A113311 -id:A113311 - cp's OEIS frontend

A360846 Array read by antidiagonals: T(m,n) is the number of dominating induced trees in the grid graph P_m X P_n.

1, 3, 3, 4, 8, 4, 4, 17, 17, 4, 4, 32, 65, 32, 4, 4, 66, 222, 222, 66, 4, 4, 130, 766, 1280, 766, 130, 4, 4, 262, 2685, 7629, 7629, 2685, 262, 4, 4, 522, 9450, 46032, 78981, 46032, 9450, 522, 4, 4, 1046, 33158, 278419, 820308, 820308, 278419, 33158, 1046, 4

Offset: 1

Andrew Howroyd, Feb 23 2023

A dominating induced tree in a graph is an acyclic connected induced subgraph whose vertices are a dominating set.

			Table starts:
=======================================================
m\n| 1   2    3      4       5         6          7 ...
---+---------------------------------------------------
1  | 1   3    4      4       4         4          4 ...
2  | 3   8   17     32      66       130        262 ...
3  | 4  17   65    222     766      2685       9450 ...
4  | 4  32  222   1280    7629     46032     278419 ...
5  | 4  66  766   7629   78981    820308    8520021 ...
6  | 4 130 2685  46032  820308  14605388  259809527 ...
7  | 4 262 9450 278419 8520021 259809527 7904828158 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435 (first 29 antidiagonals)
Eric Weisstein's World of Mathematics, Grid Graph.

Main diagonal is A360847.
Rows 1..2 are A113311(n-1), A360848.
Cf. A291872 (connected dominating sets), A360202 (induced trees).

T(n,m) = T(m,n).

A376782 Triangle read by rows: T(n,m) is the number of unlabeled graphs with n vertices having m minimum forbidden subgraphs, n >= 1, 1 <= m <= A371162(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 4, 2, 1, 4, 8, 13, 8, 1, 4, 5, 7, 20, 34, 31, 28, 12, 8, 5, 0, 1, 1, 4, 5, 13, 26, 33, 43, 59, 50, 62, 58, 60, 64, 67, 63, 70, 68, 65, 61, 60, 31, 28, 16, 8, 13, 4, 4, 4, 0, 2, 1, 0, 1, 1, 4, 6, 9, 21, 34, 39, 71, 74, 77, 99, 118, 124, 107, 129

Offset: 1

Max Alekseyev, Oct 03 2024

			Triangle starts with
n = 1: 1
n = 2: 1 1
n = 3: 1 3
n = 4: 1 4 4  2
n = 5: 1 4 8 13  8
n = 6: 1 4 5  7 20 34 31 28 12 8 5 0 1
...

Max Alekseyev, Table of k, a(k) for k = 1..209 (rows n = 1..8 flattened)
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, 2024.

Cf. A000088 (row sums), A371162 (row lengths), A000012 (column m=1), A113311 (column m=2).

Cf. A002865, A006897, A103221, A371161, A376780, A376781.

A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 0, 4, 4, 1, 0, 4, 8, 5, 1, 0, 4, 12, 13, 6, 1, 0, 4, 16, 25, 19, 7, 1, 0, 4, 20, 41, 44, 26, 8, 1, 0, 4, 24, 61, 85, 70, 34, 9, 1, 0, 4, 28, 85, 146, 155, 104, 43, 10, 1, 0, 4, 32, 113, 231, 301, 259, 147, 53, 11, 1, 0, 4, 36, 145, 344, 532, 560, 406, 200, 64, 12, 1

