A344781
Numbers k such that A070313(k) = 2^k - (2*k+1) is a prime number.
Original entry on oeis.org
4, 7, 8, 28, 32, 81, 669, 1108, 1699, 1839, 2319, 9566, 14866, 30855, 35932, 56048, 70915, 72578
Offset: 1
4 is a term since 2^4 - (2*4+1) = 16 - 9 = 7 is a prime.
7 is a term since 2^7 - (2*7+1) = 128 - 15 = 113 is a prime.
A005803
Second-order Eulerian numbers: a(n) = 2^n - 2*n.
Original entry on oeis.org
1, 0, 0, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072, 8166, 16356, 32738, 65504, 131038, 262108, 524250, 1048536, 2097110, 4194260, 8388562, 16777168, 33554382, 67108812, 134217674, 268435400, 536870854, 1073741764, 2147483586
Offset: 0
G.f. = 1 + 2*x^3 + 8*x^4 + 22*x^5 + 52*x^6 + 114*x^7 + 240*x^8 + 494*x^9 + ...
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n=0..500
- S. Bilotta, E. Grazzini, and E. Pergola, Enumeration of Two Particular Sets of Minimal Permutations, J. Int. Seq. 18 (2015) 15.10.2.
- I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
- Jim Haglund and Mirko Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations.
- Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
- Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
- Sandi Klavžar, Uroš Milutinović and Ciril Petr, Hanoi graphs and some classical numbers, Expo. Math. 23 (2005), no. 4, 371-378.
- James McClung, Constructions and Applications of W-States, Bachelor Thesis, Worcester Polytechnic Institute (2020).
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- Sam Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018 (A000246); Discrete Math, 343 (2020), article 111869.
- Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
Equivalent to second column of
A008517.
Equals for n =>3 the third right hand column of
A163936.
-
a005803 n = 2 ^ n - 2 * n
a005803_list = 1 : f 1 [0, 2 ..] where
f x (z:zs@(z':_)) = y : f y zs where y = (x + z) * 2 - z'
-- Reinhard Zumkeller, Jan 19 2014
-
[2^n-2*n: n in [0..30]]; // Wesley Ivan Hurt, Jun 04 2014
-
A005803:=-2*z/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms
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Table[2^n-2n,{n,0,50}] (* or *) LinearRecurrence[{4,-5,2},{1,0,0},51] (* Harvey P. Dale, May 21 2011 *)
-
{a(n) = if( n<0, 0, 2^n - 2*n)}; /* Michael Somos, Oct 13 2002 */
A165900
a(n) = n^2 - n - 1.
Original entry on oeis.org
-1, -1, 1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255
Offset: 0
G.f. = -1 - x + x^2 + 5*x^3 + 11*x^4 + 19*x^5 + 29*x^6 + 41*x^7 + ... - _Michael Somos_, Mar 23 2023
-
a165900 n = n * (n - 1) - 1 -- Reinhard Zumkeller, Jul 29 2012
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Table[n^2 - n - 1, {n, 0, 50}] (* Ron Knott, Oct 27 2010 *)
LinearRecurrence[{3,-3,1},{-1,-1,1},60] (* Harvey P. Dale, Jul 05 2021 *)
-
a(n)=n^2-n-1 \\ Charles R Greathouse IV, Jan 12 2012
A046739
Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.
Original entry on oeis.org
0, 1, 1, 1, 1, 7, 1, 1, 21, 21, 1, 1, 51, 161, 51, 1, 1, 113, 813, 813, 113, 1, 1, 239, 3361, 7631, 3361, 239, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 1, 2025, 141549, 1704693, 5494017
Offset: 1
Triangle starts:
0;
1;
1, 1;
1, 7, 1;
1, 21, 21, 1;
1, 51, 161, 51, 1;
1, 113, 813, 813, 113, 1;
...
From _Peter Luschny_, Sep 17 2021: (Start)
The triangle shows the coefficients of the following bivariate polynomials:
[1] 0;
[2] x*y;
[3] x^2*y + x*y^2;
[4] x^3*y + 7*x^2*y^2 + x*y^3;
[5] x^4*y + 21*x^3*y^2 + 21*x^2*y^3 + x*y^4;
[6] x^5*y + 51*x^4*y^2 + 161*x^3*y^3 + 51*x^2*y^4 + x*y^5;
[7] x^6*y + 113*x^5*y^2 + 813*x^4*y^3 + 813*x^3*y^4 + 113*x^2*y^5 + x*y^6;
...
