A047656 -id:A047656 - cp's OEIS frontend

A110147 10^((n^2-n)/2).

1, 1, 10, 1000, 1000000, 10000000000, 1000000000000000, 1000000000000000000000, 10000000000000000000000000000, 1000000000000000000000000000000000000

Offset: 0

Philippe Deléham, Sep 04 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082148 = {1, 1, 11, 131, 1661, 22101, 305151, 4335711, ...}; example: det([1, 1, 11, 131; 1, 11, 131, 1661; 11, 131, 1661, 22101; 131, 1661, 22101, 305151]) = 10^6 = 1000000.

Also the Hankel transform of A379103. - Nathaniel Johnston, Dec 16 2024

Cf. A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966.

a(n)=10^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(10i, j).

a(n)=10a(n-1)^2/a(n-2), a(0)=a(1)=1. - Michael Somos, Sep 12 2005

A294353 Product of first n terms of the binomial transform of n^n (A086331).

Original entry on oeis.org

1, 2, 14, 602, 236586, 1116922506, 78020387811618, 95634036502805444826, 2378081951650318040462277306, 1361239109900199746154166909875717978, 20062823024247092576000017563809908231829439138, 8420023655209092490508999978430595224656730339006712229850

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A002109, A047656, A086331, A086619, A294352.

Table[Product[1 + Sum[Binomial[m, k]*k^k, {k, 1, m}], {m, 0, n}], {n, 0, 12}]

a(n) ~ c * n^(n*(n+1)/2 + 1/12 + exp(-1)/2) / exp(n^2/4 - n*exp(-1)), where c = 1.981849007720509372587479129359338230641860983165241915197508762536...

A110195 a(n) = 11^((n^2-n)/2).

Original entry on oeis.org

1, 1, 11, 1331, 1771561, 25937424601, 4177248169415651, 7400249944258160101211, 144209936106499234037676064081, 30912680532870672635673352936887453361, 72890483685103052142902866787761839379440139451, 1890591424712781041871514584574319778449301246603238034051

Offset: 0

Philippe Deléham, Sep 07 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082173 = {1, 1, 12, 155, 2124, 30482, 453432, 6936799, ...}; example : det([1, 1, 12, 155; 1, 12, 155, 2124; 12, 155, 2124, 30482; 155, 2124, 30482, 453432]) = 11^6 = 1771561.

Cf. A001020, A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966, A110147, A161680.

Table[11^((n^2-n)/2),{n,0,20}] (* Harvey P. Dale, Feb 02 2012 *)
Join[{1,1},Table[Det[Table[Binomial[11i,j],{i,n},{j,n}]],{n,10}]] (* Harvey P. Dale, Apr 01 2019 *)

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(11i, j).

a(n) = A001020(A161680(n)).

a(11) from Harvey P. Dale, Feb 02 2012

a(12) from Jason Yuen, Aug 29 2025

A129147 Expansion of c(x(1+2x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 13, 52, 214, 928, 4141, 18940, 88258, 417616, 2001058, 9690184, 47348812, 233158144, 1155900541, 5764510060, 28898899594, 145556001136, 736206912982, 3737768204344, 19042072755124, 97313398530496, 498737257238482, 2562773039735896, 13200732624526804, 68148459129343648

Offset: 0

Paul Barry, Apr 01 2007

Hankel transform of a(n) is A047656(n+1)=3^C(n+1,2). In general, the Hankel transform of the expansion of c(x(1+r*x)) is (r+1)^C(n+1,2).

Number of paths weakly above X-axis from (0,0) to (0,2n) using steps (1,1), (1,-1) and two colors of (3,1). - David Scambler, Jun 21 2013