Offset: 0

Paul Curtz, Dec 16 2011

The array F(n,m), beginning with row n=0, is:
  1, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  1, 4,  8, 13,  19,  26,   34,
  0, 4, 12, 25,  44,  70,  104,
  0, 4, 16, 41,  85, 155,  259,
  0, 4, 20, 61, 146, 301,  560,
  0, 4, 24, 85, 231, 532, 1092.
Columns after A130713, A113311, A008574 have signatures (3,-3,1), (4,-6,4,-1), (5,-10,10,-5,1), (6,-15,20,-15,6,-1) (from A135278(n+3)).
Inserting columns of zeros and pushing the columns down, plus alternating sign switches defines the following triangle T(n,2m) = (-1)^(m/2)*F(n-2m,m):
  1,
  2 0,
  1 0 -1,
  0 0 -3 0,
  0 0 -4 0 1,
  0 0 -4 0 4 0,
  0 0 -4 0 8 0 -1
The row sums in the triangle are (-1)^n*A099838(n).
The companion to A201863 is
  1
  1 0
  1 0   0
  1 0  -2 0
  1 0  -4 0  1
  1 0  -6 0  5 0
  1 0  -8 0 13 0   -2
  1 0 -10 0 25 0  -12 0
  1 0 -12 0 41 0  -38 0   4
  1 0 -14 0 61 0  -88 0  28 0
  1 0 -16 0 85 0 -170 0 104 0 -8
5th column: A001844; 7th column: -A035597=-2*A005900(n+1); 9th column: 4*A006325(n+2); 11th column: -8*(1,8,34,104) (from columns 4,5,6,7 of F(n,m)).
As a triangular array, this is the Riordan array ((1+x)^2, x/(1-x)). - Philippe Deléham, Feb 21 2012

			Triangle T(n,k) begins:
  1
  2, 1
  1, 3,  1
  0, 4,  4,  1
  0, 4,  8,  5,   1
  0, 4, 12, 13,   6,   1
  0, 4, 16, 25,  19,   7,   1
  0, 4, 20, 41,  44,  26,   8,  1
  0, 4, 24, 61,  85,  70,  34,  9,  1
  0, 4, 28, 85, 146, 155, 104, 43, 10, 1
- _Philippe Deléham_, Feb 21 2012

Muniru A Asiru, Table of n, a(n) for n = 0..5151

Cf. A130713 (column 0), A113311 (column 1), A008574 (column 2), A001844 (column 3), A005900 (column 4), A006325 (column 5), A033455 (column 6).

Cf. A267633.

Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)+Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)-Binomial(n-3,n-k-3)))); # Muniru A Asiru, Mar 22 2018

A130713 := proc(n)
    if n <= 2 and n >=0 then
        op(n+1,[1,2,1]) ;
    else
        0;
    end if;
end proc:
A202241 := proc(n,m)
    option remember;
    if n < 0 then
        0 ;
    elif m = 0 then
        A130713(n);
    else
        procname(n,m-1)+procname(n-1,m) ;
    end if;
end proc:
for d from 0 to 12 do
    for m from 0 to d do
        printf("%d,",A202241(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Dec 22 2011
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
     seq(C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), k = 0..n);
end do; # Peter Bala, Mar 20 2018

rows = 12;
T[0] = PadRight[{1, 2, 1}, rows];
T[n_ /; nJean-François Alcover, Jun 29 2019 *)

def Trow(n): return [binomial(n, n-k) + binomial(n-1, n-k-1) - binomial(n-2, n-k-2) - binomial(n-3, n-k-3) for k in (0..n)]
for n in (0..9): print(Trow(n)) # Peter Luschny, Mar 21 2018

F(1,m) = m+2.
F(2,m) = A034856(m+1).
F(3,m) = A000297(m-1).
Sum_{m=0..d} F(d-m,m) = A116453(d-3), d >= 3 (antidiagonal sums).
As a triangular array T(n,k), 0 <= k <= n, satisfies: T(n,k) = T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 2, T(2,0) = 1, T(3,0) = 0. - Philippe Deléham, Feb 21 2012
Unsigned diagonals of A267633 (beginning with its main diagonal) appear to be the reverse rows of this entry's triangle beginning with the fourth row. - Tom Copeland, Jan 26 2016
T(n,k) = C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), where C(n, k) = n!/(k!*(n-k)!) if 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018

A122431 Riordan array ((1+x)^3,x).

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 0, 1, 3, 3, 1, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Sep 04 2006

Row sums are A115291. Diagonal sums are A113311. Inverse is A122432. Sequence array for C(3,n).

			Triangle begins
1,
3,1,
3,3,1,
1,3,3,1,
0,1,3,3,1,
0,0,1,3,3,1,
0,0,0,1,3,3,1,
0,0,0,0,1,3,3,1

Number triangle T(n,k)=C(3,n-k)

A171445 Expansion of g.f. (1+z)^(24)/(1-z).

