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Search: a060690 -id:a060690
Displaying 1-10 of 31 results found. page 1 2 3 4
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A014070 a(n) = binomial(2^n, n). +0
47
1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, 409663695276000, 6208116950265950720, 334265867498622145619456, 64832175068736596027448301568, 45811862025512780638750907861652480, 119028707533461499951701664512286557511680 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) is the number of n X n (0,1) matrices with distinct rows modulo rows permutations. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..60
FORMULA
G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!. - Paul D. Hanna, Dec 28 2007
a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n, k) * 2^(n*k). - Paul D. Hanna, Feb 05 2023
From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).
(End)
MAPLE
A014070:= n-> binomial(2^n, n); seq(A014070(n), n=0..20); # G. C. Greubel, Mar 14 2021
MATHEMATICA
Table[Binomial[2^n, n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)
PROG
(PARI) a(n)=binomial(2^n, n)
(PARI) /* G.f. A(x) as Sum of Series: */
a(n)=polcoeff(sum(k=0, n, log(1+2^k*x +x*O(x^n))^k/k!), n) \\ Paul D. Hanna, Dec 28 2007
(PARI) {a(n) = (1/n!) * sum(k=0, n, stirling(n, k, 1) * 2^(n*k) )}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 05 2023
(Magma) [Binomial(2^n, n): n in [0..25]]; // Vincenzo Librandi, Sep 13 2016
(Sage) [binomial(2^n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
CROSSREFS
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), this sequence (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A088229, A136393, A136555.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i. +0
17
1, 2, 5, 17, 86, 698, 9551, 226592, 9471845, 705154187, 94285792211, 22807963405043, 10047909839840456, 8110620438438750647, 12062839548612627177590, 33226539134943667506533207, 170288915434579567358828997806, 1630770670148598007261992936663653 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of n X n binary symmetric matrices with rows, considered as binary numbers, in nondecreasing order. - R. H. Hardin, May 30 2008
Also, number of (n+1) X (n+1) binary symmetric matrices with zero main diagonal and rows, considered as binary numbers, in nondecreasing order. - Max Alekseyev, Feb 06 2022
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..50
FORMULA
a(n) = Sum_{k=0..n} A097712(n, k). - Paul D. Hanna, Aug 24 2004
Equals the binomial transform of A008934 (number of tournament sequences): a(n) = Sum_{k=0..n} C(n, k)*A008934(k). - Paul D. Hanna, Sep 18 2005
MATHEMATICA
T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]];
a[n_] := Sum[T[n, k], {k, 0, n}];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 02 2019 *)
PROG
(SageMath)
@CachedFunction
def T(n, k): # T = A097712
if k<0 or k>n: return 0
elif k==0 or k==n: return 1
else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
def A016121(n): return sum(T(n, k) for k in range(n+1))
[A016121(n) for n in range(31)] # G. C. Greubel, Feb 21 2024
CROSSREFS
Cf. A008934, A060690, A089006, A351157, A351158, A351287, A351288.
Row sums of triangle A097712.
KEYWORD
nonn
AUTHOR
Jeffrey Shallit
STATUS
approved
A060336 Number of n X n {-1,0,1} matrices modulo rows permutation (by symmetry this is the same as the number of {-1,0,1} matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other. +0
3
3, 45, 3654, 1929501, 7355513529, 212787633478239, 47937678641708357304, 85524882506287709213421693, 1224201212028616655577478516173315, 142132497715474639139076246298436794277130 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..47
FORMULA
a(n) = C(3^n + n - 1, n) (where C(n, k) denotes the binomial coefficient).
a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
MATHEMATICA
Table[Binomial[3^n+n-1, n], {n, 10}] (* Harvey P. Dale, Apr 10 2012 *)
PROG
(PARI) { for (n=1, 47, write("b060336.txt", n, " ", binomial(3^n + n - 1, n)); ) } \\ Harry J. Smith, Jul 03 2009
CROSSREFS
A060690.
