A027379 -id:A027379 - cp's OEIS frontend

A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.

1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, 151, 169, 188, 208, 229, 251, 274, 298, 323, 349, 376, 404, 433, 463, 494, 526, 559, 593, 628, 664, 701, 739, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1174, 1223, 1273, 1324, 1376, 1429, 1483

Offset: 1

N. J. A. Sloane

Number of 1's in the n X n lower Hessenberg (0,1)-matrix (i.e., the matrix having 1's on or below the superdiagonal and 0's above the superdiagonal).
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Number of binary operations which have to be added to Moisil's algebras to obtain algebraic counterparts of n-valued Łukasiewicz logics. See the Wójcicki and Malinowski book, page 31. - Artur Jasinski, Feb 25 2010
Also (n + 1)!(-1)^(n + 1) times the determinant of the n X n matrix given by m(i,j) = i/(i+1) if i=j and otherwise 1. For example, (5+1)! * ((-1)^(5+1)) * Det[{{1/2, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1}, {1, 1, 3/4, 1, 1}, {1, 1, 1, 4/5, 1}, {1, 1, 1, 1, 5/6}}] = 19 = a(5), and (6+1)! * ((-1)^(6+1)) * Det[{{1/2, 1, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1, 1}, {1, 1, 3/4, 1, 1, 1}, {1, 1, 1, 4/5, 1, 1}, {1, 1, 1, 1, 5/6, 1}, {1, 1, 1, 1, 1, 6/7}}] = 26 = a(6). - John M. Campbell, May 20 2011
2*a(n-1) = n*(n+1) - 4, n>=0, with a(-1) = -2 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 17 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
a(n) is not divisible by 3, 5, 7, or 11. - Vladimir Shevelev, Feb 03 2014
With a(0) = 1 and a(1) = 2, a(n-1) is the number of distinct values of 1 +- 2 +- 3 +- ... +- n, for n > 0. - Derek Orr, Mar 11 2015
Also, numbers m such that 8*m+17 is a square. - Bruno Berselli, Sep 16 2015
Omar E. Pol's formula from Apr 23 2008 can be interpreted as the position of an element located on the third diagonal of an triangular array (read by rows) provided n > 1. - Enrique Pérez Herrero, Aug 29 2016
a(n) is the sum of the numerator and denominator of the fraction that is the sum of 2/(n-1) + 2/n; all fractions are reduced and n > 2. - J. M. Bergot, Jun 14 2017
a(n) is also the number of maximal irredundant sets in the (n+2)-path complement graph for n > 1. - Eric W. Weisstein, Apr 12 2018
From Klaus Purath, Dec 07 2020: (Start)
a(n) is not divisible by primes listed in A038890. The prime factors are given in A038889 and the prime terms of the sequence are listed in A124199.
Each odd prime factor p divides exactly 2 out of any p consecutive terms with the exception of 17, which appears only once in such an interval of terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -3 (mod p), see examples.
If A is a sequence satisfying the recurrence t(n) = 5*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 4 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i for i>0. (End)
Mark each point on a 4^n grid with the number of points that are visible from the point; for n > 1, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 23 2021
The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the external perimeter and the perimeter of inscribed squares having the cell (1,1) as a unique common vertex. See Spezia link. - Stefano Spezia, May 28 2025

			From _Bruno Berselli_, Mar 09 2015: (Start)
By the definition (first formula):
----------------------------------------------------------------------
  1       4         8           13            19              26
----------------------------------------------------------------------
                                                              X
                                              X              X X
                                X            X X            X X X
                    X          X X          X X X          X X X X
          X        X X        X X X        X X X X        X X X X X
  X      X X      X X X      X X X X      X X X X X      X X X X X X
          X        X X        X X X        X X X X        X X X X X
----------------------------------------------------------------------
(End)
From _Klaus Purath_, Dec 07 2020: (Start)
Assuming a(i) is divisible by p with 0 < i < p and a(k) is the next term divisible by p, then from i + k == -3 (mod p) follows that k = min(p*m - i - 3) != i for any integer m.
(1) 17|a(7) => k = min(17*m - 10) != 7 => m = 2, k = 24 == 7 (mod 17). Thus every a(17*m + 7) is divisible by 17.
(2) a(9) = 53 => k = min(53*m - 12) != 9 => m = 1, k = 41. Thus every a(53*m + 9) and a(53*m + 41) is divisible by 53.
(3) 101|a(273) => 229 == 71 (mod 101) => k = min(101*m - 74) != 71 => m = 1, k = 27. Thus every a(101*m + 27) and a(101*m + 71) is divisible by 101. (End)
From _Omar E. Pol_, Aug 08 2021: (Start)
Illustration of initial terms:                             _ _
.                                           _ _           |_|_|_
.                              _ _         |_|_|_         |_|_|_|_
.                   _ _       |_|_|_       |_|_|_|_       |_|_|_|_|_
.          _ _     |_|_|_     |_|_|_|_     |_|_|_|_|_     |_|_|_|_|_|_
.   _     |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.  |_|    |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.
.   1       4         8          13            19              26
------------------------------------------------------------------------ (End)

