A077866 -id:A077866 - cp's OEIS frontend

A005803 Second-order Eulerian numbers: a(n) = 2^n - 2*n.

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

N. J. A. Sloane, Simon Plouffe

Starting with n=2, a(n) is the second-order Eulerian number <> (see A008517).
Also, number of 3 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
This is the number of target DNA sequences of the correct length present after each successive cycle of the Polymerase Chain Reaction (PCR). The first two cycles produce 0 target sequences, there are 2 target sequences present after the third cycle, then 8 after the fourth cycle, and so forth. - A. Timothy Royappa, Jun 16 2012
a(n+2) = the row sums of A222403. - J. M. Bergot, Apr 04 2018

			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.
a(n) = A070313 + 1 = A052515 + 2. Bisection of A077866.
Equals for n =>3 the third right hand column of A163936.
Cf. A000918 (first differences).
Cf. A000079, A000325, A005843, A130102.

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

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 */

G.f.: 1 + 2*x^3/((1-x)^2*(1-2*x)). a(n) = A008517(n-1, 2). - Michael Somos, Oct 13 2002
Equals binomial transform of [1, -1, 1, 1, 1, ...]. - Gary W. Adamson, Jul 14 2008
a(0) = 1 and a(n) = Sum_{k=0..n-3} ((-1)^(n+k+1)*binomial(2*n-1,k)*stirling1(2*n-k-3,n-k-2)), n=>1. - Johannes W. Meijer, Oct 16 2009
a(0)=1, a(1)=0, a(2)=0, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, May 21 2011
a(n) = A000225(n+1) - A081494(n+1), n > 1. In other words, a(n) equals the sum of the elements in a Pascal triangle of depth n+1 minus the sum of the elements on its perimeter. - Ivan N. Ianakiev, Jun 01 2014
a(n) = A165900(n-1) + Sum_{i=0..n-1} a(i), for n > 0. - Ivan N. Ianakiev, Nov 24 2014
a(n) = A000225(n) - A005408(n-1). - Miquel Cerda, Nov 25 2016
E.g.f.: exp(x)*(exp(x) - 2*x). - Ilya Gutkovskiy, Nov 25 2016

A276677 Number of squares added at the n-th generation of a symmetric (with 45-degree angles), non-overlapping Pythagoras tree.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 48, 76, 120, 180, 272, 396, 584, 836, 1216, 1724, 2488, 3508, 5040, 7084, 10152, 14244, 20384, 28572, 40856, 57236, 81808, 114572, 163720, 229252, 327552, 458620, 655224, 917364, 1310576, 1834860, 2621288, 3669860, 5242720, 7339868

Offset: 0

Ernst van de Kerkhof, Sep 13 2016

The auxiliary sequence C(n), which appears in the recurrence relation for a(n), is defined as the number of collisions (squares touching each other, halting tree growth at that point) in generation n.

Colin Barker, Table of n, a(n) for n = 0..1000
Ernst van de Kerkhof, Illustration of a(7)
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

With an offset of 4, auxiliary sequence C(n) is equal to A077866: C(n+4) = A077866(n).

Partial sums give A276647.

TableForm[Table[{n, 6*2^Floor[n/2] + 8*2^Floor[(n-1)/2] - (4n + 8)}, {n, 1, 100, 1}], TableSpacing -> {1, 5}]
LinearRecurrence[{2,1,-4,2},{1,2,4,8,16},70] (* Harvey P. Dale, Jan 21 2019 *)

Vec((1+x)^2*(1-2*x+2*x^2)/((1-x)^2*(1-2*x^2)) + O(x^50)) \\ Colin Barker, Sep 20 2016

a(0) = 1, a(n) = 2*a(n-1) - 4*C(n-1), where:
C(0) = 0; for n >= 1, C(n) = C(n-1) + 2^(floor(n/2)-1) - 1. Also:
C(0) = 0; for n >= 1, C(n) = 2^floor(n/2) + 2^floor((n-1)/2) - (n+1).
a(0) = 1; for n >= 1, a(n) = 6*2^floor(n/2) + 8*2^floor((n-1)/2) - (4*n+8).
All formulas are proved.
From Colin Barker, Sep 20 2016: (Start)
G.f.: (1 + x)^2*(1 - 2*x + 2*x^2) / ((1 - x)^2*(1 - 2*x^2)).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + 2*a(n-4) for n>4.
a(n) = -4+2^((n-1)/2)*(7-7*(-1)^n+5*sqrt(2)+5*(-1)^n*sqrt(2))-4*(1+n) for n>0. Therefore:
a(n) = 5*2^(n/2+1)-8-4*n for n>0 and even;
a(n) = 7*2^((n+1)/2)-8-4*n for n>0 and odd. (End)

A069023 Define a subset of divisors of n to be a dedicated subset if the product of any two members is also a divisor of n. 1 is not allowed as a member as it gives trivially 1*d = d a divisor. a(n) is the number of dedicated subsets of divisors of n with at least two members.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 9, 0, 1, 1, 3, 0, 7, 0, 5, 1, 1, 1, 9, 0, 1, 1, 9, 0, 7, 0, 3, 3, 1, 0, 17, 0, 3, 1, 3, 0, 9, 1, 9, 1, 1, 0, 20, 0, 1, 3, 8, 1, 7, 0, 3, 1, 7, 0, 28, 0, 1, 3, 3, 1, 7, 0, 17, 2, 1, 0, 20, 1, 1, 1, 9, 0, 20, 1, 3, 1, 1, 1, 35, 0, 3, 3, 9, 0, 7

Offset: 1

Amarnath Murthy, Apr 02 2002

nonn

a(n) is determined by the prime signature of n.

			a(12) = 3. The divisors of 12 are 1,2,3,4,6,12. The divisor subsets (2,3),(2,6) and (3,4) are such that their product is also a divisor of 12. a(24) = 9 and the dedicated divisor subsets are (2,3),(2,4),(2,6),(2,12),(3,4),(3,8),(4,6),(2,3,4),(2,4,6).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Scheme-program for computing this sequence
Index entries for sequences computed from exponents in factorization of n

Cf. A077866.