KEYWORD
nonn
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
EXTENSIONS
More terms from Harry J. Smith, Jul 03 2009
STATUS
approved
A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows. +0
8
1, 2, 10, 176, 16456, 6710912, 11453291200, 80421421917440, 2305843009750581376, 268650182136584290872320, 126765060022823052739661424640, 241677817415439249618874010960064512, 1858395433210885261795036719974526548094976 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Also the number of digraphical necklaces with n vertices. A digraphical necklace is defined to be a directed graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of directed graphs under rotation of the vertices. These are a kind of partially labeled digraphs. - Gus Wiseman, Mar 04 2019
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..57
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
FORMULA
a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n^2/d) for n > 0, a(0) = 1.
EXAMPLE
From Gus Wiseman, Mar 04 2019: (Start)
Inequivalent representatives of the a(2) = 10 digraphical necklace edge-sets:
{}
{(1,1)}
{(1,2)}
{(1,1),(1,2)}
{(1,1),(2,1)}
{(1,1),(2,2)}
{(1,2),(2,1)}
{(1,1),(1,2),(2,1)}
{(1,1),(1,2),(2,2)}
{(1,1),(1,2),(2,1),(2,2)}
(End)
MATHEMATICA
Table[Fold[ #1+EulerPhi[ #2] 2^(n^2 /#2)&, 0, Divisors[n]]/n, {n, 16}]
(* second program *)
rotdigra[g_, m_]:=Sort[g/.k_Integer:>If[k==m, 1, k+1]];
Table[Length[Select[Subsets[Tuples[Range[n], 2]], #=={}||#==First[Sort[Table[Nest[rotdigra[#, n]&, #, j], {j, n}]]]&]], {n, 0, 4}] (* Gus Wiseman, Mar 04 2019 *)
CROSSREFS
Cf. A000031 (binary necklaces), A000939 (cycle necklaces), A008965, A060690, A061417 (permutation necklaces), A184271, A192332 (graphical necklaces), A275527 (path necklaces), A323858 (toroidal necklaces), A323870.
Cf. A324463, A324464, A324513, A324514.
KEYWORD
nonn,easy
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 27 2003
EXTENSIONS
More terms from Wouter Meeussen, Jul 29 2003
a(0)=1 prepended by Gus Wiseman, Mar 04 2019
STATUS
approved
A089006 Number of distinct n X n (0,1) matrices after double sorting: by row, by column, by row .. until reaching a fixed point. +0
6
1, 2, 7, 45, 650, 24520, 2625117, 836488618, 818230288201, 2513135860300849, 24686082394548211147, 787959836124458000837941, 82905574521614049485027140026 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Also, number of n X n binary matrices with both rows and columns, considered as binary numbers, in nondecreasing order. (Ordering only rows gives A060690.) - R. H. Hardin, May 08 2008
A result of Adolf Mader and Otto Mutzbauer shows that the two definitions are equivalent. - Victor S. Miller, Feb 03 2009
For n=5, only 0.07% remain distinct. Sorting columns and\or rows does not change the permanent of the matrix and leaves the absolute value of the determinant unchanged.
Diagonal of A180985.
REFERENCES
Adolf Mader and Otto Mutzbauer, "Double Orderings of (0,1) Matrices", Ars Combinatoria v. 61 (2001) pp 81-95.
LINKS
Table of n, a(n) for n=0..12.
R. H. Hardin, Binary arrays with both rows and cols sorted, symmetries
M. Werner, An Algorithmic Approach for the Zarankiewicz Problem, Slides, 2012. - From N. J. A. Sloane, Jan 01 2013
EXAMPLE
The 7 (2 X 2)-matrices are {{0,0},{0,0}}, {{0,0},{0,1}}, {{0,0},{1,1}}, {{0,1},{0,1}}, {{0,1},{1,0}}, {{0,1},{1,1}} and {{1,1},{1,1}}.
MATHEMATICA
baseform[li_List] := FixedPoint[Sort[Transpose[Sort[Transpose[Sort[ #1]]]]]&, li]; Table[Length@Split[Sort[baseform/@(Partition[ #, n]&/@(IntegerDigits[Range[0, -1+2^n^2], 2, n^2]))]], {n, 4}]
CROSSREFS
Cf. A088672, A087981, A180985.