A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).
G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.
Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 471.
Milan Janjic, Two Enumerative Functions.
W. F. Klostermeyer, M. E. Mays, L. Soltes, and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, Vol. 35, No. 4 (1997), pp. 318-328.
Stavros Konstantinidis, António Machiavelo, Nelma Moreira, and Rogério Reis, On the size of partial derivatives and the word membership problem, Acta Informatica Vol. 58, 357-375.
G. C. Moisil, Recherches sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 26 (1940), 431-466.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
D. D. Olesky, B. L. Shader, and P. van den Driessche, Permanents of Hessenberg (0,1)-matrices, Electronic Journal of Combinatorics, 12 (2005), #R70.
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev., Vol. 4, No. 5 (1960), pp. 473-478.
Stefano Spezia, Illustrations for n = 1..9.
Renzo Sprugnoli, Alternating Weighted Sums of Inverses of Binomial Coefficients, J. Integer Sequences, 15 (2012), #12.6.3. - From _N. J. A. Sloane_, Nov 29 2012
Eric Weisstein's World of Mathematics, Maximal Irredundant Set.
Eric Weisstein's World of Mathematics, Path Complement Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to Łukasiewicz.

Subsequence of A165157.
Triangular numbers (A000217) minus two.
Third diagonal of triangle in A059317.
Cf. A000096, A000124, A000217, A000290, A005408, A005843, A008277, A024206, A027379.
Cf. A028347, A038889, A038890, A049600, A091823, A101881, A113452, A113453, A113454.
Cf. A113455, A124199, A130296, A131818, A134225, A161680, A168244, A253909, A267633.

a034856 = subtract 1 . a000096 -- Reinhard Zumkeller, Feb 20 2015

[Binomial(n + 1, 2) + n - 1: n in [1..60]]; // Vincenzo Librandi, May 21 2011

a := n -> hypergeom([-2, n-1], [1], -1);
seq(simplify(a(n)), n=1..53); # Peter Luschny, Aug 02 2014

f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] (* Robert G. Wilson v, Jun 11 2010 *)
Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Aug 29 2016 *)

