ Altug Alkan, Apr 05 2016

def A024206(n): return (n+1)**2//4 - 1 # Ya-Ping Lu, Jan 01 2024

G.f.: x^2*(1+x-x^2)/((1-x^2)*(1-x)^2) = x^2*(1+x-x^2) / ( (1+x)*(1-x)^3 ).
a(n+1) = A002623(n) - A002623(n-1) - 1.
a(n) = A002620(n+1) - 1 = A014616(n-2) + 1.
a(n+1) = A002620(n) + n, n >= 0. - Philippe Deléham, Feb 27 2004
a(0)=0, a(n) = floor(a(n-1) + sqrt(a(n-1)) + 1) for n > 0. - Gerald McGarvey, Jul 30 2004
a(n) = floor((n+1)^2/4) - 1. - Franz Vrabec, Feb 22 2008
a(n) = A005744(n-1) - A005744(n-2). - R. J. Mathar, Nov 04 2008
a(n) = a(n-1) + [side length of the least square > a(n-1) ], that is a(n) = a(n-1) + ceiling(sqrt(a(n-1) + 1)). - Ctibor O. Zizka, Oct 06 2009
For a(1)=0, a(2)=1, a(n) = 2*a(n-1) - a(n-2) + 1 if n is odd; a(n) = 2*a(n-1) - a(n-2) if n is even. - Vincenzo Librandi, Dec 23 2010
a(n) = A181971(n, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4); a(1)=0, a(2)=1, a(3)=3, a(4)=5. - Harvey P. Dale, Jun 14 2013
a(n) = floor( (n-1)*(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013
a(n) = (2*n^2 + 4*n - 7 - (-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014
a(n) = a(-n-2) = n-1 + floor( (n-1)^2/4 ). - Bruno Berselli, Feb 03 2015
a(n) = (1/4)*(n+3)^2 - (1/8)*(1 + (-1)^n) - 1. - Serkan Sonel, Aug 26 2020
a(n) + a(n+1) = A034856(n). - R. J. Mathar, Mar 13 2021
a(2*n) = n^2 + n - 1, a(2*n+1) = n^2 + 2*n. - Greg Dresden and Zijie He, Jun 28 2022
Sum_{n>=2} 1/a(n) = 7/4 + tan(sqrt(5)*Pi/2)*Pi/sqrt(5). - Amiram Eldar, Dec 10 2022
E.g.f.: (4 + (x^2 + 3*x - 4)*cosh(x) + (x^2 + 3*x - 3)*sinh(x))/4. - Stefano Spezia, Aug 06 2024

Corrected and extended by Vladeta Jovovic, Jun 02 2000

A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5, 1, 1, 12, 51, 65, 28, 6, 1, 1, 16, 103, 230, 156, 43, 7, 1, 1, 20, 196, 736, 863, 336, 62, 8, 1, 1, 25, 348, 2197, 4571, 2864, 664, 86, 9, 1, 1, 30, 590, 6093, 22952, 25326, 8609, 1229, 115, 10, 1, 1, 36, 960

Offset: 1

Vladeta Jovovic, Jun 13 2000

Also number of unlabeled split graphs on n vertices and with a k-element clique (cf. A048194).

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   9,   4,   1;
  1,  9,  23,  17,   5,   1;
  1, 12,  51,  65,  28,   6,  1;
  1, 16, 103, 230, 156,  43,  7, 1;
  1, 20, 196, 736, 863, 336, 62, 8, 1;
  ...
There are four minimal covers of an unlabeled 3-set: one 1-cover {{1,2,3}}, two 2-covers {{1,2},{3}}, {{1,2},{1,3}} and one 3-cover {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
G. F. Royle, Counting Set Covers and Split Graphs, J. Integer Seqs., 3 (2000), #00.2.6.
Eric Weisstein's World of Mathematics, Minimal covers

Row sums give A048194.
Columns are A000012, A087811, A005783, A005784, A005785 A005786, A055066.
Diagonals are A000012, A000027, A005744, A005745, A005746, A005771, A005747, A005748.
Cf. A035348 for labeled case.
Cf. A002620, A028657.

