A005993 -id:A005993 - cp's OEIS frontend

A212686 Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|=n+|y-z|.

0, 0, 4, 8, 24, 40, 76, 112, 176, 240, 340, 440, 584, 728, 924, 1120, 1376, 1632, 1956, 2280, 2680, 3080, 3564, 4048, 4624, 5200, 5876, 6552, 7336, 8120, 9020, 9920, 10944, 11968, 13124, 14280, 15576, 16872, 18316, 19760, 21360, 22960

Offset: 0

Clark Kimberling, May 25 2012

a(n)=4*A005993(n-2) for n>=2.

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 Abs[w - x] == n + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212686 *)
%/4  (* A005993 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 4, 8, 24, 40}, 40]

a(n)=2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).

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

A301739 The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.

Original entry on oeis.org

2, 4, 10, 17, 30, 44, 67, 91, 126, 163, 213, 265, 333, 403, 491, 582, 693, 807, 944, 1084, 1249, 1418, 1614, 1814, 2044, 2278, 2544, 2815, 3120, 3430, 3777, 4129, 4520, 4917, 5355, 5799, 6287, 6781, 7321, 7868, 8463, 9065, 9718, 10378, 11091, 11812, 12588, 13372, 14214, 15064

Offset: 4

R. J. Mathar, Mar 26 2018

Computed by the sum over the A000055(4)=2 shapes of the trees: the linear graph of the n-Butane, and the star graph of (1)-Methyl-Propane.

			a(4)=2 because there is a linear tree with all labels equal 1 and the star tree with all labels equal to 1.

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

4th column of A303841.

x^4*(2+2*x+2*x^2+x^3+x^4)/(1+x)^2/(x-1)^4/(1+x+x^2) ;
taylor(%,x=0,80) ;
gfun[seriestolist](%) ;

a(n) = A005993(n-4)+A000601(n-4).

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

A317714 Chessboard rectangles sequence (see Comments), also A037270 interleaved with A163102.

Original entry on oeis.org

0, 0, 1, 2, 10, 18, 45, 72, 136, 200, 325, 450, 666, 882, 1225, 1568, 2080, 2592, 3321, 4050, 5050, 6050, 7381, 8712, 10440, 12168, 14365, 16562, 19306, 22050, 25425, 28800, 32896, 36992, 41905, 46818, 52650, 58482, 65341, 72200, 80200, 88200, 97461, 106722, 117370

Offset: 1

Ivan N. Ianakiev, Aug 05 2018

Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite rectangles of p X q unit squares whose vertices are covered by black unit squares (1 < p <= n, 1 < q <= n).

			In a 4 X 4 chessboard there are two such rectangles (for both p = q = 3) and the coordinates of their lower left vertices are a1 and b2. Therefore, a(4) = 2.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A005993, A037270, A163102.

[(5-5*(-1)^n-12*n+12*(-1)^n*n+14*n^2-6*(-1)^n*n^2-8*n^3+2*n^4)/64: n in [1..50]]; // Vincenzo Librandi, Aug 05 2018

CoefficientList[Series[-((x^2 (1+4 x^2+x^4))/((-1+x)^5 (1+x)^3)),{x,0,44}],x]
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 1, 2, 10, 18, 45, 72}, 80] (* Vincenzo Librandi, Aug 06 2018 *)

a(n) = sum(i = 1, n-1, floor(i/2)^3); \\ Jinyuan Wang, Aug 12 2019

n, a = 0, 0
while n < 10:
    print(n,a)
n, a = n+1, a+((n+1)//2)**3 # A.H.M. Smeets, Aug 09 2019