A034856(n) := block(
        n-1+(n+1)*n/2
)$ /* R. J. Mathar, Mar 19 2012 */

A034856(n)=(n+3)*n\2-1 \\ M. F. Hasler, Jan 21 2015

def A034856(n): return n*(n+3)//2 -1 # G. C. Greubel, Jun 15 2025

G.f.: A(x) = x*(1 + x - x^2)/(1 - x)^3.
a(n) = A049600(3, n-2).
a(n) = binomial(n+2, 2) - 2. - Paul Barry, Feb 27 2003
With offset 5, this is binomial(n, 0) - 2*binomial(n, 1) + binomial(n, 2), the binomial transform of (1, -2, 1, 0, 0, 0, ...). - Paul Barry, Jul 01 2003
Row sums of triangle A131818. - Gary W. Adamson, Jul 27 2007
Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007
Row sums of triangle A134225. - Gary W. Adamson, Oct 14 2007
a(n) = A000217(n+1) - 2. - Omar E. Pol, Apr 23 2008
From Jaroslav Krizek, Sep 05 2009: (Start)
a(n) = a(n-1) + n + 1 for n >= 1.
a(n) = n*(n-1)/2 + 2*n - 1.
a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). (End)
a(n) = Hyper2F1([-2, n-1], [1], -1). - Peter Luschny, Aug 02 2014
a(n) = floor[1/(-1 + Sum_{m >= n+1} 1/S2(m,n+1))], where S2 is A008277. - Richard R. Forberg, Jan 17 2015
a(n) = A101881(2*(n-1)). - Reinhard Zumkeller, Feb 20 2015
a(n) = A253909(n+3) - A000217(n+3). - David Neil McGrath, May 23 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - David Neil McGrath, May 23 2015
For n > 1, a(n) = 4*binomial(n-1,1) + binomial(n-2,2), comprising the third column of A267633. - Tom Copeland, Jan 25 2016
From Klaus Purath, Dec 07 2020: (Start)
a(n) = A024206(n) + A024206(n+1).
a(2*n-1) = -A168244(n+1).
a(2*n) = A091823(n). (End)
Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Jan 06 2021
a(n) + a(n+1) = A028347(n+2). - R. J. Mathar, Mar 13 2021
a(n) = A000290(n) - A161680(n-1). - Omar E. Pol, Mar 26 2021
E.g.f.: 1 + exp(x)*(x^2 + 4*x - 2)/2. - Stefano Spezia, Jun 05 2021
a(n) = A024916(n) - A244049(n). - Omar E. Pol, Aug 01 2021
a(n) = A000290(n) - A000217(n-2). - Omar E. Pol, Aug 05 2021

More terms from Zerinvary Lajos, May 12 2006

A055998 a(n) = n*(n+5)/2.

Original entry on oeis.org

0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272

Offset: 0

Barry E. Williams, Jun 14 2000

If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Bisection of A165157. - Jaroslav Krizek, Sep 05 2009
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012
Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017
a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017
a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020
Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
Milan Janjic, Two Enumerative Functions.
Kival Ngaokrajang, Illustration from A000027 (contains errors).
Linhui Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv:2003.07901 [math.RT], 2020. See p. 8.
Leo Tavares, Illustration: Truncated Point Triangles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
Cf. A000096, A000217, A001477, A002522.
Row n=2 of A255961.
Cf. other rows, columns and diagonals of A000027 written as a table: A034856, A046691, A052905, A055999, A155212, A051936, A056000, A183897, A056115, A051938; A000124, A022856, A152950, A145018, A077169, A166136, A167487, A173036; A059993, A090288, A054000, A142463, A056220, A001105, A001844, A058331, A051890, A097080, A093328, A137882.

[n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019

f[n_]:=n*(n+5)/2; f[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019

G.f.: x*(3-2*x)/(1-x)^3.
a(n) = A027379(n), n > 0.
a(n) = A126890(n,2) for n > 1. - Reinhard Zumkeller, Dec 30 2006
a(n) = A000217(n) + A005843(n). - Reinhard Zumkeller, Sep 24 2008
If we define f(n,i,m) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-m-j), then a(n) = -f(n,n-1,3), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = A167544(n+8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n + 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+2). - Gary Detlefs, Aug 10 2010
a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - Jaroslav Krizek, Sep 05 2009
Sum_{n>=1} 1/a(n) = 137/150. - R. J. Mathar, Jul 14 2012
a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=3..n+2} i. - Wesley Ivan Hurt, Jun 28 2013
a(n) = 3*A000217(n) - 2*A000217(n-1). - Bruno Berselli, Dec 17 2014
a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - Andrey Zabolotskiy, May 18 2016
E.g.f.: x*(6+x)*exp(x)/2. - G. C. Greubel, Apr 05 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)

A046691 a(n) = (n^2 + 5*n - 2)/2.

Original entry on oeis.org

-1, 2, 6, 11, 17, 24, 32, 41, 51, 62, 74, 87, 101, 116, 132, 149, 167, 186, 206, 227, 249, 272, 296, 321, 347, 374, 402, 431, 461, 492, 524, 557, 591, 626, 662, 699, 737, 776, 816, 857, 899, 942, 986, 1031, 1077, 1124, 1172, 1221, 1271, 1322

Offset: 0

N. J. A. Sloane

If Y_i (i=1,2,3,4) are 2-blocks of an n-set X then, for n>=8, a(n-3) is the number of (n-2)-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Nov 09 2007
Numbers m > -3 such that 8*m + 33 is a square. - Bruno Berselli, Aug 20 2015
a(n-1) yields the second Betti number of a path graph on n vertices. - Samuel J. Bevins, Nov 27 2022

G. C. Greubel, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Triangular numbers (A000217) minus 4. Cf. A027379.

Cf. A000096, A000217, A002522, A028884.