\\ Needs A(n,m) from A028657.
T(n,k) = A(n-k, k) - if(kAndrew Howroyd, Feb 28 2023

T(n,k) = A028657(n,k) - A028657(n-1,k). - Andrew Howroyd, Feb 28 2023

A181971 Triangle read by rows: T(n,0) = 1, T(n,n) = floor((n+3)/2) and T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 5, 3, 1, 5, 9, 8, 3, 1, 6, 14, 17, 11, 4, 1, 7, 20, 31, 28, 15, 4, 1, 8, 27, 51, 59, 43, 19, 5, 1, 9, 35, 78, 110, 102, 62, 24, 5, 1, 10, 44, 113, 188, 212, 164, 86, 29, 6, 1, 11, 54, 157, 301, 400, 376, 250, 115, 35, 6, 1, 12, 65, 211, 458, 701, 776, 626, 365, 150, 41, 7

Offset: 0

Reinhard Zumkeller, Jul 09 2012

Another variant of Pascal's triangle;
row sums: A081254; central terms: T(2*n,n) = A128082(n+1);
T(n,0) = 1;
T(n,1) = n + 1 for n > 0;
T(n,2) = A000096(n-1) for n > 1;
T(n,3) = A105163(n-2) for n > 2;
T(n,n-2) = A005744(n-1) for n > 1;
T(n,n-1) = A024206(n) for n > 0;
T(n,n) = A008619(n+1).

			The triangle begins:
.  0:                              1
.  1:                           1     2
.  2:                        1     3     2
.  3:                     1     4     5     3
.  4:                  1     5     9     8     3
.  5:               1     6    14    17    11     4
.  6:            1     7    20    31    28    15     4
.  7:         1     8    27    51    59    43    19     5
.  8:      1     9    35    78   110   102    62    24     5
.  9:   1    10    44   113   188   212   164    86    29     6.

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A035317, A007318, A074909.

a181971 n k = a181971_tabl !! n !! k
a181971_row n = a181971_tabl !! n
a181971_tabl = map snd $ iterate f (1, [1]) where
   f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i]))

T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = If[n == k, Quotient[n + 3, 2], If[k == 0, 1, If[n > k, T[n - 1, k - 1] + T[n - 1, k]]]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

{T(n,k)=if(n==k,(n+3)\2,if(k==0,1,if(n>k,T(n-1,k-1)+T(n-1,k))))}
for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ Paul D. Hanna, Jul 18 2012

A003453 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 3, 6, 11, 17, 26, 36, 50, 65, 85, 106, 133, 161, 196, 232, 276, 321, 375, 430, 495, 561, 638, 716, 806, 897, 1001, 1106, 1225, 1345, 1480, 1616, 1768, 1921, 2091, 2262, 2451, 2641, 2850, 3060, 3290, 3521, 3773, 4026

Offset: 5

N. J. A. Sloane, Simon Plouffe

In other words, the number of 2-dissections of an n-gon modulo the dihedral action.
John W. Layman observes that this appears to be the alternating sum transform (PSumSIGN) of A005744.
Row 2 of the convolution array A213847. - Clark Kimberling, Jul 05 2012
Number of nonisomorphic outer planar graphs of order n >= 3 and size n+2. - Christian Barrientos and Sarah Minion, Feb 27 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=5..1000
Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012. See Theorem 5(2).
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619-630.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Column 3 of A295634.

Cf. A005744, A213847, A295419.

