Original entry on oeis.org
0, 1, 2, 16, 162, 3600, 147456, 12320100, 2058386904, 701841817600, 488286500625000, 696425232679321600, 2038348954317776486400, 12259459134020160144810000, 151596002479762016373851690400, 3855806813438155578522841251840000
Offset: 0
a(0) = 0 + 0 = 0
a(1) = (0+1) * (1+0) = 1
a(2) = (0+1) * (1+1) * (1+0) = 2
a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
As noted above, a(2*k+1) is a square for k>=0. The first 5 squares are 1, 16, 3600, 12320100, 701841817600, with corresponding square roots 1, 4, 60, 3510, 837760.
If n = 2*k, then s**s(n) has the form 2*F(k)*m^2, where m is an integer and F(k) is the k-th Fibonacci number; e.g., a(6) = 2*F(3)*(192)^2.
-
a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
seq(a(n), n=0..15); # Alois P. Heinz, Aug 02 2024
-
s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
-
a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024
A053290
Number of nonsingular n X n matrices over GF(3).
Original entry on oeis.org
1, 2, 48, 11232, 24261120, 475566474240, 84129611558952960, 134068444202678083338240, 1923442429811445711790394572800, 248381049201184165590947520186915225600, 288678833735376059528974260112416365258106470400
Offset: 0
-
[1] cat [&*[(3^n - 3^k): k in [0..n-1]]: n in [1..9]]; // Bruno Berselli, Jan 28 2013
-
Table[Product[3^n - 3^k, {k, 0, n - 1}], {n, 0, 10}] (* Geoffrey Critzer, Jan 26 2013; edited by Vincenzo Librandi, Jan 28 2013 *)
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for(n=0,10, print1(prod(k=0,n-1, 3^n - 3^k), ", ")) \\ G. C. Greubel, May 31 2018
A027871
a(n) = Product_{i=1..n} (3^i - 1).
Original entry on oeis.org
1, 2, 16, 416, 33280, 8053760, 5863137280, 12816818094080, 84078326697164800, 1654829626053597593600, 97714379759212830706892800, 17309711516825516108403231948800
Offset: 0
-
[1] cat [&*[ 3^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015
-
A027871 := proc(n)
mul( 3^i-1,i=1..n) ;
end proc:
seq(A027871(n),n=0..8) ; # R. J. Mathar, Jul 13 2017
-
Table[Product[(3^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[3, 3, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
-
a(n) = prod(i=1, n, 3^i-1); \\ Michel Marcus, Nov 21 2015
A001174
Number of oriented graphs (i.e., digraphs with no bidirected edges) on n unlabeled nodes. Also number of complete digraphs on n unlabeled nodes. Number of antisymmetric relations (i.e., oriented graphs with loops) on n unlabeled nodes is A083670.
Original entry on oeis.org
1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, 816007449011040, 4374406209970747314, 64539836938720749739356, 2637796735571225009053373136, 300365896158980530053498490893399
Offset: 1
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 133, c_p.
- M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Andrew Howroyd, Table of n, a(n) for n = 1..50
- P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
- R. L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
- Musa Demirci, Ugur Ana, and Ismail Naci Cangul, Properties of Characteristic Polynomials of Oriented Graphs, Proc. Int'l Conf. Adv. Math. Comp. (ICAMC 2020) Springer, see p. 60.
- F. Harary and E. M. Palmer, Enumeration of mixed graphs, Proc. Amer. Math. Soc., 17 (1966), 682-687.
- T. R. Hoffman and J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014-2017.
- M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
- G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
- Eric Weisstein's World of Mathematics, Oriented Graph
-
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v - 1, 2];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 15] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)
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permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Oct 23 2017
-
from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A001174(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q-1>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 15 2024
A015001
q-factorial numbers for q=3.
Original entry on oeis.org
1, 1, 4, 52, 2080, 251680, 91611520, 100131391360, 328430963660800, 3232089113385932800, 95424198983606279987200, 8452007576574959037306265600, 2245867453247498115393020895232000, 1790317944898228845164815929864036352000
Offset: 0
-
[n le 1 select 1 else (3^n-1)*Self(n-1)/2: n in [1..15]]; // Vincenzo Librandi, Oct 22 2012
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RecurrenceTable[{a[1]==1, a[n]==((3^n - 1) * a[n-1])/2}, a, {n,15}] (* Vincenzo Librandi, Oct 27 2012 *)
Table[QFactorial[n, 3], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)
A195248
T(n,k) = Number of lower triangles of an n X n 0..k array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by two or less.
Original entry on oeis.org
2, 3, 8, 4, 27, 64, 5, 46, 729, 1024, 6, 65, 1682, 59049, 32768, 7, 84, 2729, 190514, 14348907, 2097152, 8, 103, 3776, 357847, 67379894, 10460353203, 268435456, 9, 122, 4823, 533142, 147824001, 74236765958, 22876792454961, 68719476736, 10, 141
Offset: 1
Some solutions for n=6, k=5
..4............1............5............0............0............1
..5.3..........1.3..........5.4..........1.2..........0.2..........0.0
..5.4.5........1.2.4........3.4.4........1.3.1........1.0.0........2.1.2
..4.3.4.4......3.3.3.3......5.3.5.4......2.1.1.2......0.1.1.0......0.0.2.2
..5.4.2.2.4....4.5.4.5.5....4.3.5.4.3....2.3.1.3.1....2.0.2.2.0....2.0.1.2.4
..4.3.2.2.4.4..3.5.3.3.5.5..3.5.5.3.3.1..4.2.2.1.1.0..0.2.0.2.0.1..1.1.0.2.4.4
A054941
Number of weakly connected oriented graphs on n labeled nodes.