These polynomials are the permanents of the n X n matrices with all entries above the main antidiagonal set to 'x' and all entries below the main antidiagonal set to 'y'. The main antidiagonals consist only of zeros. Substituting x <- 1 and y <- -1 generates the Euler secant numbers A122045. (Compare with A081658.)
(End)
- Robert Israel, Table of n, a(n) for n = 1..10012 (rows 0 to 142, flattened)
- L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]
- L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975
- N. Gustafsson and L. Solus. Derangements, Ehrhart theory, and local h-polynomials, arXiv:1807.05246 [math.CO], 2018.
- R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]
- Lili Mu and Volkmar Welker, On a question about real rooted polynomials and f-polynomials of simplicial complexes, arXiv:2503.24076 [math.CO], 2025. See p. 8.
- D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
- D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16.
- R. P. Stanley, Subdivisions and local h-vectors, J. Amer. Math. Soc., 5 (1992), 805-851.
- John D. Wiltshire-Gordon, Alexander Woo, and Magdalena Zajaczkowska, Specht Polytopes and Specht Matroids, arXiv:1701.05277 [math.CO], 2017. [See Conjecture 6.2]
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G := (1-t)*exp(-t*z)/(1-t*exp((1-t)*z)): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form # Emeric Deutsch, May 25 2009
-
max = 12; f[t_, z_] := (1-t)*(Exp[-t*z]/(1 - t*Exp[(1-t)*z])); se = Series[f[t, z], {t, 0, max}, {z, 0, max}];
coes = Transpose[ #*Range[0, max]! & /@ CoefficientList[se, {t, z}]]; Join[{0}, Flatten[ Table[ coes[[n, k]], {n, 2, max}, {k, 2, n-1}]]] (* Jean-François Alcover, Oct 24 2011, after g.f. *)
E1[n_ /; n >= 0, 0] = 1; (* E1(n,k) are the Eulerian numbers *)
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
Table[T[n, k], {n, 1, 100}, {k, 1, n-1}] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 31 2020, after Peter Luschny in A271697 *)
Table[Expand[n!Factor[SeriesCoefficient[(x-y)/(x Exp[y t]-y Exp[x t]),{t,0,n}]]],{n,0,12}]//TableForm (* Mamuka Jibladze, Nov 26 2024 *)
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T(n)={my(x='x+O('x^(n+1))); concat([[0]], [Vecrev(p/y) | p<-Vec(-1+serlaplace((y-1)/(y*exp(x)-exp(x*y))))])}
{ my(A=T(10));for(i=1,#A,print(A[i])) } \\ Andrew Howroyd, Nov 13 2024
More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000
A052515
Number of ordered pairs of complementary subsets of an n-set with both subsets of cardinality at least 2.
Original entry on oeis.org
0, 0, 0, 0, 6, 20, 50, 112, 238, 492, 1002, 2024, 4070, 8164, 16354, 32736, 65502, 131036, 262106, 524248, 1048534, 2097108, 4194258, 8388560, 16777166, 33554380, 67108810, 134217672, 268435398, 536870852, 1073741762
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
-
m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(x)-1-x)^2 )); [0,0,0,0] cat [Factorial(n+3)*b[n]: n in [1..m-5]]; // G. C. Greubel, May 13 2019
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Pairs spec := [S,{S=Prod(B,B),B=Set(Z,2 <= card)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
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Join[{0,0,0}, LinearRecurrence[{4,-5,2}, {0,6,20}, 35]] (* G. C. Greubel, May 13 2019 *)
With[{nn=30},CoefficientList[Series[(Exp[x]-x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 29 2023 *)
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concat([0,0,0,0],Vec((6-4*x)/(1-x)^2/(1-2*x)+O(x^35))) \\ Charles R Greathouse IV, Apr 03 2012
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x='x+O('x^35); concat([0,0,0,0],Vec(serlaplace((exp(x)-x-1)^2))) \\ Joerg Arndt, Apr 10 2013
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(2*x^4*(3-2*x)/((1-x)^2*(1-2*x))).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, May 13 2019
A271697
Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 7, 1, 0, 0, 1, 21, 21, 1, 0, 0, 1, 51, 161, 51, 1, 0, 0, 1, 113, 813, 813, 113, 1, 0, 0, 1, 239, 3361, 7631, 3361, 239, 1, 0, 0, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 0, 0, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 0
Offset: 0
Triangle starts:
1;
0, 0;
0, 1, 0;
0, 1, 1, 0;
0, 1, 7, 1, 0;
0, 1, 21, 21, 1, 0;
0, 1, 51, 161, 51, 1, 0;
0, 1, 113, 813, 813, 113, 1, 0;
...