T52:= proc(n)
if n mod 2 = 0 then (n-4)*(n-2)*(n+3)/24;
else (n-3)*(n^2-13)/24; fi end;
[seq(T52(n),n=5..80)]; # N. J. A. Sloane, Dec 28 2012

nd[n_]:=If[EvenQ[n],(n-4)(n-2) (n+3)/24,(n-3) (n^2-13)/24]; Array[nd,50,5] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{1,3,6,11,17,26},50] (* Harvey P. Dale, Jan 28 2013 *)

\\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

G.f.: (1+x-x^2) / ((1-x)^4*(1+x)^2).
See also the Maple code.
a(5)=1, a(6)=3, a(7)=6, a(8)=11, a(9)=17, a(10)=26, a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a (n-6). - Harvey P. Dale, Jan 28 2013
a(n) = (2*n^3-6*n^2-23*n+63+3*(n-5)*(-1)^n)/48, for n>=5. - Luce ETIENNE, Apr 07 2015
a(n) = (1/2) * Sum_{i=1..n-4} floor((i+1)*(n-i-2)/2). - Wesley Ivan Hurt, May 07 2016

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012

Name clarified by Andrew Howroyd, Nov 24 2017

A213778 Rectangular array: (row n) = bc, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 4, 2, 9, 6, 2, 17, 13, 7, 3, 28, 23, 15, 9, 3, 43, 37, 27, 19, 10, 4, 62, 55, 43, 33, 21, 12, 4, 86, 78, 64, 52, 37, 25, 13, 5, 115, 106, 90, 76, 58, 43, 27, 15, 5, 150, 140, 122, 106, 85, 67, 47, 31, 16, 6, 191, 180, 160, 142, 118, 97, 73, 53, 33, 18, 6, 239

Offset: 1

Clark Kimberling, Jun 21 2012

Principal diagonal: A213779.
Antidiagonal sums: A213780.
Row 1,  (1,2,3,4,5,...)**(1,2,2,3,3,4,4,...): A005744.
Row 2,  (1,2,3,4,5,...)**(2,2,3,3,4,4,...)
Row 3,  (1,2,3,4,5,...)**(3,4,4,5,5,...)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...6....13...23...37...55....78
2...7....15...27...43...64....90
3...9....19...33...52...76....106
3...10...21...37...58...85....118
4...12...25...43...67...97....134
4...13...27...47...73...106...146

Clark Kimberling, Antidiagonals n=1..80, flattened

Cf. A213500.

b[n_] := n; c[n_] := 1 + Floor[n/2];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213778 *)
Table[t[n, n], {n, 1, 40}] (* A213779 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213780 *)

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = x*(1 + [n/2] + d(n)*x - [(n+1)/2]*x^2), g(x) = (1 + x)*(1 - x)^4, d(n) = (n mod 2) and [] = floor.

A005745 Number of n-covers of an unlabeled 3-set.

Original entry on oeis.org

1, 6, 23, 65, 156, 336, 664, 1229, 2159, 3629, 5877, 9221, 14070, 20951, 30530, 43634, 61283, 84725, 115461, 155294, 206368, 271210, 352784, 454550, 580509, 735280, 924163, 1153207, 1429292, 1760218, 2154776, 2622859, 3175555, 3825247

Offset: 1

N. J. A. Sloane, Simon Plouffe

Number of n X 3 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Vladeta Jovovic, Binary matrices up to row and column permutations

A diagonal of A055080.
First differences give A055609.
Cf. A002623, A002727.
Cf. A005744, A005746, A005747, A005748, A005771.

Vec(G(3, x) - G(2, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

a(n) = A002727(n) - A002623(n).

G.f.: -x*(x^8-x^7-x^6-2*x^5+2*x^4+x^3-3*x^2-2*x-1)/((x^3-1)^2*(x^2-1)^2*(x-1)^4).

More terms from Vladeta Jovovic, May 26 2000

A005746 Number of n-covers of an unlabeled 4-set.

Original entry on oeis.org

1, 9, 51, 230, 863, 2864, 8609, 23883, 61883, 151214, 350929, 778113, 1656265, 3398229, 6743791, 12983181, 24311044, 44377016, 79124476, 138048542, 236050912, 396137492, 653285736, 1059923072, 1693592112, 2667563553, 4145373780, 6360553548, 9643151582

Offset: 1

N. J. A. Sloane, Simon Plouffe

Number of n X 4 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations

A diagonal of A055080.
First differences give A055082.
Cf. A002727, A006148.
Cf. A005744, A005745, A005747, A005748, A005771.