Original entry on oeis.org
1, 2, 20, 624, 55248, 13982208, 10358360640, 22792648882176, 149888345786341632, 2952810709943411146752, 174416705255313941476193280, 30901060796613886817249881227264, 16422801513633911416125344647746244608, 26183660776604240464418800095675915958222848
Offset: 1
-
m:=30;
f:= func< x | (&+[3^Binomial(n,2)*x^n/Factorial(n) : n in [0..m+3]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( Log(f(x)) ))); // G. C. Greubel, Apr 28 2023
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nn=20; s=Sum[3^Binomial[n,2]x^n/n!,{n,0,nn}];
Drop[Range[0,nn]! CoefficientList[Series[Log[s]+1,{x,0,nn}],x],1] (* Geoffrey Critzer, Oct 22 2012 *)
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N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, 3^binomial(k, 2)*x^k/k!)))) \\ Seiichi Manyama, May 18 2019
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m=30
def f(x): return sum(3^binomial(n,2)*x^n/factorial(n) for n in range(m+4))
def A054941_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( log(f(x)) ).egf_to_ogf().list()
a=A054941_list(40); a[1:] # G. C. Greubel, Apr 28 2023
A109345
a(n) = 5^((n^2 - n)/2).
Original entry on oeis.org
1, 1, 5, 125, 15625, 9765625, 30517578125, 476837158203125, 37252902984619140625, 14551915228366851806640625, 28421709430404007434844970703125
Offset: 0
Cf.
A006125 (number of graphs on n labeled nodes),
A047656 (number of semi-complete digraphs on n labeled nodes),
A053763 (number of simple digraphs on n labeled nodes),
A053764.
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List([0..12], n -> 5^Binomial(n,2)); # G. C. Greubel, Feb 09 2019
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[5^Binomial(n,2): n in [0..12]]; // G. C. Greubel, Feb 09 2019
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seq(5^(binomial(2+n,n)), n=-2..8); # Zerinvary Lajos, Jun 12 2007
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5^Binomial[Range[0, 12], 2] (* G. C. Greubel, Feb 09 2019 *)
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a(n)=5^binomial(n,2) \\ Charles R Greathouse IV, Jan 11 2012
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[5^binomial(n,2) for n in (0..12)] # G. C. Greubel, Feb 09 2019
A117262
Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: [T^-1](n+1,n) = -3^n, with all 1's in the main diagonal and zeros elsewhere.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 27, 27, 9, 1, 729, 729, 243, 27, 1, 59049, 59049, 19683, 2187, 81, 1, 14348907, 14348907, 4782969, 531441, 19683, 243, 1, 10460353203, 10460353203, 3486784401, 387420489, 14348907, 177147, 729, 1
Offset: 0
Triangle T begins:
1;
1,1;
3,3,1;
27,27,9,1;
729,729,243,27,1;
59049,59049,19683,2187,81,1;
14348907,14348907,4782969,531441,19683,243,1;
10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;
Matrix inverse T^-1 has -3^n in the 2nd diagonal:
1,
-1,1,
0,-3,1,
0,0,-9,1,
0,0,0,-27,1,
0,0,0,0,-81,1,
0,0,0,0,0,-243,1, ...
Cf.
A047656 (column 0),
A117263 (row sums); variants:
A117250 (p=q=2),
A117252 (p=q=3),
A117254 (p=q=4),
A117256 (p=q=5),
A117258 (p=2, q=4),
A117260 (p=-1, q=2),
A117265 (p=-2, q=2).
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{T(n,k)=local(m=1,p=-1,q=3,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
A025237
Expansion of (1 -x -sqrt(1-2*x-11*x^2))/(6*x^2).
Original entry on oeis.org
1, 1, 4, 10, 37, 121, 451, 1639, 6259, 23923, 93502, 367852, 1465003, 5874103, 23740276, 96503554, 394542379, 1620716251, 6687296308, 27700303510, 115152607831, 480244735171, 2008802728819, 8425318166635, 35425680021397, 149296062114181, 630526903497706, 2668194946794124, 11311786743536125
Offset: 0
G.f.: 1 + x + 4*x^2 + 10*x^3 + 37*x^4 + 121*x^5 + 451*x^6 + 1639*x^7 + ...
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.
- Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
- Stefano Capparelli and Alberto Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, 18 (2015), #15.8.5.
- Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
- Serkan Demiriz, Adem Şahin, and Sezer Erdem, Some topological and geometric properties of novel generalized Motzkin sequence spaces, Rendiconti Circ. Mat. Palermo Ser. 2 (2025) Vol. 74, No. 136. See p. 4.
- Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
-
CoefficientList[Series[(1 - x - Sqrt[1 - 2*x - 11*x^2])/(6*x^2), {x, 0, 50}], x] (* G. C. Greubel, Feb 07 2017 *)
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{a(n) = polcoeff((1 - x - sqrt(1 - 2*x - 11*x^2 + x^3*O(x^n))) / (6*x^2), n)}; /* Michael Somos, Sep 23 2003 */
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