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8), with a(1)=0, a(2)=0, a(3)=1, a(4)=2, a(5)=10, a(6)=18, a(7)=45, a(8)=72.
G.f.: -(x^3*(1+ 4*x^2 + x^4))/((-1+x)^5*(1+x)^3).
a(n) = (5 - 5*(-1)^n - 12*n + 12*(-1)^n*n + 14*n^2 - 6*(-1)^n*n^2 - 8*n^3 + 2*n^4)/64.
a(n) = Sum_{i=1..n-1} floor(i/2)^3. - Ridouane Oudra, Jul 24 2019
E.g.f.: (1/64)*exp(-x)*(-5-6*x-6*x^2+exp(2*x)*(5-4*x+4*x^2+4*x^3+2*x^4)). - Stefano Spezia, Aug 14 2019
a(2*n) = A163102(n-1) and a(2*n+1) = A037270(n). - Ridouane Oudra, Mar 24 2024
Sum_{n>=3} 1/a(n) = Pi^2 - Pi*coth(Pi) - 5. - Amiram Eldar, Jul 04 2025

A089351 Number of planar partitions of n with trace 4.

Original entry on oeis.org

1, 2, 6, 14, 33, 64, 127, 228, 404, 672, 1100, 1724, 2661, 3974, 5849, 8402, 11911, 16556, 22751, 30772, 41198, 54436, 71283, 92316, 118609, 150950, 190753, 239090, 297783, 368236, 452782, 553240, 672532, 812980, 978211, 1171144, 1396235

Offset: 4

Wouter Meeussen and Vladeta Jovovic, Dec 26 2003

Also number of partitions of n objects of 2 colors into 4 parts, each part containing at least one black object.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (Ch. XI, exercise 5 and Ch. XII, exercise 5).

Column 4 of A089353. Cf. A000219, A005380, A005993 (trace 2), A050531 (trace 3).

G.f.: q^4*(q^12+q^10+2*q^9+4*q^8+2*q^7+4*q^6+2*q^5+4*q^4+2*q^3+q^2+1) / ((-1+q^4)^2*(-1+q^3)^2*(-1+q^2)^2*(-1+q)^2).

Edited and extended by Christian G. Bower, Jan 08 2004

A143901 Rectangular array R by antidiagonals: R(m,n) = floor((m*n+1)/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 3, 4, 5, 4, 3, 3, 5, 6, 6, 5, 3, 4, 6, 8, 8, 8, 6, 4, 4, 7, 9, 10, 10, 9, 7, 4, 5, 8, 11, 12, 13, 12, 11, 8, 5, 5, 9, 12, 14, 15, 15, 14, 12, 9, 5, 6, 10, 14, 16, 18, 18, 18, 16, 14, 10, 6, 6, 11, 15, 18, 20, 21, 21, 20, 18, 15, 11, 6, 7, 12, 17, 20, 23, 24, 25, 24

Offset: 1

Clark Kimberling, Sep 04 2008

Old name was: R(m,n) = number of white squares.

			Northwest corner:
1 1 2 2 3 3 4 4
1 2 3 4 5 6 7 8
2 3 5 6 8 9 11 12
2 4 6 8 10 12 14 16

Cf. A143902.
Antidiagonal sums: (1,2,6,10,19,...)=A005993.
Diagonals: A000982, A000217, A007590, A000096, A116940.
Rows and columns: A004526, A000027, A007494, A005843, A047218 et al.

Entry revised by N. J. A. Sloane, Jun 12 2015

A280186 Number of 3-element subsets of S = {1..n} whose sum is odd.

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320, 5740, 6160, 6622, 7084, 7590, 8096, 8648, 9200

Offset: 0

Necip Fazil Patat, Dec 28 2016

The same as A006584 (apart from the offset). - R. J. Mathar, Jan 15 2017
There are two cases: n is odd and n is even.
Let n be an odd integer and n > 3, the sum of 3 integers is odd when all of them are odd or one is odd and the others are even. Number of ways to choose 3 odd numbers: C((n+1)/2, 3). Number of ways to choose 2 even numbers and 1 odd: C((n-1)/2, 2)*C((n+1)/2, 1). Total number of ways: C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1).
Let n be an even integer and n > 3. Number of ways to choose 3 odd numbers: C(n/2, 3). Number of ways to choose 2 even numbers and 1 odd: C(n/2, 2)*C(n/2, 1). Total number of ways: C(n/2, 3) + C(n/2, 2)*C(n/2, 1).
Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite squares having white unit squares on their vertices. For the number of composite squares having black unit squares on their vertices see A005993. - Ivan N. Ianakiev, Aug 19 2018

			For n = 5 then a(5) = 4. The subsets are: {1, 2, 4}, {1, 3, 5}, {2, 3, 4}, {2, 4, 5}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Essentially 2*A006918.