Rest@ CoefficientList[Series[x (1 + 3 x + 9 x^2 + 26 x^3 + 35 x^4 + 92 x^5 + 127 x^6 + 201 x^7 + 242 x^8 + 253 x^9 + 248 x^10 + 205 x^11 + 123 x^12 + 86 x^13 + 31 x^14 + 24 x^15 + 19 x^16 + 5 x^17 + 3 x^18 -
2 x^19 - 4 x^20 + 2 x^21 - 4 x^22 + 3 x^23 - x^25 + 2 x^26 - x^27)/((1 - x)^16 (1 + x)^6 (1 + x^2)^3 (1 + x + x^2)^4), {x, 0, 29}], x] (* Michael De Vlieger, Aug 23 2016 *)

Vec(x*(1 +3*x +9*x^2 +26*x^3 +35*x^4 +92*x^5 +127*x^6 +201*x^7 +242*x^8 +253*x^9 +248*x^10 +205*x^11 +123*x^12 +86*x^13 +31*x^14 +24*x^15 +19*x^16 +5*x^17 +3*x^18 -2*x^19 -4*x^20 +2*x^21 -4*x^22 +3*x^23 -x^25 +2*x^26 -x^27) / ((1 -x)^16*(1 +x)^6*(1 +x^2)^3*(1 +x +x^2)^4) + O(x^40)) \\ Colin Barker, Aug 23 2016

Vec(G(4, x) - G(3, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

a(n) = A006148(n) - A002727(n).

G.f.: x*(1 +3*x +9*x^2 +26*x^3 +35*x^4 +92*x^5 +127*x^6 +201*x^7 +242*x^8 +253*x^9 +248*x^10 +205*x^11 +123*x^12 +86*x^13 +31*x^14 +24*x^15 +19*x^16 +5*x^17 +3*x^18 -2*x^19 -4*x^20 +2*x^21 -4*x^22 +3*x^23 -x^25 +2*x^26 -x^27) / ((1 -x)^16*(1 +x)^6*(1 +x^2)^3*(1 +x +x^2)^4). - Corrected by Colin Barker, Aug 23 2016

More terms and g.f. from Vladeta Jovovic, May 26 2000

a(19) onwards corrected by Sean A. Irvine, Aug 22 2016

A213781 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 4, 2, 9, 7, 3, 17, 14, 10, 4, 28, 25, 19, 13, 5, 43, 39, 33, 24, 16, 6, 62, 58, 50, 41, 29, 19, 7, 86, 81, 73, 61, 49, 34, 22, 8, 115, 110, 100, 88, 72, 57, 39, 25, 9, 150, 144, 134, 119, 103, 83, 65, 44, 28, 10, 191, 185, 173, 158, 138, 118, 94, 73, 49, 31

Offset: 1

Clark Kimberling, Jun 22 2012

Principal diagonal: A213782.
Antidiagonal sums: A005712.
row 1,  (1,2,2,3,3,4,4,...)**(1,2,3,4,5,6,7,...): A005744.
row 2,  (1,2,2,3,3,4,4,...)**(2,3,4,5,6,7,8,...).
row 3,  (1,2,2,3,3,4,4,...)**(3,4,5,6,7,8,9,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...7....14...25...39...58....81
3...10...19...33...50...73....100
4...13...24...41...61...88....119
5...16...29...49...72...103...138
6...19...34...57...83...118...157
7...22...39...65...94...133...176

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500, A213778.

b[n_] := Floor[(n + 2)/2]; c[n_] := n;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213781 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005712 *)

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = x*(n + x - (2*n - 1)*x^2 + (n -1)*x^3) and g(x) = (1 + x)(1 - x)^4.

