A006331 -id:A006331 - cp's OEIS frontend

A185355 Number of n X n symmetric (0,1)-matrices containing four ones.

0, 1, 12, 52, 150, 345, 686, 1232, 2052, 3225, 4840, 6996, 9802, 13377, 17850, 23360, 30056, 38097, 47652, 58900, 72030, 87241, 104742, 124752, 147500, 173225, 202176, 234612, 270802, 311025, 355570, 404736, 458832, 518177, 583100, 653940, 731046, 814777

Offset: 1

L. Edson Jeffery, Feb 29 2012

nonn

Based on equation (11) from the Cameron et al., reference.

G. C. Greubel, Table of n, a(n) for n = 1..1000
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.

Cf. A002378, A006331.

Column m=4 of A184948.

a:= n-> (7+(5*n-12)*n)*n^2/12:
seq (a(n), n=1..40);

Table[n^2*(n - 1)*(5*n - 7)/12, {n, 1, 50}] (* G. C. Greubel, Jun 28 2017 *)

for(n=1,25, print1(n^2*(n-1)*(5*n-7)/12, ", ")) \\ G. C. Greubel, Jun 28 2017

a(n) = Sum_{k=0..2} C(C(n,2),k)*C(n,4-2*k).
a(n) = n^2*(n-1)*(5*n-7)/12.
G.f.: x^2*(1+7*x+2*x^2)/(1-x)^5.

A189711 Number of non-monotonic functions from [k] to [n-k].

Original entry on oeis.org

2, 10, 8, 28, 54, 22, 60, 190, 204, 52, 110, 490, 916, 676, 114, 182, 1050, 2878, 3932, 2118, 240, 280, 1988, 7278, 15210, 16148, 6474, 494, 408, 3444, 15890, 45738, 77470, 65210, 19576, 1004, 570, 5580, 31192, 115808, 278358, 389640, 261708, 58920, 2026, 770, 8580, 56484, 258720, 820118, 1677048, 1951700, 1048008, 176994, 4072, 1012, 12650, 96006, 525444, 2090296, 5758802, 10073698, 9763628, 4193580, 531262, 8166

Offset: 5

Dennis P. Walsh, Apr 25 2011

Triangle T(n,k), 3<=k<=n-2, given by (n-k)^k-2*C(n-1,k)+(n-k) is derived using inclusion/exclusion. The triangle contains several other listed sequences: T(2n,n) is sequence A056174(n), number of monotonic functions from [n] to [n]; T(n+2,n) is sequence A005803(n), second-order Eulerian numbers; and T(n,3) is A006331(n-4), maximum accumulated number of electrons at energy level n.

			Triangle T(n,k) begins
  n\k    3     4     5     6     7     8     9
   5     2
   6    10     8
   7    28    54    22
   8    60   190   204    52
   9   110   490   916   676   114
  10   182  1050  2878  3932  2118   240
  11   280  1988  7278 15210 16148  6474   494
  ...
For n=6 and k=4, T(6,4)=8 since there are 8 non-monotonic functions f from [4] to [2], namely, f = <f(1),f(2),f(3),f(4)> given by <1,1,2,1>, <1,2,1,1>, <1,2,2,1>, <1,2,1,2>, <2,2,1,2>, <2,1,2,2>, <2,1,1,2>, and <2,1,2,1>.

Reinhard Zumkeller, Rows n=5..100 of triangle, flattened
Dennis Walsh, Notes on finite monotonic and non-monotonic functions

Cf. A007318.

a189711 n k = (n - k) ^ k - 2 * a007318 (n - 1) k + n - k
a189711_row n = map (a189711 n) [3..n-2]
a189711_tabl = map a189711_row [5..]
-- Reinhard Zumkeller, May 16 2014

seq(seq((n-k)^k-2*binomial(n-1,k)+(n-k),k=3..(n-2)),n=5..15);

nmax = 15; t[n_, k_] := (n-k)^k-2*Binomial[n-1, k]+(n-k); Flatten[ Table[ t[n, k], {n, 5, nmax}, {k, 3, n-2}]](* Jean-François Alcover, Nov 18 2011, after Maple *)

T(n,k)=(n-k)^k-2*C(n-1,k)+(n-k).
T(n,3) = A006331(n-4) for n>=5.
T(n+2,n) = A005803(n) for n>=3.
T(2n,n) = A056174(n) for n>=3.