Same as A006584.

Table[Binomial[(n + #)/2, 3] + Binomial[(n - #)/2, 2] Binomial[(n + #)/2, 1] &@ Boole@ OddQ@ n, {n, 0, 49}] (* or *)
CoefficientList[Series[2 x^4/((1 - x)^4*(1 + x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jan 07 2017 *)

concat(vector(4), Vec(2*x^4 / ((1-x)^4*(1+x)^2) + O(x^60))) \\ Colin Barker, Dec 28 2016

a(n) = C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1) when n is odd.
a(n) = C(n/2, 3) + C(n/2, 2)*C(n/2, 1) when n is even.
From Colin Barker, Dec 28 2016: (Start)
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>5.
a(n) = n*(n - 1)*(n - 2)/12 for n even.
a(n) = (n - 1)*(n + 1)*(n - 3)/12 for n odd.
G.f.: 2*x^4 / ((1-x)^4*(1+x)^2). (End)
a(n) = ((-1)^n)*(-1+n)*(3 - 3*(-1)^n - 4*((-1)^n)*n + 2*((-1)^n)*n^2)/24. - Ivan N. Ianakiev, Aug 19 2018

More terms from Colin Barker, Dec 28 2016

A361275 Number of 1423-avoiding even Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 17, 29, 41, 61, 81, 111, 141, 183, 225, 281, 337, 409, 481, 571, 661, 771, 881, 1013, 1145, 1301, 1457, 1639, 1821, 2031, 2241, 2481, 2721, 2993, 3265, 3571, 3877, 4219, 4561, 4941, 5321, 5741, 6161, 6623, 7085, 7591, 8097, 8649, 9201, 9801, 10401

Offset: 0

Juan B. Gil, Mar 10 2023

A permutation is said to be Grassmannian if it has at most one descent. A permutation is even if it has an even number of inversions.

Avoiding any of the patterns 2314 or 3412 gives the same sequence.

			For n=4 the a(4) = 5 permutations are 1234, 1342, 2314, 3124, 3412.

Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A356185, A361272, A361273, A361274.

For the corresponding odd permutations, cf. A005993.

seq(1 - 5*n/24 + n^3/12 - (-1)^n * n/8, n = 0 .. 100); # Robert Israel, Mar 10 2023

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

a(n) = 1 - 5*n/24 + n^3/12 - (-1)^n * n/8. - Robert Israel, Mar 10 2023

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.

A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)(2n + 1) - i*j.

Original entry on oeis.org

1, 1, 17, 1177, 210249, 76961257, 50203153993, 53127675356625, 85252003916011889, 197131843368693693937, 631233222450168374457057

Offset: 0

Stefano Spezia, Apr 30 2023

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(2) = 17:
    [4, 3, 2, 1]
    [3, 6, 4, 2]
    [2, 4, 6, 3]
    [1, 2, 3, 4]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Hafnian
Wikipedia, Special unitary group
Wikipedia, Symmetric matrix

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985, A362679 (permanent).

M[i_, j_, n_]:=Part[Part[Table[Min[r,c](n+1)-r c, {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, min(i, j)*(n + 1) - i*j);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

cp's OEIS Frontend

A212686 Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|=n+|y-z|.

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A301739 The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.

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Maple

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A317714 Chessboard rectangles sequence (see Comments), also A037270 interleaved with A163102.

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A089351 Number of planar partitions of n with trace 4.

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A143901 Rectangular array R by antidiagonals: R(m,n) = floor((m*n+1)/2).

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A280186 Number of 3-element subsets of S = {1..n} whose sum is odd.

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Mathematica

PARI

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Extensions

A361275 Number of 1423-avoiding even Grassmannian permutations of size n.

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Maple

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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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A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)(2n + 1) - i*j.