A213783 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 3, 1, 6, 4, 2, 11, 8, 6, 2, 17, 14, 11, 7, 3, 26, 22, 19, 13, 9, 3, 36, 32, 28, 22, 16, 10, 4, 50, 45, 41, 33, 27, 18, 12, 4, 65, 60, 55, 47, 39, 30, 21, 13, 5, 85, 79, 74, 64, 56, 44, 35, 23, 15, 5, 106, 100, 94, 84, 74, 62, 50, 38, 26, 16, 6, 133, 126, 120, 108

Offset: 1

Clark Kimberling, Jun 22 2012

Principal diagonal: A213759.
Antidiagonal sums: A213760.
Row 1, (1,2,2,3,3,4,4,...)**(1,1,2,2,3,3,4,...): A005744.
Row 2, (1,2,2,3,3,4,4,5,...)**(1,2,2,3,3,4,4,5,...).
Row 3, (1,2,2,3,3,4,4,5,...)**(2,2,3,3,4,4,5,5,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...3....6....11...17...26...36....50
1...4....8....14...22...32...45....60
2...6....11...19...28...41...55....74
2...7....13...22...33...47...64....84
3...9....16...27...39...56...74....98
3...10...18...30...44...62...83....108
4...12...21...35...50...71...93....122
4...13...23...38...55...77...102...132

Clark Kimberling, Antidiagonals n = 1..80, flattened

Cf. A213500.

b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/2];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213760 *)

T(n,k) = 2*T(n,k-1) -T(n,k-2) - 4*T(n,k-3) +T(n,k-4) +2*T(n,k-5) -T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = [(n+1)/2] + [(n+2)/2]*x + ([(n-1)/2] + [(n+1)/2])*x^2 - (1+[n/2]-(n mod 2))*x^3 + [n/2]*x^4 and g(x) = (1 + x)^2 *(1 - x)^4, where [ ] = floor.

A355754 Irregular triangle read by rows: T(n,k) is the number of unlabeled n-node graphs with intersection number (or edge clique cover number) k; n >= 1, 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 1, 4, 9, 10, 7, 2, 1, 1, 5, 17, 36, 46, 30, 14, 4, 2, 1, 1, 6, 28, 97, 219, 281, 226, 116, 45, 18, 5, 1, 1, 1, 7, 43, 226, 872, 2104, 3170, 2927, 1774, 793, 290, 87, 37, 9, 3, 2, 1, 1, 8, 62, 472, 2966, 12882, 36595, 63842, 69294, 48881, 24939, 9808, 3387, 1059, 313, 107, 37, 9, 4, 1, 1

Offset: 1

Pontus von Brömssen, Jul 16 2022

			Triangle begins:
  n\k | 0  1  2   3   4    5    6    7    8   9  10 11 12 13 14 15 16
  ----+--------------------------------------------------------------
   1  | 1
   2  | 1  1
   3  | 1  2  1
   4  | 1  3  4   2   1
   5  | 1  4  9  10   7    2    1
   6  | 1  5 17  36  46   30   14    4    2   1
   7  | 1  6 28  97 219  281  226  116   45  18   5  1  1
   8  | 1  7 43 226 872 2104 3170 2927 1774 793 290 87 37  9  3  2  1

Eric W. Weisstein, Table of n, a(n) for n = 1..108
Paul Erdős, A. W. Goodman, and Louis Pósa, The representation of a graph by set intersections, Canadian Journal of Mathematics 18 (1966), 106-112.
Eric Weisstein's World of Mathematics, Intersection Number
Wikipedia, Intersection number

Cf. A000088 (row sums), A005744 (column k=2), A355755.

T(n,0) = 1.
T(n,1) = n-1.
T(n,2) = floor((n-2)*(2*n^2+7*n-12)/24) = A005744(n-2) = (4*n^3+6*n^2-52*n+45+3*(-1)^n)/48.

cp's OEIS Frontend

A024206 Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2).

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A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

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A181971 Triangle read by rows: T(n,0) = 1, T(n,n) = floor((n+3)/2) and T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

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A003453 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation and reflection.

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A213778 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

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A005745 Number of n-covers of an unlabeled 3-set.

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A024206 Expansion of x^2(1+x-x^2)/((1-x^2)(1-x)^2).

A213778 Rectangular array: (row n) = bc, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.

A213781 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and = convolution.

A213783 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.