Column 0 of A374525.
KEYWORD
nonn,hard,more
AUTHOR
Wouter Meeussen, Nov 03 2003
EXTENSIONS
a(6)-a(12) found by R. H. Hardin, May 08 2008. These terms were found using bdd's (binary decision diagrams), just setting up the logical relations between bits in a gigantic bdd expression and using that to count the satisfying states.
Edited by N. J. A. Sloane, Feb 05 2009 at the suggestion of Victor S. Miller
STATUS
approved
A092056 Square table by antidiagonals where T(n,k) = binomial(n+2^k-1,n). +0
4
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 10, 4, 1, 1, 16, 36, 20, 5, 1, 1, 32, 136, 120, 35, 6, 1, 1, 64, 528, 816, 330, 56, 7, 1, 1, 128, 2080, 5984, 3876, 792, 84, 8, 1, 1, 256, 8256, 45760, 52360, 15504, 1716, 120, 9, 1, 1, 512, 32896, 357760, 766480, 376992, 54264, 3432, 165, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Each column is convolution of preceding column starting from the all 1's sequence.
LINKS
Table of n, a(n) for n=0..65.
FORMULA
T(n,k) = Sum_{i=0..n} T(i,k-1)*T(n-i,k-1) starting with T(n,0) = 1 for n>=0.
EXAMPLE
Rows start:
1, 1, 1, 1, 1, 1, 1,...
1, 2, 4, 8, 16, 32, 64,...
1, 3, 10, 36, 136, 528, 2080,...
1, 4, 20, 120, 816, 5984, 45760,...
1, 5, 35, 330, 3876, 52360, 766480,...
...
CROSSREFS
Columns include (essentially) A000012, A000027, A000292, A000580, A010968, etc. Rows include A000012, A000079, A007582, A092056.
Main diagonal gives A060690.
Cf. A137153 (same with reflected antidiagonals).
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Feb 19 2004
STATUS
approved
A093048 a(n) = n minus exponent of 2 in n, with a(0) = 0. +0
4
0, 1, 1, 3, 2, 5, 5, 7, 5, 9, 9, 11, 10, 13, 13, 15, 12, 17, 17, 19, 18, 21, 21, 23, 21, 25, 25, 27, 26, 29, 29, 31, 27, 33, 33, 35, 34, 37, 37, 39, 37, 41, 41, 43, 42, 45, 45, 47, 44, 49, 49, 51, 50, 53, 53, 55, 53, 57, 57, 59, 58, 61, 61, 63, 58, 65, 65, 67, 66, 69 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..1000
FORMULA
Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n + 1.
G.f.: Sum_{k>=0} (t*(t^3 + t^2 + 1)/(1 - t^2)^2), with t = x^2^k.
a(n) = Sum_{k=1..n} sign(n mod 2^k). - Wesley Ivan Hurt, May 09 2021
EXAMPLE
G.f. = x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 7*x^7 + 5*x^8 + 9*x^9 + ... - Michael Somos, Jan 25 2020
MAPLE
A093048 := proc(n)
n-A007814(n) ;
end proc: # R. J. Mathar, Jul 24 2014
MATHEMATICA
a[ n_] := If[ n == 0, n - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)
PROG
(PARI) a(n) = if(n<1, 0, if(n%2==0, a(n/2) + n/2 - 1, n))
(PARI) a(n) = n - valuation(n, 2) \\ Jianing Song, Oct 24 2018
(Python)
def A093048(n): return n-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022
CROSSREFS
a(n) = n - A007814(n) = A093049(n) + 1, n > 0.
a(n) is the exponent of 2 in A002689(n-1), A014070(n), A060690(n), A075101(n).
See also A084623.
