A081494 -id:A081494 - 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

A081495 Start with Pascal's triangle; form a rhombus by sliding down n steps from top on both sides then sliding down inwards to complete the rhombus and then deleting the inner numbers; a(n) = sum of entries on perimeter of rhombus.

Original entry on oeis.org

1, 5, 17, 55, 189, 681, 2519, 9451, 35765, 136153, 520695, 1998745, 7696467, 29716025, 115000947, 445962899, 1732525861, 6741529113, 26270128535, 102501265057, 400411345659, 1565841089321, 6129331763923, 24014172955545, 94163002754699, 369507926510401

Offset: 1

Amarnath Murthy, Mar 25 2003

nonn

			The rhombus pertaining to n = 4 is obtained from the solid rhombus
.....1
...1...1
.1...2...1
1..3...3...1
..4..6...4
...10..10
.....20
giving
.....1
...1...1
.1.......1
1..........1
..4......4
...10..10
.....20
and the sum of all the numbers is 55, a(4) = 55.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A081494, A081496, A081497.

B:=Binomial;; Concatenation([1], List([2..25], n-> B(2*n, n)-B(2*(n-1), n-1) +2*n -3)); # G. C. Greubel, Aug 13 2019

C:=Catalan; [1] cat [(n+1)*C(n) -n*C(n-1) +2*n-3: n in [2..25]]; // G. C. Greubel, Aug 13 2019

seq(coeff(series(((1-x)^3 - (1-2*x-x^3)*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x) ), x, n+1), x, n), n = 1..25); # G. C. Greubel, Aug 13 2019

With[{C = CatalanNumber}, Table[If[n==1, 1, (n+1)*C[n] -n*C[n-1] +2*n-3], {n, 25}]] (* G. C. Greubel, Aug 13 2019 *)

vector(25, n, b=binomial; if(n==1,1,b(2*n, n)-b(2*(n-1), n-1) +2*n -3)) \\ G. C. Greubel, Aug 13 2019

b=binomial; [1]+[b(2*n, n)-b(2*(n-1), n-1) +2*n -3 for n in (2..25)] # G. C. Greubel, Aug 13 2019

a(0)=1 for n>0 a(n)=binomial(2*n, n)-binomial(2*n-2, n-1)+2*n-3. - Benoit Cloitre, Sep 10 2003

G.f.: ((1-x)^3 - (1-2*x-x^3)*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). - G. C. Greubel, Aug 13 2019

More terms from Benoit Cloitre, Sep 10 2003

A081496 Start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central rhombus containing exactly 4 numbers.

Original entry on oeis.org

5, 14, 46, 160, 574, 2100, 7788, 29172, 110110, 418132, 1595620, 6113744, 23505356, 90633800, 350351640, 1357278300, 5268292830, 20483876820, 79765662900, 311038321440, 1214362277700, 4746455801880, 18570960418920, 72728638093800

Offset: 1

Amarnath Murthy, Mar 25 2003

nonn

			The first three rhombuses are
...1...........2.........6
.1...1.......3...3.....10..10
...2......,....6.....,...20
and the corresponding sums are a(1) =5, a(2) =14 and a(3) =46.

Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019. See Example 5.1 and table page 17.

Cf. A081494, A081495, A081497, A029635.

seq((9*n-4)*binomial(2*(n-1),(n-1))/n,n=1..26); # C. Ronaldo, Dec 20 2004

{ A029635(n, k) = if( k<0 || k>n, 0, (n==0) + binomial(n, k) + binomial(n-1, k-1))}; \\ program from Michael Somos in A029635
{a(n) = sum(k=0,n,A029635(n, k)^2)} \\ Paul D. Hanna, Oct 17 2017
for(n=1,30,print1(a(n),", "))

a(n) = (9*n-4)*Catalan(n-1) = (9*n-4)*binomial(2*(n-1), (n-1))/n. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

a(n) = Sum_{k=0..n} A029635(n,k)^2 for n>=1, where A029635 is the Lucas triangle. - Paul D. Hanna, Oct 17 2017

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

A028270 Central elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

1, 3, 8, 26, 90, 322, 1176, 4356, 16302, 61490, 233376, 890188, 3409588, 13104756, 50517200, 195234120, 756197910, 2934686610, 11408741520, 44420399100, 173191792620, 676104403260, 2642356838160, 10337529691320, 40481034410700

Offset: 0

Mohammad K. Azarian

Or, start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central triangle containing exactly 3 numbers. The first three triangles are
...1...........2.........6
.1...1.......3...3.....10..10
and the corresponding sums are 3, 8 and 26. - Amarnath Murthy, Mar 25 2003
This sequence starting at a(n+2) has Hankel transform A000032(2n+1)*2^n (empirical observation). - Tony Foster III, May 20 2016

Cf. A081494, A081495, A081496, A000984.

Cf. A000108.

seq(binomial(2*n,n)+binomial(2*n-2,n-1),n=0..24);
seq(2*binomial(2*n-1,n-1)+binomial(2*n-2,n-1),n=1..24);

G.f.: (x+1)/sqrt(1-4*x). - Vladeta Jovovic, Jan 08 2004
a(n) = binomial(2n, n)+binomial(2n-2, n-1)=A000984(n)+A000984(n-1). - Emeric Deutsch, Apr 20 2004
a(n) = 2binomial(2n-1, n-1)+binomial(2n-2, n-1). - Emeric Deutsch, Apr 20 2004
a(n) = (n+1)*C(n) + n*C(n-1), C = Catalan number (A000108). - Gary W. Adamson, Dec 28 2007
G.f.: G(0) where G(k)= 1 + x/(1 - (4*k+2)/((4*k+2) + (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012
D-finite with recurrence n*a(n) -3*n*a(n-1) +2*(-2*n+5)*a(n-2)=0. - R. J. Mathar, May 01 2024

More terms from James Sellers

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

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A081495 Start with Pascal's triangle; form a rhombus by sliding down n steps from top on both sides then sliding down inwards to complete the rhombus and then deleting the inner numbers; a(n) = sum of entries on perimeter of rhombus.

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A081496 Start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central rhombus containing exactly 4 numbers.

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A028270 Central elements in 3-Pascal triangle A028262 (by row).

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