A033484 -id:A033484 - cp's OEIS frontend

A089591 "Lazy binary" representation of n. Also called redundant binary representation of n.

0, 1, 10, 11, 20, 101, 110, 111, 120, 201, 210, 1011, 1020, 1101, 1110, 1111, 1120, 1201, 1210, 2011, 2020, 2101, 2110, 10111, 10120, 10201, 10210, 11011, 11020, 11101, 11110, 11111, 11120, 11201, 11210, 12011, 12020, 12101, 12110, 20111

Offset: 0

Jeff Erickson, Dec 29 2003

Let a(0) = 0 and construct a(n) from a(n-1) by (i) incrementing the rightmost digit and (ii) if any digit is 2, replace the rightmost 2 with a 0 and increment the digit immediately to its left. (Note that changing "if" to "while" in this recipe gives the standard binary representation of n, A007088(n)).
Equivalently, a(2n+1) = a(n):1 and a(2n+2) = b(n):0, where b(n) is obtained from a(n) by incrementing the least significant digit and : denotes string concatenation.
If the digits of a(n) are d_k, d_{k-1}, ..., d_2, d_1, d_0, then n = Sum_{i=0..k} d_i*2^i, just as in standard binary notation.  The difference is that here we are a bit lazy, and allow a few digits to be 2's. The number of 2's in a(n) appears to be A037800(n+1). - N. J. A. Sloane, Jun 03 2023
Every pair of 2's is separated by a 0 and every pair of significant 0's is separated by a 2.
a(n) has exactly floor(log_2((n+2)/3))+1 digits [cf. A033484] and their sum is exactly floor(log_2(n+1)) [A000523].
The i-th digit of a(n) is ceiling( floor( ((n+1-2^i) mod 2^(i+1))/2^(i-1) ) / 2).
A137951 gives values of terms interpreted as ternary numbers, a(n)=A007089(A137951(n)). - Reinhard Zumkeller, Feb 25 2008

			a(8) = 120 -> 121 -> 201 = a(9); a(9) = 201 -> 202 -> 210 = a(10).

Gerth S. Brodal, Worst-case efficient priority queues, SODA 1996.
Michael J. Clancy and D. E. Knuth, A programming and problem-solving seminar, Technical Report STAN-CS-77-606, Department of Computer Science, Stanford University, Palo Alto, 1977.
Haim Kaplan and Robert E. Tarjan, Purely functional representations of catenable sorted lists, STOC 1996.
Chris Okasaki, Purely Functional Data Structures, Cambridge, 1998.

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000523, A033484, A072998, A024629, A089600, A089601, A089604, A037800.

A158582: lazy binary different from regular binary, A089633: lazy binary and regular binary agree.

A089591 := proc(n) option remember ; local nhalf ; if n <= 1 then RETURN(n) ; else nhalf := floor(n/2) ; if n mod 2 = 1 then RETURN(10*A089591(nhalf) +1) ; else RETURN(10*(A089591(nhalf-1)+1)) ; fi ; fi ; end: for n from 0 to 200 do printf("%d, ",A089591(n)) ; od ; # R. J. Mathar, Mar 11 2007

a[n_] := a[n] = Module[{nhalf}, If[n <= 1, Return[n], nhalf = Floor[n/2]; If[Mod[n, 2]==1, Return[10*a[nhalf]+1], Return[10*(a[nhalf-1]+1)]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 19 2016, after R. J. Mathar *)

More terms from R. J. Mathar, Mar 11 2007
Edited by Charles R Greathouse IV, Apr 30 2010
Edited by N. J. A. Sloane, Jun 03 2023

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 16, 60, 228, 864, 3276, 12420, 47088, 178524, 676836, 2566080, 9728748, 36884484, 139839696, 530172540, 2010036708, 7620627744, 28891993356, 109537863300, 415289569968, 1574482299804, 5969315609316, 22631393727360, 85802128010028, 325300565212164