A233295 Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

Original entry on oeis.org

1, 4, 2, 9, 10, 4, 16, 28, 24, 8, 25, 60, 80, 56, 16, 36, 110, 200, 216, 128, 32, 49, 182, 420, 616, 560, 288, 64, 64, 280, 784, 1456, 1792, 1408, 640, 128, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256, 100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

Offset: 0

Philippe Deléham, Dec 07 2013

Subtriangle of the triangle in A208532.
Row sums are A060188(n+2).
Diagonal sums are A000295(n+2)=A125128(n+1)=A130103(n+2).

			Triangle begins :
1
4, 2
9, 10, 4
16, 28, 24, 8
25, 60, 80, 56, 16
36, 110, 200, 216, 128, 32
49, 182, 420, 616, 560, 288, 64
64, 280, 784, 1456, 1792, 1408, 640, 128
81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

Cf. Columns: A000290, A006331, A112742.

Cf. Diagonal: A000079.

G.f. for the column k: 2^k*(1+x)/(1-x)^(k+3).
T(n,k) = 2^k*(binomial(n,k)+3*binomial(n,k+1)+2*binomial(n,k+2)), 0<=k<=n.
T(n,0) = 2*T(n-1,0)-T(n-2,0)+2, T(n,k)=2*T(n-1,k)+2*T(n-1,k-1)-2*T(n-2,k-1)-T(n-2,k) for k>=1, T(0,0)=1, T(1,0)=4, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k) = A060188(n+2).
Sum_{k=0..n} T(n,k)*(-1)^k = n+1.
T(n,k) = 2*sum_{j=1..n-k+1} T(n-j,k-1).
T(n,k) = 2^k*A125165(n,k).
T(n,n) = 2^n=A000079(n).
T(n,0) = (n+1)^2=A000290(n+1).
exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(16 + 28*x + 24*x^2/2! + 8*x^3/3!) = 16 + 60*x + 200*x^2/2! + 616*x^3/3! + 1792*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 2*x/(1 - x) ). Cf. A125165. - Peter Bala, Dec 21 2014

A296636 Sequences n(n+1)(6n+1)/2 and n(n+1)(7n+1)/2 interleaved.

Original entry on oeis.org

0, 7, 8, 39, 45, 114, 132, 250, 290, 465, 540, 777, 903, 1204, 1400, 1764, 2052, 2475, 2880, 3355, 3905, 4422, 5148, 5694, 6630, 7189, 8372, 8925, 10395, 10920, 12720, 13192, 15368, 15759, 18360, 18639, 21717, 21850, 25460, 25410, 29610, 29337, 34188, 33649, 39215, 38364, 44712

Offset: 0

Luce ETIENNE, Dec 17 2017

Difference between these subsequences is A002411.

This sequence gives numbers of triangles all sizes in every n-th stage [of what? - N. J. A. Sloane, Feb 09 2018].

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration
Eric Weisstein's World of Mathematics, Polyiamond
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A002411, A002717, A006331, A033581, A033582, A255211.

List([0..50], n -> (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128); # Bruno Berselli, Feb 12 2018

[(2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128: n in [0..50]]; // Bruno Berselli, Feb 12 2018

CoefficientList[Series[x (7 + 8 x + 11 x^2 + 13 x^3)/((1 - x)^4*(1 + x)^4), {x, 0, 46}], x] (* Michael De Vlieger, Dec 18 2017 *)
LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,7,8,39,45,114,132,250},50] (* Harvey P. Dale, May 01 2018 *)
Rest[Flatten[Table[With[{c=(n(n+1))/2},{c*(6n+1),c*(7n+1)}],{n,0,30}]]] (* Harvey P. Dale, Oct 11 2020 *)

concat(0, Vec(x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4) + O(x^80))) \\ Colin Barker, Dec 18 2017

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9).
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128.
From Colin Barker, Dec 18 2017: (Start)
G.f.: x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4).
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>7.
(End)

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).

Original entry on oeis.org

0, 0, 2, 0, 6, 4, 0, 10, 12, 6, 0, 14, 20, 18, 8, 0, 18, 28, 30, 24, 10, 0, 22, 36, 42, 40, 30, 12, 0, 26, 44, 54, 56, 50, 36, 14, 0, 30, 52, 66, 72, 70, 60, 42, 16, 0, 34, 60, 78, 88, 90, 84, 70, 48, 18, 0, 38, 68, 90, 104, 110, 108, 98, 80, 54, 20, 0, 42, 76, 102

Offset: 0

Philippe Deléham, Dec 05 2013

Row sums are A006331(n).
Diagonal sums are A212964(n+1).
T(2n,n)=A002943(n).

			Triangle begins:
  0
  0, 2
  0, 6, 4
  0, 10, 12, 6
  0, 14, 20, 18, 8
  0, 18, 28, 30, 24, 10

Cf. columns: A000004, A016825, A017113, A017593, A051062, 10*A005408, A073762.

