A222403 -id:A222403 - 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

A222405 Triangle read by rows: left and right edges are A002061 (1,3,7,13,21,...), interior entries are filled in using the Pascal triangle rule.

Original entry on oeis.org

1, 3, 3, 7, 6, 7, 13, 13, 13, 13, 21, 26, 26, 26, 21, 31, 47, 52, 52, 47, 31, 43, 78, 99, 104, 99, 78, 43, 57, 121, 177, 203, 203, 177, 121, 57, 73, 178, 298, 380, 406, 380, 298, 178, 73, 91, 251, 476, 678, 786, 786, 678, 476, 251, 91, 111, 342, 727, 1154, 1464, 1572, 1464, 1154, 727, 342, 111

Offset: 0

N. J. A. Sloane, Feb 18 2013

			Triangle begins:
1
3, 3
7, 6, 7
13, 13, 13, 13
21, 26, 26, 26, 21
31, 47, 52, 52, 47, 31
43, 78, 99, 104, 99, 78, 43
57, 121, 177, 203, 203, 177, 121, 57
73, 178, 298, 380, 406, 380, 298, 178, 73
...

Cf. A007318, A002061, A222403, A222404.

Row sums are A027178.

d:=[seq(n*(n+1)+1,n=0..14)];
f:=proc(d) local T,M,n,i;
M:=nops(d);
T:=Array(0..M-1,0..M-1);
for n from 0 to M-1 do T[n,0]:=d[n+1]; T[n,n]:=d[n+1]; od:
for n from 2 to M-1 do
for i from 1 to n-1 do T[n,i]:=T[n-1,i-1]+T[n-1,i]; od: od:
lprint("triangle:");
for n from 0 to M-1 do lprint(seq(T[n,i],i=0..n)); od:
lprint("row sums:");
lprint([seq( add(T[i,j],j=0..i), i=0..M-1)]);
end;
f(d);

t[n_, n_] := n^2+n+1; t[n_, 0] := n^2+n+1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

A222404 Triangle read by rows: left and right edges are A002378, interior entries are filled in using the Pascal triangle rule.

Original entry on oeis.org

0, 2, 2, 6, 4, 6, 12, 10, 10, 12, 20, 22, 20, 22, 20, 30, 42, 42, 42, 42, 30, 42, 72, 84, 84, 84, 72, 42, 56, 114, 156, 168, 168, 156, 114, 56, 72, 170, 270, 324, 336, 324, 270, 170, 72, 90, 242, 440, 594, 660, 660, 594, 440, 242, 90, 110, 332, 682, 1034, 1254, 1320, 1254, 1034, 682, 332, 110

Offset: 0

N. J. A. Sloane, Feb 18 2013

			Triangle begins:
0
2, 2
6, 4, 6
12, 10, 10, 12
20, 22, 20, 22, 20
30, 42, 42, 42, 42, 30
42, 72, 84, 84, 84, 72, 42
56, 114, 156, 168, 168, 156, 114, 56
...

Cf. A007318, A002378, A222403, A222405.

Row sums are 4*A000295.

d:=[seq(n*(n+1),n=0..14)];
f:=proc(d) local T,M,n,i;
M:=nops(d);
T:=Array(0..M-1,0..M-1);
for n from 0 to M-1 do T[n,0]:=d[n+1]; T[n,n]:=d[n+1]; od:
for n from 2 to M-1 do
for i from 1 to n-1 do T[n,i]:=T[n-1,i-1]+T[n-1,i]; od: od:
lprint("triangle:");
for n from 0 to M-1 do lprint(seq(T[n,i],i=0..n)); od:
lprint("row sums:");
lprint([seq( add(T[i,j],j=0..i), i=0..M-1)]);
end;
f(d);

t[n_, n_] := n*(n+1); t[n_, 0] := n*(n+1); t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)

A134634 Triangle formed by Pascal's rule with borders = A000108.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 4, 4, 5, 14, 9, 8, 9, 14, 42, 23, 17, 17, 23, 42, 132, 65, 40, 34, 40, 65, 132, 429, 197, 105, 74, 74, 105, 197, 429, 1430, 626, 302, 179, 148, 179, 302, 626, 1430, 4862, 2056, 928, 481, 327, 327, 481, 928, 2056, 4862, 16796, 6918, 2984, 1409, 808, 654, 808, 1409, 2984, 6918, 16796, 58786, 23714, 9902, 4393, 2217, 1462, 1462, 2217, 4393, 9902, 23714, 58786

Offset: 0

Gary W. Adamson, Nov 04 2007

Row sums = A134635: (1, 2, 6, 18, 54, 164, ...).

			First few rows of the triangle:
    1;
    1,  1;
    2,  2,  2;
    5,  4,  4,  5;
   14,  9,  8,  9, 14;
   42, 23, 17, 17, 23, 42;
  132, 65, 40, 34, 40, 65, 132;
  ...

Cf. A000108, A134635, A222403, A222404, A222405.

Triangle, given right and left borders consist of the Catalan sequence, A000108; then T(n,k) = T(n-1,k) + T(n-1,k-1).

Recomputed by N. J. A. Sloane, Feb 18 2013

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

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A222405 Triangle read by rows: left and right edges are A002061 (1,3,7,13,21,...), interior entries are filled in using the Pascal triangle rule.

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A222404 Triangle read by rows: left and right edges are A002378, interior entries are filled in using the Pascal triangle rule.

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A134634 Triangle formed by Pascal's rule with borders = A000108.

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