[Binomial(n+3,2) -4: n in [0..50]]; // G. C. Greubel, Jul 31 2022

seq(binomial(n+3, 2)-4, n=0..49); # Zerinvary Lajos, Jan 13 2007

Table[(n^2 +5n -2)/2, {n, 0, 50}] (* Bruno Berselli, Dec 17 2014 *)

a(n)=(n^2+5*n-2)/2 \\ Charles R Greathouse IV, Oct 07 2015

[(n^2 +5*n -2)/2 for n in (0..50)] # G. C. Greubel, Jul 31 2022

G.f.: (-1 + 5*x - 3*x^2)/(1 - x)^3.
a(n) = a(n-1) + n + 2 with a(0) = -1. - Vincenzo Librandi, Nov 18 2010
a(n) = 3*A000096(n-1) - 2*A000096(n-2), with A000096(-2)=A000096(-1)=-1. - Bruno Berselli, Dec 17 2014
a(n) = 2*A000217(n) - A000217(n-2), with A000217(-2)=1, A000217(-1)=0. - Bruno Berselli, Oct 13 2016
E.g.f.: (1/2)*(x^2 + 6*x - 2)*exp(x). - G. C. Greubel, Jul 13 2017
a(n) + a(n+1) = A028884(n). - R. J. Mathar, Mar 13 2021
a(n) = A000217(n+2) - 4. - G. C. Greubel, Jul 31 2022
Sum_{n>=0} 1/a(n) = 7/12 + 2*Pi*tan(sqrt(33)*Pi/2)/sqrt(33). - Vaclav Kotesovec, Dec 31 2022

A098832 Square array read by antidiagonals: even-numbered rows of the table are of the form n(n+m) and odd-numbered rows are of the form n(n+m)/2.

Original entry on oeis.org

1, 3, 3, 6, 8, 2, 10, 15, 5, 5, 15, 24, 9, 12, 3, 21, 35, 14, 21, 7, 7, 28, 48, 20, 32, 12, 16, 4, 36, 63, 27, 45, 18, 27, 9, 9, 45, 80, 35, 60, 25, 40, 15, 20, 5, 55, 99, 44, 77, 33, 55, 22, 33, 11, 11, 66, 120, 54, 96, 42, 72, 30, 48, 18, 24, 6, 78, 143, 65, 117, 52, 91, 39, 65, 26, 39, 13, 13

Offset: 1

Eugene McDonnell (eemcd(AT)mac.com), Nov 02 2004

The rows of this table and that in A098737 are related. Given a function f = n/( 1 + (1+n) mod(2) ), row n of A098737 can be derived from row n of T by multiplying the latter by f(n); row n of T can be derived from row n of A098737 by dividing the latter by f(n).

			Array begins as:
  1,  3,  6, 10, 15, 21,  28,  36,  45 ... A000217;
  3,  8, 15, 24, 35, 48,  63,  80,  99 ... A005563;
  2,  5,  9, 14, 20, 27,  35,  44,  54 ... A000096;
  5, 12, 21, 32, 45, 60,  77,  96, 117 ... A028347;
  3,  7, 12, 18, 25, 33,  42,  52,  63 ... A027379;
  7, 16, 27, 40, 55, 72,  91, 112, 135 ... A028560;
  4,  9, 15, 22, 30, 39,  49,  60,  72 ... A055999;
  9, 20, 33, 48, 65, 84, 105, 128, 153 ... A028566;
  5, 11, 18, 26, 35, 45,  56,  68,  81 ... A056000;
Antidiagonals begin as:
   1;
   3,  3;
   6,  8,  2;
  10, 15,  5,  5;
  15, 24,  9, 12,  3;
  21, 35, 14, 21,  7,  7;
  28, 48, 20, 32, 12, 16,  4;
  36, 63, 27, 45, 18, 27,  9,  9;
  45, 80, 35, 60, 25, 40, 15, 20,  5;
  55, 99, 44, 77, 33, 55, 22, 33, 11, 11;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Row m of array: A000217 (m=1), A005563 (m=2), A000096 (m=3), A028347 (m=4), A027379 (m=5), A028560 (m=6), A055999 (m=7), A028566 (m=8), A056000 (m=9), A098603 (m=10), A056115 (m=11), A098847 (m=12), A056119 (m=13), A098848 (m=14), A056121 (m=15), A098849 (m=16), A056126 (m=17), A098850 (m=18), A051942 (m=19).
Column m of array: A026741 (m=1), A022998 (m=2), A165351 (m=3).
Cf. A098737, A168509, A181900.