KEYWORD
nonn
AUTHOR
Ralf Stephan, Mar 16 2004
STATUS
approved
A094223 Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows). +0
13
1, 2, 7, 68, 2251, 247016, 89254228, 108168781424, 451141297789858, 6625037125817801312, 348562672319990399962384, 66545827618461283102105245248, 46543235997095840080425299916917968, 120155975713532210671953821005746669185792, 1152009540439950050422144845158703009569109376384 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*binomial(2^k, n).
a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k+n-1, n).
MATHEMATICA
a[n_] := Sum[(-1)^(n - k)*StirlingS1[n, k]*Binomial[2^k, n], {k, 0, n}]; (* or *) a[n_] := Sum[ StirlingS1[n, k]*Binomial[2^k + n - 1, n], {k, 0, n}]; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, May 29 2004 *)
PROG
(PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022
CROSSREFS
Main diagonal of A059584 and A059587, A060690, A088309.
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763
KEYWORD
easy,nonn
AUTHOR
Goran Kilibarda and Vladeta Jovovic, May 28 2004
EXTENSIONS
More terms from Robert G. Wilson v, May 29 2004
a(13) onwards from Andrew Howroyd, Jan 20 2024
STATUS
approved
A132683 a(n) = binomial(2^n + n, n). +0
13
1, 3, 15, 165, 4845, 435897, 131115985, 138432467745, 525783425977953, 7271150092378906305, 368539102493388126164865, 68777035446753808820521420545, 47450879627176629761462147774626305 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..50
FORMULA
a(n) = [x^n] 1/(1-x)^(2^n + 1).
G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)*n!). - Paul D. Hanna, Feb 25 2009
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
EXAMPLE
From Paul D. Hanna, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 3*x + 15*x^2 + 165*x^3 + 4845*x^4 + 435897*x^5 + ...
A(x) = 1/(1-x) - log(1-2x)/(1-2x) + log(1-4x)^2/((1-4x)*2!) - log(1-8x)^3/((1-8x)*3!) +- ... (End)
MAPLE
A132683:= n-> binomial(2^n +n, n); seq(A132683(n), n=0..20); # G. C. Greubel, Mar 14 2021
MATHEMATICA
Table[Binomial[2^n+n, n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
PROG
(PARI) a(n)=binomial(2^n+n, n)
(PARI) {a(n)=polcoeff(sum(m=0, n, (-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))*m!)), n)} \\ Paul D. Hanna, Feb 25 2009
(Sage) [binomial(2^n +n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +n, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
CROSSREFS
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), this sequence (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A136555.
Cf. A066384. - Paul D. Hanna, Feb 25 2009
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 26 2007
STATUS
approved
A132684 a(n) = binomial(2^n + n + 1, n). +0
13
1, 4, 21, 220, 5985, 501942, 143218999, 145944307080, 542150225230185, 7398714129087308170, 372134605932348010322571, 69146263065062394421802892300, 47589861944854471977019273909187085 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..50
FORMULA
a(n) = [x^n] 1/(1-x)^(2^n + 2).
G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)^2*n!). - Paul D. Hanna, Feb 25 2009
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
EXAMPLE
From Paul D. Hanna, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 4*x + 21*x^2 + 220*x^3 + 5985*x^4 + 501942*x^5 +...
A(x) = 1/(1-x)^2 - log(1-2x)/(1-2x)^2 + log(1-4x)^2/((1-4x)^2*2!) - log(1-8x)^3/((1-8x)^2*3!) +- ... (End)
MAPLE
A132684:= n-> binomial(2^n +n+1, n); seq(A132684(n), n=0..20); # G. C. Greubel, Mar 14 2021
MATHEMATICA
Table[Binomial[2^n+n+1, n], {n, 0, 20}] (* Harvey P. Dale, Nov 10 2011 *)
PROG
(PARI) a(n)=binomial(2^n+n+1, n)
(PARI) {a(n)=polcoeff(sum(m=0, n, (-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))^2*m!)), n)} \\ Paul D. Hanna, Feb 25 2009
(Sage) [binomial(2^n +n+1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +n+1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
CROSSREFS
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), this sequence (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A136555.
Cf. A066384. - Paul D. Hanna, Feb 25 2009
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 26 2007
STATUS
approved
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Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)