Barry, Paul; Hennessy, Aoife Four-term recurrences, orthogonal polynomials and Riordan arrays. J. Integer Seq. 15 (2012), no. 4, Article 12.4.2, 19 pp.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[(1-Sqrt[1-4*x*(1+2*x)])/(2*x*(1+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

x='x+O('x^66);
C(x)=(1-sqrt(1-4*x))/(2*x);
Vec(C(x*(1+2*x))) \\ Joerg Arndt, May 15 2013

a(n)=sum{k=0..n, C(k,n-k)*2^(n-k)*C(k)};
a(n)=(1/(2*pi))*int(x^n*sqrt(8+4x-x^2)/(x+2),x,2-2*sqrt(3),2+2*sqrt(3));
Conjecture: (n+1)*a(n) +2*(2-n)*a(n-1) +4*(5-4n)*a(n-2) +16*(2-n)*a(n-3)=0. - R. J. Mathar, Dec 14 2011
G.f.: Q(0), where Q(k)= 1 + (4*k+1)*x*(1+2*x)/(k + 1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
a(n) ~ sqrt(3-sqrt(3)) * (2*(1+sqrt(3)))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014

A341471 Number of antisymmetric, antitransitive relations on n labeled nodes.

Original entry on oeis.org

1, 1, 3, 21, 317, 9735, 583907, 66226033, 13837055261

Offset: 0

Peter Kagey, Feb 13 2021

An antisymmetric, antitransitive relation is one where xRy implies "not yRx" and xRy and yRz implies "not xRz". All antitransitive relations are irreflexive, so this sequence is counting "anti-equivalence relations".
a(n) < A047656(n).
Idea thanks to Richard Arratia, who saw, verbatim in an editorial, "False equivalences? There were almost too many to count."

			There are a(3) = 21 antisymmetric, antitransitive relations on n = 3 letters:
  - the empty relation,
  - all six relations containing only a single pair (x,y) (with x != y),
  - all twelve relations {(x1,y1), (x2,y2)} containing exactly two ordered pairs, neither of which is (y1,x1) or (y2,x2), and
  - two relations containing three ordered pairs: {(1,2), (2,3), (3,1)} and {(1,3), (3,2), (2,1)}.

Wikipedia, Binary relation

Number of relations on labeled nodes: A000110 (equivalence), A001831 (transitive and antitransitive), A002416 (unrestricted), A006125 (symmetric), A006905 (transitive), A047656 (reflexive and antisymmetric), A083667 (antisymmetric), A341473 (antitransitive).

a(6)-a(8) from Bert Dobbelaere, Feb 27 2021

A341473 The number of antitransitive relations on n labeled nodes.

Original entry on oeis.org

1, 1, 4, 39, 921, 47462, 5205915, 1161039833, 516101770210

Offset: 0

Peter Kagey, Feb 13 2021

A relation is antitransitive if xRy and yRz implies "not xRz". As such, antitransitive relations are always irreflexive.

Wikipedia, Intransitivity.

Number of relations on labeled nodes: A000110 (equivalence), A002416 (unrestricted), A006125 (symmetric), A006905 (transitive), A047656 (reflexive and antisymmetric), A083667 (antisymmetric), A341471 (antisymmetric and antitransitive).

a(6)-a(8) from Bert Dobbelaere, Feb 27 2021

A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 12, 8, 1, 12, 60, 160, 240, 192, 64, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024, 1, 30, 420, 3640, 21840, 96096, 320320, 823680, 1647360, 2562560, 3075072, 2795520, 1863680, 860160, 245760, 32768

Offset: 0

Andrew Howroyd, Feb 15 2022

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1,  2;
  [3] 1,  6,  12,   8;
  [4] 1, 12,  60, 160,  240,  192,    64;
  [5] 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Row sums are A047656.
The unlabeled version is A350733.
Cf. A013609, A350732 (weakly connected), A350731 (strongly connected).

PARI
```
T(n,k) = 2^k * binomial(n*(n-1)/2, k)
```

row(n) = {Vecrev((1+2*y)^(n*(n-1)/2))}
{ for(n=0, 6, print(row(n))) }

T(n,k) = 2^k * binomial(n*(n-1)/2, k) = A013609(n*(n-1)/2, k).

T(n,k) = [y^k] (1+2*y)^(n*(n-1)/2).

A353041 G.f. A(x) satisfies: A(x) = 1 + x * A(3x/(1 + 2x)) / (1 - x).