A098832:= func< n,k | (1/4)*(3+(-1)^k)*(n+1)*(n-k+1) >;
[A098832(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 31 2022

A098832[n_, k_]:= (1/4)*(3+(-1)^k)*(n+1)*(n-k+1);
Table[A098832[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 31 2022 *)

def A098832(n,k): return (1/4)*(3+(-1)^k)*(n+1)*(n-k+1)
flatten([[A098832(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jul 31 2022

Item m of row n of T is given (in infix form) by: n T m = n * (n + m) / (1 + m (mod 2)). E.g. Item 4 of row 3 of T: 3 T 4 = 14.
From G. C. Greubel, Jul 31 2022: (Start)
A(n, k) = (1/4)*(3 + (-1)^n)*k*(k+n) (array).
T(n, k) = (1/4)*(3 + (-1)^k)*(n+1)*(n-k+1) (antidiagonal triangle).
Sum_{k=1..n} T(n, k) = (1/8)*(n+1)*( (3*n-1)*(n+1) + (1+(-1)^n)/2 ).
T(2*n-1, n) = A181900(n).
T(2*n+1, n) = 2*A168509(n+1). (End)

Missing terms added by G. C. Greubel, Jul 31 2022

A227025 T(n,k)=Number of nXk (0,1,2) arrays of permanents of 2X2 subblocks of some (n+1)X(k+1) binary array with rows and columns of the latter in lexicographically nondecreasing order.

Original entry on oeis.org

3, 7, 7, 12, 26, 12, 18, 72, 72, 18, 25, 171, 335, 171, 25, 33, 368, 1366, 1366, 368, 33, 42, 729, 4948, 10050, 4948, 729, 42, 52, 1343, 16115, 65317, 65317, 16115, 1343, 52, 63, 2325, 47659, 375270, 786154, 375270, 47659, 2325, 63, 75, 3819, 129463, 1924848

Offset: 1

R. H. Hardin Jun 27 2013

Table starts
..3....7.....12.......18.........25...........33............42............52
..7...26.....72......171........368..........729..........1343..........2325
.12...72....335.....1366.......4948........16115.........47659........129463
.18..171...1366....10050......65317.......375270.......1924848.......8908719
.25..368...4948....65317.....786154......8379285......79224749.....668706345
.33..729..16115...375270....8379285....168973605....3034194621...48518491730
.42.1343..47659..1924848...79224749...3034194621..104606988712.3220901001021
.52.2325.129463..8908719..668706345..48518491730.3220901001021
.63.3819.326522.37616613.5081503151.694140800875

			Some solutions for n=4 k=4
..0..0..0..2....0..0..1..1....1..0..0..0....0..0..1..1....0..0..1..0
..0..0..1..1....1..0..1..1....0..0..0..0....0..2..1..1....0..1..0..0
..1..0..1..1....1..0..1..1....1..1..0..1....0..2..2..1....2..0..0..1
..0..0..1..0....2..2..2..1....2..1..0..1....1..2..1..0....2..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A027379

Empirical for column k:
k=1: a(n) = (1/2)*n^2 + (5/2)*n
k=2: [polynomial of degree 5] for n>2
k=3: [polynomial of degree 11] for n>6

A111373 A generalized Pascal triangle.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 0, 4, 0, 0, 1, 0, 7, 0, 0, 5, 0, 0, 1, 0, 0, 12, 0, 0, 6, 0, 0, 1, 12, 0, 0, 18, 0, 0, 7, 0, 0, 1, 0, 30, 0, 0, 25, 0, 0, 8, 0, 0, 1, 0, 0, 55, 0, 0, 33, 0, 0, 9, 0, 0, 1, 55, 0, 0, 88, 0, 0, 42, 0, 0, 10, 0, 0, 1, 0, 143, 0, 0, 130, 0, 0, 52, 0, 0, 11, 0, 0, 1