Original entry on oeis.org

1, 25, 301, 2325, 12951, 55455, 190051, 536155, 1271626, 2579130, 4540386, 7036530, 9740686, 12236830, 14198086, 15505590, 16241061, 16587165, 16721761, 16764265, 16774891, 16776915, 16777191, 16777215, 16777216, 16777216

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^(24)=16777216 for n>=24. We observe that this sequence is the transform of A171443 by the iterated T^(16) of T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

			a(3) = C(25,3)+C(25,3-2) = 2325.

Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171440, A171441, A171442, A171443.

m:=25:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

CoefficientList[Series[(1+x)^24/(1-x),{x,0,30}],x] (* Harvey P. Dale, Jun 11 2019 *)

With m=25, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

A207327 Riordan array (1, x*(1+x)^2/(1-x)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 4, 17, 9, 1, 0, 4, 32, 39, 12, 1, 0, 4, 48, 111, 70, 15, 1, 0, 4, 64, 240, 268, 110, 18, 1, 0, 4, 80, 432, 769, 530, 159, 21, 1, 0, 4, 96, 688, 1792, 1905, 924, 217, 24, 1, 0, 4

Offset: 0

Philippe Deléham, Feb 17 2012

Triangle T(n,k), read by rows, given by (0, 3, -5/3, 4/15, -3/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Row sums are A077995(n).

			Triangle begins :
1
0, 1
0, 3, 1
0, 4, 6, 1
0, 4, 17, 9, 1
0, 4, 32, 39, 12, 1
0, 4, 48, 111, 70, 15, 1
0, 4, 64, 240, 268, 110, 18, 1
0, 4, 80, 432, 769, 530, 159, 21, 1
0, 4, 96, 688, 1792, 1905, 924, 217, 24, 1
0, 4, 112, 1008, 3584, 5503, 3999, 1477, 284, 27, 1
0, 4, 128, 1392, 6400, 13440, 13842, 7483, 2216, 360, 30, 1

Cf. Diagonals : A000012, A008585, A022266, A000007, A113311

T(2*n,n) = A119259(n).
G.f.: (1-x)/(1-(1+y)*x-2*y*x^2-y*x^3).
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1) + T(n-3,k-1), T(0,0) = 1, T(1,0) = 0.

A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Natan Arie Consigli, Jan 19 2016

Decimal expansion of 101/450.
Also list of smallest n-composites.
A hyperoperator aggregation b[n]c is n-composite if b,c are positive non-right-identity elements.
The identity elements are:
Hyper-0 (zeration): none.
Hyper-1 (addition): 0.
Hyper-2 (multiplication): 1.
Hyper-3 (exponentiation): 1.
Hyper-n (n>2): 1.
For more information on hyperoperations see A054871.
Essentially the same as A255176, A151798, A123932, A113311, A040002 and A010709. - R. J. Mathar, May 25 2023
Continued fraction expansion of 2 + sqrt(1/5) = 2 + sqrt(5)/5. - Elmo R. Oliveira, Aug 06 2024

			a(0) = 2 because 1 is the smallest non-identity element in zeration and 1[0]1=2;
a(1) = 2 because 1 is the smallest non-identity element in addition and 1[1]1=2;
a(2) = 4 because 2 is the smallest non-identity element in multiplication and 2[2]2=4;
a(3) = 4 because 2 is the smallest non-identity element in exponentiation and 2[2]2=4;
a(4) = 4 because 2 is the smallest non-identity element in titration and 2[2]2=4;
Etc.

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

Cf. A000027 (1-composites), A002808 (composites), A267647 (3-composites), A097374 (4-composites).

a(n) = a[n]b where a,b are the positive smallest non-right-identity elements.
From Elmo R. Oliveira, Aug 06 2024: (Start)
G.f.: 4/(1 - x) - 2*(1 + x).
E.g.f.: 4*exp(x) - 2*(1 + x). (End)

cp's OEIS Frontend

A360846 Array read by antidiagonals: T(m,n) is the number of dominating induced trees in the grid graph P_m X P_n.

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A376782 Triangle read by rows: T(n,m) is the number of unlabeled graphs with n vertices having m minimum forbidden subgraphs, n >= 1, 1 <= m <= A371162(n).

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A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

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A122431 Riordan array ((1+x)^3,x).

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A171445 Expansion of g.f. (1+z)^(24)/(1-z).

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Maple

Mathematica

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A207327 Riordan array (1, x*(1+x)^2/(1-x)).

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A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

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