Offset: 0

N. J. A. Sloane, Nov 20 2006

From Johannes W. Meijer, Aug 14 2010: (Start)
A berserker sequence, see A180141. For the corner squares 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the side squares to A180142 and for the central square to A155116.
This sequence belongs to a family of sequences with GF(x) = (1+x+k*x^2)/(1-3*x+(k-4)*x^2). Berserker sequences that are members of this family are 4*A055099(n) (k=2; with leading 1 added), A123620 (k=1; this sequence), A000302 (k=0), 4*A179606 (k=-1; with leading 1 added) and A180141 (k=-2). Some other members of this family are 4*A003688 (k=3; with leading 1 added), 4*A003946 (k=4; with leading 1 added), 4*A002878 (k=5; with leading 1 added) and 4*A033484 (k=6; with leading 1 added).
(End)
a(n) is the number of length n sequences on an alphabet of 4 letters that do not contain more than 2 consecutive equal letters. For example, a(3)=60 because we count all 4^3=64 words except: aaa, bbb, ccc, ddd. - Geoffrey Critzer, Mar 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 205
Index entries for linear recurrences with constant coefficients, signature (3,3).

Column 4 in A265584.

[1] cat [Round(((2^(1-n)*(-(3-Sqrt(21))^(1+n) + (3+Sqrt(21))^(1+n))))/(3*Sqrt(21))): n in [1..50]]; // G. C. Greubel, Oct 26 2017

nn=25;CoefficientList[Series[(1-z^(m+1))/(1-r z +(r-1)z^(m+1))/.{r->4,m->2},{z,0,nn}],z] (* Geoffrey Critzer, Mar 12 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1 - 3 x - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{3,3},{1,4,16},30] (* Harvey P. Dale, Jul 14 2023 *)

my(x='x+O('x^50)); Vec((1+x+x^2)/(1-3*x-3*x^2)) \\ G. C. Greubel, Oct 16 2017

a(0)=1, a(1)=4, a(2)=16, a(n)=3*a(n-1)+3*a(n-2) for n>2. - Philippe Deléham, Sep 18 2009

a(n) = ((2^(1-n)*(-(3-sqrt(21))^(1+n) + (3+sqrt(21))^(1+n)))) / (3*sqrt(21)) for n>0. - Colin Barker, Oct 17 2017

A176449 a(n) = 9*2^n - 2.

Original entry on oeis.org

7, 16, 34, 70, 142, 286, 574, 1150, 2302, 4606, 9214, 18430, 36862, 73726, 147454, 294910, 589822, 1179646, 2359294, 4718590, 9437182, 18874366, 37748734, 75497470, 150994942, 301989886, 603979774, 1207959550, 2415919102

Offset: 0

Vincenzo Librandi, Apr 18 2010

			For n = 1, a(1) = 2*(7+1) = 16;
for n = 2, a(2) = 2*(16+1) = 34;
for n = 3, a(3) = 2*(34+1) = 70.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Magma
```
[9*2^n-2: n in [0..100]]
```

A176449:=n->9*2^n-2; seq(A176449(n), n=0..100); # Wesley Ivan Hurt, Nov 08 2013

Table[9*2^n-2, {n,0,100}] (* Wesley Ivan Hurt, Nov 08 2013 *)
LinearRecurrence[{3,-2},{7,16},40] (* Harvey P. Dale, Jan 12 2024 *)

apply( A176449(n)=9<M. F. Hasler, Dec 11 2018

a(n) = 2*(a(n-1)+1) with a(0) = 7.
a(n) = 3*a(n-1) -2*a(n-2). G.f.: (7-5*x) / ((2*x-1)*(x-1)). - R. J. Mathar, May 02 2010
a(n) = 2*A052996(n+1) for n > 0. - Bruno Berselli and Vincenzo Librandi, Aug 27 2010
a(n) = A033484(n+2) - A007283(n). - M. F. Hasler, Dec 11 2018

Edited by M. F. Hasler, Dec 11 2018

A196305 a(n) = 15*2^n - 1.

Original entry on oeis.org

14, 29, 59, 119, 239, 479, 959, 1919, 3839, 7679, 15359, 30719, 61439, 122879, 245759, 491519, 983039, 1966079, 3932159, 7864319, 15728639, 31457279, 62914559, 125829119, 251658239, 503316479, 1006632959, 2013265919, 4026531839, 8053063679, 16106127359, 32212254719, 64424509439

Offset: 0

Brad Clardy, Oct 07 2011

Primes of this sequence are in A196940.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A033484, A083705, A110286 (first differences), A196940.