Offset: 0

N. J. A. Sloane, Nov 09 2005

First diagonal is A000012, the all 1's sequence. Second nonzero diagonal is A000027 = n. Third nonzero diagonal is A027379 = n*(n+5)/2 for n>=1, or essentially A000217(n) - 3. Fourth nonzero diagonal is A111396. - Jonathan Vos Post, Nov 10 2005

Row sums are A126042. - Paul Barry, Dec 16 2006

			Triangle begins:
   1;
   0,  1;
   0,  0,  1;
   1,  0,  0,  1;
   0,  2,  0,  0,  1;
   0,  0,  3,  0,  0,  1;
   3,  0,  0,  4,  0,  0,  1;
   0,  7,  0,  0,  5,  0,  0,  1;
   0,  0, 12,  0,  0,  6,  0,  0,  1;
  12,  0,  0, 18,  0,  0,  7,  0,  0,  1;
   0, 30,  0,  0, 25,  0,  0,  8,  0,  0,  1;
   0,  0, 55,  0,  0, 33,  0,  0,  9,  0,  0,  1;
  55,  0,  0, 88,  0,  0, 42,  0,  0, 10,  0,  0,  1;
Production matrix is
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1, 0, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 7, 10.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

First column is A001764. Bears same relation to A001764 as A053121 does to A000108.

Cf. A000012, A000027, A027379, A111396, A126042 (row sums), A167543.

function A111373(n,k)
  if k eq n then return 1;
  elif ((n-k) mod 3) eq 0 then return (3/(n-k))*Binomial(n, Floor((n-k-3)/3))*(k+1);
  else return 0;
  end if; return A111373;
end function;
[A111373(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 30 2022

T[n_, k_]= If[k==n, 1, If[Mod[n-k, 3]==0, (3/(n-k))*Binomial[n, (n-k)/3 -1]*(k + 1), 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 30 2022 *)

def A111373(n,k):
    if(k==n): return 1
    elif ((n-k)%3==0): return (3/(n-k))*binomial(n, (n-k)/3 -1)*(k+1)
    else: return 0
flatten([[A111373(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 30 2022

Each term is the sum of the two terms above it to the left and two steps to the right.
From Paul Barry, Dec 16 2006: (Start)
Riordan array (g(x^3),x*g(x^3)) where g(x)=(2/sqrt(3x))*sin(asin(sqrt(27x/4))/3), the g.f. of A001764;
Number triangle T(n,k) = C(3*floor((n+2k)/3)-2k,floor((n+2k)/3)-k)*(k+1)/(2*floor((n+2k)/3)-k+ 1)(2*cos(2*pi*(n-k)/3)+1)/3. (End)
Inverse of Riordan array (1/(1+x^3), x/(1+x^3)), A126030. - Paul Barry, Dec 16 2006
G.f. (x*A(x))^k=sum{n>=k, T(n,k)*x^n}, where A(x)=1+x^3*A(x)^3. - Vladimir Kruchinin, Feb 18 2011
From G. C. Greubel, Jul 30 2022: (Start)
T(n, k) = (3/(n-k))*binomial(n, (n-k)/3 -1)*(k+1) for ( (n-k) mod 3 ) = 0, otherwise 0, with T(n, n) = 1.
T(n, n-3) = A000027(n-2), n >= 3.
T(n, n-6) = A027379(n-5), n >= 6.
T(n, n-9) = A111396(n-8), n >= 9.
T(n, n-12) = A167543(n+5), n >= 12.
Sum_{k=0..n} T(n, k) = A126042(n). (End)

More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Jan 23 2006

cp's OEIS Frontend

A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.

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A055998 a(n) = n*(n+5)/2.

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A046691 a(n) = (n^2 + 5*n - 2)/2.

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A098832 Square array read by antidiagonals: even-numbered rows of the table are of the form n*(n+m) and odd-numbered rows are of the form n*(n+m)/2.

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A227025 T(n,k)=Number of nXk (0,1,2) arrays of permanents of 2X2 subblocks of some (n+1)X(k+1) binary array with rows and columns of the latter in lexicographically nondecreasing order.

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A111373 A generalized Pascal triangle.

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A098832 Square array read by antidiagonals: even-numbered rows of the table are of the form n(n+m) and odd-numbered rows are of the form n(n+m)/2.