Variant:
A046739 (main entry for this triangle).
Cf.
A000166 (row sums),
A122045 (Euler numbers are the alternating row sums),
A070313 (col. 2) and (diag. n,n-2).
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A271697 := (n,k) -> add(binomial(-j-1,-n-1)*combinat:-eulerian1(j,k), j=0..n):
seq(seq(A271697(n, k), k=0..n), n=0..11);
-
<= 0, 0] = 1;
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2020 *)
-
T(n)={my(x='x+O('x^(n+1)), v=Vec(serlaplace((y-1)/(y*exp(x)-exp(x*y))))); vector(#v,n,Vecrev(v[n],n))}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Nov 13 2024
A209248
a(n) = 2^(2^n - 2*n - 1).
Original entry on oeis.org
2, 128, 2097152, 2251799813685248, 10384593717069655257060992658440192, 883423532389192164791648750371459257913741948437809479060803100646309888
Offset: 3
a(5) = 2^(2^5 - 2*5 - 1) = 2^21 = 2097152.
A347502
Number of dominating sets in the n-cycle complement graph.
Original entry on oeis.org
0, -1, -1, 1, 9, 21, 51, 113, 239, 493, 1003, 2025, 4071, 8165, 16355, 32737, 65503, 131037, 262107, 524249, 1048535, 2097109, 4194259, 8388561, 16777167, 33554381, 67108811, 134217673, 268435399, 536870853, 1073741763, 2147483585, 4294967231, 8589934525
Offset: 0
A355659
Martingale win/loss triangle, read by rows: T(n,k) = total number of dollars won (or lost) using the martingale method on all possible n trials that have exactly k losses and n-k wins, for 0 <= k <= n.
Original entry on oeis.org
0, 1, -1, 2, 1, -3, 3, 5, -1, -7, 4, 11, 7, -7, -15, 5, 19, 24, 4, -21, -31, 6, 29, 53, 38, -12, -51, -63, 7, 41, 97, 111, 41, -57, -113, -127, 8, 55, 159, 243, 187, 5, -163, -239, -255, 9, 71, 242, 458, 500, 248, -130, -394, -493, -511, 10, 89, 349, 784, 1084, 874, 202, -488, -878, -1003, -1023
Offset: 0
Triangle T(n,k) begins:
n\k| 0 1 2 3 4 5 6 7 8 9
---+-------------------------------------------
0| 0
1| 1 -1
2| 2 1 -3
3| 3 5 -1 -7
4| 4 11 7 -7 -15
5| 5 19 24 4 -21 -31
6| 6 29 53 38 -12 -51 -63
7| 7 41 97 111 41 -57 -113 -127
8| 8 55 159 243 187 5 -163 -239 -255
9| 9 71 242 458 500 248 -130 -394 -493 -511
Examples from triangle:
T(4,3) = -7: In this example, we consider all possibilities with 4 trials that result in 3 losses and one win. There are binomial(4,3) = 4 different combinations to consider (lllw, llwl, lwll, and wlll), which have net earnings of +1, 0, -2, -6 respectively when using the martingale method, giving a total of -7.
T(6,2) = 53: In this example, we have 6 trials and we consider the results with 2 losses and 4 wins. There are binomial(6,2) = 15 such combinations to consider (wwwwll, wwwlwl, wwwllw, wwlwwl, wwlwlw, wwllww, wlwwwl, wlwwlw, wlwlww, wllwww, lwwwwl, lwwwlw, lwwlww, lwlwww, llwwww), and summing over all 15 earnings gives us a total of 53.
T(2,0) = 2: In this example, we have 2 trials, with 0 losses and 2 wins. In this one single case, the martingale method gives us earnings of +1 and +1 with a total of 2.
Showing 1-9 of 9 results.
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