Cf. diagonals: A005843, A008588, A008592, A008596, A008600, A008604.

T(n+k,k) = A005843(k)*A005408(n).

Sum_{k=0..n} T(n,k) = n*(n+1)*(2*n+1)/3 = A006331(n).

A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 10, 6, 14, 42, 28, 10, 42, 198, 165, 60, 15, 132, 1001, 1092, 455, 110, 21, 429, 5304, 7752, 3876, 1020, 182, 28, 1430, 29070, 57684, 35420, 10626, 1995, 280, 36, 4862, 163438, 444015, 339300, 118755, 24570, 3542, 408, 45, 16796, 937365, 3506100, 3362260, 1391280, 324632, 50344, 5850, 570, 55

Offset: 1

Jean-François Alcover, Apr 18 2014

About the "root estimation" question asked in MathOverflow, one can check (at least numerically) that, for instance with k = 4 and a = 1/11, the series a^-1 + (k - 1) + Sum_{n>=} (-1)^n*binomial(n*k, n+1)/n*a^n evaluates to the positive solution of x^k = (x+1)^(k-1).
Row 1 is A000217 (triangular numbers),
Row 2 is A006331 (twice the square pyramidal numbers),
Row 3 is A067047(3n) = lcm(3n, 3n+1, 3n+2, 3n+3)/12 (from column r=4 of A067049),
Row 4 is A222715(2n) = (n-1)*n*(2n-1)*(4n-3)*(4n-1)/15,
Row 5 is not in the OEIS.
Column 1 is A000108 (Catalan numbers),
Column 2 is A007226 left shifted 1 place,
Column 4 is A007228 left shifted 1 place,
Column 5 is A124724 left shifted 1 place,
Column 6 is not in the OEIS.

			Array begins:
    1,    3,     6,     10,      15,      21, ...
    2,   10,    28,     60,     110,     182, ...
    5,   42,   165,    455,    1020,    1995, ...
   14,  198,  1092,   3876,   10626,   24570, ...
   42, 1001,  7752,  35420,  118755,  324632, ...
  132, 5304, 57684, 339300, 1391280, 4496388, ...
  etc.

N. S. S. Gu, H. Prodinger, S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010) 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2

MathOverflow, Root estimation

Cf. A000217, A006331, A000108, A007226, A007228, A067047, A067049, A124724, A222715.

t[n_, k_] := Binomial[n*k, n+1]/n; Table[t[n-k+2, k], {n, 1, 10}, {k, 2, n+1}] // Flatten

A354127 Triangle read by rows: T(n, k) is the number of graphs obtained by adding k pierced circles to a path graph P_n.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 12, 10, 3, 0, 82, 82, 28, 4, 0, 646, 738, 315, 60, 5, 0, 5574, 7198, 3636, 900, 110, 6, 0, 51386, 74086, 43225, 13020, 2135, 182, 7, 0, 498026, 793490, 524784, 185920, 37940, 4452, 280, 8, 0, 5019720, 8761906, 6475959, 2634912, 642180, 95508, 8442, 408, 9, 0

Offset: 0

Stefano Spezia, May 18 2022

			The triangle begins
      1;
      1,   0;
      2,   2,   0;
     12,  10,   3,   0;
     82,  82,  28,   4,   0;
    646, 738, 315,  60,   5,   0;
    ...

Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.

Cf. A000007 (k = n), A000027 (k = n - 1), A000108, A001246 (row sums), A006331, A007318, A052553.

bigO[k_,s_]:=Binomial[2s-k-1,k]CatalanNumber[s-k]^2; T[n_,k_]:=Sum[(-1)^(m+k)Binomial[m,k]bigO[m,n],{m,k,n}];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]]

T(n, k) = Sum_{m=k..n} (-1)^(m+k)*binomial(m, k)*O(m, n), with O(k, s) = binomial(2*s-k-1, k)*C(s-k)^2 (see Lemma 3.3 at page 7 in Owad and Tsvietkova).

T(n, n-2) = A006331(n-1).

cp's OEIS Frontend

A185355 Number of n X n symmetric (0,1)-matrices containing four ones.

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A189711 Number of non-monotonic functions from [k] to [n-k].

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A233295 Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

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A296636 Sequences n*(n+1)*(6*n+1)/2 and n*(n+1)*(7*n+1)/2 interleaved.

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A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2*k*(2*n+1).

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A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

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A354127 Triangle read by rows: T(n, k) is the number of graphs obtained by adding k pierced circles to a path graph P_n.

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A296636 Sequences n(n+1)(6n+1)/2 and n(n+1)(7n+1)/2 interleaved.

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).