Magma
```
[15*2^n -1 : n in [0..50]];
```

a(n)=15<Charles R Greathouse IV, Oct 08 2011

a(n) = 15*2^n - 1.
a(n) = A033484(n+1) + A083705(n).
From Philippe Deléham, Feb 17 2014: (Start)
a(n) = 2*a(n-1) + 1.
a(n) = 3*a(n-2) - 2*a(n-2).
a(n) = A110286(n) - 1. (End)
From Elmo R. Oliveira, Sep 14 2024: (Start)
G.f.: (14 - 13*x)/((1 - x)*(1 - 2*x)).
E.g.f.: exp(x)*(15*exp(x) - 1). (End)

A238275 a(n) = (4*7^n - 1)/3.

Original entry on oeis.org

1, 9, 65, 457, 3201, 22409, 156865, 1098057, 7686401, 53804809, 376633665, 2636435657, 18455049601, 129185347209, 904297430465, 6330082013257, 44310574092801, 310174018649609, 2171218130547265, 15198526913830857, 106389688396816001, 744727818777712009

Offset: 0

Philippe Deléham, Feb 21 2014

Sum of n-th row of triangle of powers of 7: 1; 1 7 1; 1 7 49 7 1; 1 7 49 343 49 7 1; ...

Number of cubes in the crystal structure cubic carbon CCC(n+1), defined in the Baig et al. and in the Gao et al. references. - Emeric Deutsch, May 28 2018

			a(0) = 1;
a(1) = 1 + 7 + 1 = 9;
a(2) = 1 + 7 + 49 + 7 + 1 = 65;
a(3) = 1 + 7 + 49 + 343 + 49 + 7 + 1 = 457; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, Molecular description of carbon graphite and crystal cubic carbon structures, Canadian J. Chem., 95, 674-686, 2017.
W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, Topological characterization of carbon graphite and crystal cubic carbon structures, Molecules, 22, 1496, 1-12, 2017.
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, this sequence, A238276, A138894, A090843, A199023.

Cf. A112468, A112739.

[(4*7^n - 1)/3: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Table[(4 7^n - 1)/3, {n, 0, 40}] (* Vincenzo Librandi, Feb 23 2014 *)

G.f.: (1+x)/((1-x)*(1-7*x)).
a(n) = 7*a(n-1) + 2, a(0) = 1.
a(n) = 8*a(n-1) - 7*a(n-2), a(0) = 1, a(1) = 9.
a(n) = Sum_{k=0..n} A112468(n,k)*8^k.
E.g.f.: exp(x)*(4*exp(6*x) - 1)/3. - Stefano Spezia, Feb 12 2025

A238276 a(n) = (9*8^n - 2)/7.

Original entry on oeis.org

1, 10, 82, 658, 5266, 42130, 337042, 2696338, 21570706, 172565650, 1380525202, 11044201618, 88353612946, 706828903570, 5654631228562, 45237049828498, 361896398627986, 2895171189023890, 23161369512191122, 185290956097528978, 1482327648780231826

Offset: 0

Philippe Deléham, Feb 21 2014

Sum of n-th row of triangle of powers of 8: 1; 1 8 1; 1 8 64 8 1; 1 8 64 512 64 8 1; ...

			a(0) = 1;
a(1) = 1 + 8 + 1 = 10;
a(2) = 1 + 8 + 64 + 8 + 1 = 82;
a(3) = 1 + 8 + 64 + 512 + 64 + 8 + 1 = 658; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9, -8).

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, this sequence, A138894, A090843, A199023.

Cf. A112468, A112739.

[(9*8^n - 2)/7: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Table[(9 8^n - 2)/7, {n, 0, 50}] (* Vincenzo Librandi, Feb 23 2014 *)

G.f.: (1+x)/((1-x)*(1-8*x)).
a(n) = 8*a(n-1) + 2, a(0) = 1.
a(n) = 9*a(n-1) - 8*a(n-2), a(0) = 1, a(1) = 10.
a(n) = Sum_{k=0..n} A112468(n,k)*9^k.