\\ The following program is very inefficient:
A069023(n) = { if(bigomega(n)<2,return(0)); my(pds=(divisors(n)[2..numdiv(n)]), subsets = select(v -> (length(v)>=2),powerset(pds)), pair_products = apply(ss -> podp(ss), subsets), prodsmodn = apply(pps -> vector(#pps, i, n%pps[i]),pair_products)); length(select(s -> 0==vecsum(s),prodsmodn)); };
powerset(v) = { my(siz=2^length(v),pv=vector(siz)); for(i=0,siz-1,pv[i+1] = choosebybits(v,i)); pv; };
choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };
podp(v) = { my(siz=binomial(length(v),2),rv=vector(siz),k=0); for(i=1,length(v)-1,for(j=i+1,length(v),k++;rv[k] = v[i]*v[j])); rv; }; \\ podp = product of distinct pairs
\\ Antti Karttunen, Nov 24 2017

Scheme
```
;; See in the links-section.
```

It seems that for n >= 3, a(p^n) = A077866(n-3). - Antti Karttunen, Nov 24 2017

Edited by David Wasserman, Mar 26 2003

A143086 Triangle read by rows: T(n,k) = 2^(k + 1) - 1 if k < = floor(n/2), otherwise 2^(n - k + 1) - 1, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 7, 3, 1, 1, 3, 7, 7, 3, 1, 1, 3, 7, 15, 7, 3, 1, 1, 3, 7, 15, 15, 7, 3, 1, 1, 3, 7, 15, 31, 15, 7, 3, 1, 1, 3, 7, 15, 31, 31, 15, 7, 3, 1, 1, 3, 7, 15, 31, 63, 31, 15, 7, 3, 1, 1, 3, 7, 15, 31, 63, 63, 31, 15, 7, 3, 1, 1, 3, 7, 15, 31, 63, 127, 63, 31, 15, 7, 3, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 16 2008

			Triangle begins:
  {1},
  {1,  1},
  {1,  3,  1},
  {1,  3,  3,  1},
  {1,  3,  7,  3,  1},
  {1,  3,  7,  7,  3,  1},
  {1,  3,  7, 15,  7,  3,  1},
  {1,  3,  7, 15, 15,  7,  3,  1},
  {1,  3,  7, 15, 31, 15,  7,  3,  1},
  {1,  3,  7, 15, 31, 31, 15,  7,  3,  1},
  {1,  3,  7, 15, 31, 63, 31, 15,  7,  3,  1}

Cf. A077866 (row sums).

Table[Table[If[m <= Floor[n/2], 2^(m + 1) - 1, 2^(n - m + 1) - 1], {m, 0, n}], {n, 0, 10}] // Flatten

Offset changed to 0 by Georg Fischer, Jun 08 2023

A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 4, 3, 5, 3, 1, 0, 7, 8, 8, 4, 1, 8, 7, 15, 16, 12, 5, 1, 0, 15, 22, 31, 28, 17, 6, 1, 16, 15, 37, 53, 59, 45, 23, 7, 1, 0, 31, 52, 90, 112, 104, 68, 30, 8, 1, 32, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 0, 63, 114, 225

Offset: 0

Philippe Deléham, Jan 11 2014

Row sums are A007179(n+1).

			Triangle begins (0<=k<=n):
1
0, 1
2, 1, 1
0, 3, 2, 1
4, 3, 5, 3, 1
0, 7, 8, 8, 4, 1
8, 7, 15, 16, 12, 5, 1
0, 15, 22, 31, 28, 17, 6, 1

Cf. Columns: A077957, A052551, A077866.
Diagonals: A000012, A001477, A022856.
Cf. Similar sequences: A059260, A191582.

T(n,n)=1, T(2n,0)=2^n, T(2n+1,0)=0, T(n,k)=T(n-1,k-1)+T(n-1,k) for 0

T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-3,k)-2*T(n-3,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.

T(n,n)=1, T(n+1,n)=n, T(n+2,n)=n*(n+1)/2 + 2.

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3*x + 2*x^2/2! + x^3/3!) = 3*x + 8*x^2/2! + 16*x^3/3! + 28*x^4/4! + 45*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

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A005803 Second-order Eulerian numbers: a(n) = 2^n - 2*n.

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A276677 Number of squares added at the n-th generation of a symmetric (with 45-degree angles), non-overlapping Pythagoras tree.

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A069023 Define a subset of divisors of n to be a dedicated subset if the product of any two members is also a divisor of n. 1 is not allowed as a member as it gives trivially 1*d = d a divisor. a(n) is the number of dedicated subsets of divisors of n with at least two members.

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A143086 Triangle read by rows: T(n,k) = 2^(k + 1) - 1 if k < = floor(n/2), otherwise 2^(n - k + 1) - 1, for 0 <= k <= n.

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A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

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