Original entry on oeis.org

1, 1, 4, 34, 820, 62140, 14651728, 10547347384, 22950318347248, 150277943334242320, 2955664382713520203072, 174478760893191691170298912, 30905073486465684713191125079360, 16423574117627547687292156418920831936, 26184104208316120602662312616366633316565248

Offset: 0

Ilya Gutkovskiy, Apr 19 2022

nonn

Cf. A006898, A047656, A135755 (partial sums), A353042.

nmax = 14; A[] = 0; Do[A[x] = 1 + x A[3 x/(1 + 2 x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] 3^(k (k - 1)/2), {k, 0, n}], {n, 0, 14}]

G.f.: Sum_{k>=0} 3^(k*(k-1)/2) * (x/(1 - x))^k.

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * 3^(k*(k-1)/2).

A206601 3^(n(n+1)/2) - 1.

Original entry on oeis.org

0, 2, 26, 728, 59048, 14348906, 10460353202, 22876792454960, 150094635296999120, 2954312706550833698642, 174449211009120179071170506, 30903154382632612361920641803528, 16423203268260658146231467800709255288, 26183890704263137277674192438430182020124346

Offset: 0

Ivan N. Ianakiev, Feb 10 2012

There are n cities located on the vertices of a convex n-gon and 2 types of communication lines available. Any city can be connected to any other by only one communication line (that can be of any type). A network exists if at least 2 cities are connected by a communication line. The sequence shows how many different networks a(n) can be built. In general, if the number of communication-line types is c, then a(n) = (c+1)^(n(n+1)/2)-1. Thus other sequences of this type can be generated.

			In the case of 2 different types of communication lines and 4 cities, the number of different networks (connecting at least 2 cities) is 728.

Cf. A000217, A126883.

a(n) = (3^A000217) - 1.

a(n) = A047656(n+1) - 1. - Omar E. Pol, Feb 18 2012

A209916 Kolmogorov's button, 2-color generic convex polygon version.

Original entry on oeis.org

0, 0, 2, 26, 1457, 1889567, 470184984575, 359414999291950792703, 27008149481218253520093899825086463, 12768639440249474099578561928613102801011591209543532543

Offset: 0

Ivan N. Ianakiev, Mar 15 2012

This sequence shows the number of distinct patterns that can be created with threads of 2 colors while sewing on a button with n buttonholes located on the vertices of a generic convex n-gon, i.e., a convex n-gon with no more than two diagonals intersecting at any point in its interior. The number of all distinct patterns due to intersections made by differently colored diagonals of the n-gon, equaling 2^A000332(n), is taken into account (as red-diagonal-over-green-diagonal, for instance, is a different pattern from green-diagonal-over-red-diagonal). In general, if the number of colors is c, then a(n) = ((c+1)^(n-1)*n/2)*((c-1)*c)^A000332(n)-1.

Kolmogorov's button problem is briefly mentioned in the book by Gessen.

			For the classic 4-hole button (where n=4 and c=2) the number of distinct patterns is a(n) = A047656(4)*2^A000332(4) - 1 = 729*2 - 1 = 1457. The "-1" stands for the case where the threads are missing, i.e., where the button is unattached to the cloth.

Masha Gessen, Perfect Rigor, A Genius and the Mathematical Breakthrough of the Century, Houghton Mifflin Harcourt, 2009, page 38.

Cf. A000332, A047656.

[3^((n^2-n) div 2)*2^Binomial(n,4)-1: n in [0..10]]; // Vincenzo Librandi, Dec 29 2015

Table[-1+(3^Binomial[n,2])*(2^Binomial[n,4]),{n,0,9}] (* Ivan N. Ianakiev, Dec 29 2015 *)

a(n) = A047656(n)*2^A000332(n) - 1.

cp's OEIS Frontend

A110147 10^((n^2-n)/2).

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A294353 Product of first n terms of the binomial transform of n^n (A086331).

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A110195 a(n) = 11^((n^2-n)/2).

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A129147 Expansion of c(x(1+2x)), c(x) the g.f. of A000108.

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A341471 Number of antisymmetric, antitransitive relations on n labeled nodes.

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A341473 The number of antitransitive relations on n labeled nodes.

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A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

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A353041 G.f. A(x) satisfies: A(x) = 1 + x * A(3*x/(1 + 2*x)) / (1 - x).

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A206601 3^(n(n+1)/2) - 1.

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A353041 G.f. A(x) satisfies: A(x) = 1 + x * A(3x/(1 + 2x)) / (1 - x).