Corrected by Vincenzo Librandi, Feb 23 2014

A382673 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 52, 22, 1, 1, 46, 208, 208, 46, 1, 1, 94, 736, 1372, 736, 94, 1, 1, 190, 2440, 7516, 7516, 2440, 190, 1, 1, 382, 7792, 37012, 60316, 37012, 7792, 382, 1, 1, 766, 24328, 170668, 418996, 418996, 170668, 24328, 766, 1, 1, 1534, 74896, 754132, 2653036, 3964684, 2653036, 754132, 74896, 1534, 1

Offset: 0

Seiichi Manyama, Apr 03 2025

			Square array begins:
  1,  1,    1,     1,      1,       1, ...
  1,  4,   10,    22,     46,      94, ...
  1, 10,   52,   208,    736,    2440, ...
  1, 22,  208,  1372,   7516,   37012, ...
  1, 46,  736,  7516,  60316,  418996, ...
  1, 94, 2440, 37012, 418996, 3964684, ...
  ...

Columns k=0..2 give A000012, A033484, A382675.
Main diagonal gives A382676.
Cf. A099594, A136126, A382674.
Cf. A382735.

a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));

E.g.f.: exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).

A075779 Triangle T(n,k) = f(n,k,n-1), n >= 2, 1 <= k <= n-1, where f is given below.

Original entry on oeis.org

2, 6, 6, 12, 16, 12, 20, 35, 35, 20, 30, 66, 84, 66, 30, 42, 112, 175, 175, 112, 42, 56, 176, 328, 400, 328, 176, 56, 72, 261, 567, 819, 819, 567, 261, 72, 90, 370, 920, 1540, 1820, 1540, 920, 370, 90, 110, 506, 1419, 2706, 3696, 3696, 2706, 1419, 506, 110, 132, 672

Offset: 2

N. J. A. Sloane, Oct 17 2002

Row sums give sequence A033484(n)*(n+2). Essentially same triangle as A051597(n,k)*(n+2). - Philippe Deléham, Oct 01 2003

			2; 6,6; 12,16,12; 20,35,35,20; ...

Michel Lassalle, A new family of positive integers

Cf. A014410 and A007318 for f(n, k, n), A075779 and A075798 for f(n, k, n-1) and A075780 and A075837 for f(n, k, n-2).

Cf. A033484 A051597.

f := proc(n,p,k) convert( binomial(n,k)*hypergeom([1-k,-p,p-n],[1-n,1],1), `StandardFunctions`); end;

t[n_, k_] := n*HypergeometricPFQ[{-k, 2-n, k-n}, {1, 1-n}, 1]; Table[t[n, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

f(n, p, k) = binomial(n, k)*hypergeom([1-k, -p, p-n], [1-n, 1], 1).

A082560 a(1)=1, a(n)=2*a(n-1) if n is odd, or a(n)=a(n/2)+1 if n is even.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 4, 8, 7, 14, 6, 12, 11, 22, 5, 10, 9, 18, 8, 16, 15, 30, 7, 14, 13, 26, 12, 24, 23, 46, 6, 12, 11, 22, 10, 20, 19, 38, 9, 18, 17, 34, 16, 32, 31, 62, 8, 16, 15, 30, 14, 28, 27, 54, 13, 26, 25, 50, 24, 48, 47, 94, 7, 14, 13, 26, 12, 24, 23, 46, 11, 22, 21, 42, 20

Offset: 1

Benoit Cloitre, May 04 2003

b(1)=1, b(n)=2*b(n/2) if n is even, or b(n)=b(n-1)+1 if n is odd produces the sequence of natural numbers.

Seen as a triangle read by rows: T(1,1) = 1; T(n+1,2*k-1) = T(n,k)+1 and T(n+1,2*k) = 2*T(n,k)+2, 1 <= k <= 2^n. - Reinhard Zumkeller, May 13 2015

			.  1:                                 1
.  2:                 2                                4
.  3:        3               6                5                10
.  4:    4       8       7       14       6       12       11       22
.  5:  5  10   9  18   8  16  15   30   7  14  13   26  12   24  23   46

Reinhard Zumkeller, Rows n = 1..13 of triangle, flattened

Cf. A000079 (row lengths), A033484 (right edges), A166060 (row sums), A232642 (duplicates removed).

a082560 n k = a082560_tabf !! (n-1) !! (k-1)
a082560_row n = a082560_tabf !! (n-1)
a082560_tabf = iterate (concatMap (\x -> [x + 1, 2 * x + 2])) [1]
a082560_list = concat a082560_tabf
-- Reinhard Zumkeller, May 13 2015

a(n)=if(n<2,1,if(n%2,2*a(n-1),1+a(n/2)))

if n is in A010737 : a(n)=n-1

A121692 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and vertical height (i.e., number of rows) k (1 <= k <= n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 12, 1, 1, 22, 57, 39, 1, 1, 46, 216, 293, 163, 1, 1, 94, 741, 1651, 1664, 888, 1, 1, 190, 2412, 8181, 12458, 11143, 5934, 1, 1, 382, 7617, 37739, 81255, 102558, 87066, 46261, 1, 1, 766, 23616, 166573, 489753, 823597, 941572, 773772, 409149, 1

Offset: 1

Emeric Deutsch, Aug 17 2006

Row sums are the factorials (A000142).
T(n,1) = 1,
T(n,2) = 3*2^(n-2) - 2 = A033484(n-2) for n >= 2.
T(n,3) = A121693(n).
Sum_{k=1..n} k*T(n,k) = A121694(n).

			T(2,1)=1 and T(2,2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.
Triangle starts:
  1;
  1,  1;
  1,  4,   1;
  1, 10,  12,   1;
  1, 22,  57,  39,   1;
  1, 46, 216, 293, 163, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Elena Barcucci, Sara Brunetti and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Cf. A000142, A033484, A121693, A121694.

T(2n,n) gives A374794.

T:=proc(n,k) option remember; if k=1 then 1 elif k=n then 1 elif k>n then 0 else k*T(n-1,k)+2*T(n-1,k-1)+add(T(n-1,j),j=1..k-2) fi: end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
with(linalg): a:=proc(i,j) if i=j then i elif i>j then 1 else 0 fi end: p:=proc(Q) local n,A,b,w,QQ: n:=degree(Q): A:=matrix(n,n,a): b:=j->coeff(Q,t,j): w:=matrix(n,1,b): QQ:=multiply(A,w): sort(expand(add(QQ[k,1]*t^k,k=1..n)+t*Q)): end: P[1]:=t: for n from 2 to 11 do P[n]:=p(P[n-1]) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

T[n_, k_] := T[n, k] = Which[k == 1, 1, k == n, 1, k > n, 0, True, k*T[n - 1, k] + 2*T[n - 1, k - 1] + Sum[T[n - 1, j], {j, 1, k - 2}]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 20 2024 *)

Rec. relation: T(n,1) = 1; T(n,n) = 1; T(n,k) = k*T(n-1,k) + 2*T(n-1,k-1) + Sum_{j =1..k-2} T(n-1,j) for k <= n; T(n,k) = 0 for k > n.
Rec. relation for the row generating polynomials P[n](t): P[1] = t, P[n] = tP[n-1] + (t+t^2+...+t^(n-1))#P[n-1] for n >= 2. Here # stands for the "max-multiplication" of polynomials, a distributive operation, following the rule t^a # t^b = t^max(a,b).
The second Maple program is based on these polynomials.

cp's OEIS Frontend

A089591 "Lazy binary" representation of n. Also called redundant binary representation of n.

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A123620 Expansion of (1 + x + x^2) / (1 - 3*x - 3*x^2).

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A176449 a(n) = 9*2^n - 2.

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A196305 a(n) = 15*2^n - 1.

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A238275 a(n) = (4*7^n - 1)/3.

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A238276 a(n) = (9*8^n - 2)/7.

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A382673 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

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Keywords

Examples

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).