A108411 -id:A108411 - cp's OEIS frontend

A208329 Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.

1, 0, 3, 0, 2, 5, 0, 2, 4, 11, 0, 2, 4, 14, 21, 0, 2, 4, 18, 32, 43, 0, 2, 4, 22, 44, 82, 85, 0, 2, 4, 26, 56, 130, 188, 171, 0, 2, 4, 30, 68, 186, 324, 438, 341, 0, 2, 4, 34, 80, 250, 492, 834, 984, 683, 0, 2, 4, 38, 92, 322, 692, 1374, 2028, 2202, 1365, 0, 2, 4, 42

Offset: 1

Clark Kimberling, Feb 26 2012

Row sums, u(n,1):  A000129
Row sums, v(n,1):  A001333
As triangle T(n,k) with 0 <= k <= n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 27 2012

			First five rows:
  1;
  0,  3;
  0,  2,  5;
  0,  2,  4, 11;
  0,  2,  4, 14, 21;
First five polynomials u(n,x):
  1
     3x
     2x + 5x^2
     2x + 4x^2 + 11x^3
     2x + 4x^2 + 14x^3 + 21x^4.

Cf. A208328.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208328 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208329 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 27 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-(1-2*y)*x)/(1-(1+y)*x+y*(1-2*y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A108411(n+1), A000007(n), A001333(n+1) for x = -1, 0, 1 respectively. (End)

A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.

Original entry on oeis.org

1, 6, 38, 248, 1644, 10984, 73672, 495072, 3329936, 22407776, 150819168, 1015220608, 6834184384, 46006990464, 309717848192, 2085024691712, 14036454256896, 94493999351296, 636137999861248, 4282512012883968

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Fifth binomial transform of A108411. Binomial transform of A162557. Inverse binomial transform of A162757.

2nd binomial transform of A086405. - R. J. Mathar, Jul 17 2009

Index entries for linear recurrences with constant coefficients, signature (10, -22).

Cf. A108411 (powers of 3 repeated), A162557, A162757.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1-4*x)/(1-10*x+22*x^2).
From R. J. Mathar, Jul 17 2009: (Start)
a(n) = 10*a(n-2) - 22*a(n-2).
G.f.: (1-4*x)/(1-10*x+22*x^2). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 13 2009

More terms from R. J. Mathar, Jul 17 2009

A162757 a(n) = 12a(n-1)-33a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 51, 381, 2889, 22095, 169803, 1308501, 10098513, 78001623, 602768547, 4659169005, 36018666009, 278471414943, 2153041001019, 16646935319109, 128712870795681, 995205584017575, 7694942271953427, 59497522990861149

Offset: 0

Klaus Brockhaus, Jul 13 2009

nonn

Sixth binomial transform of A108411. Binomial transform of A162558. Inverse binomial transform of A162758.

Index entries for linear recurrences with constant coefficients, signature (12, -33).

Cf. A108411 (powers of 3 repeated), A162558, A162758.

[ n le 2 select 6*n-5 else 12*Self(n-1)-33*Self(n-2): n in [1..20] ];

a(n) = ((3+sqrt(3))*(6+sqrt(3))^n+(3-sqrt(3))*(6-sqrt(3))^n)/6.

G.f.: (1-5*x)/(1-12*x+33*x^2).

A162758 a(n) = 14a(n-1)-46a(n-2) for n > 1; a(0) = 1, a(1) = 8.

Original entry on oeis.org

1, 8, 66, 556, 4748, 40896, 354136, 3076688, 26783376, 233439616, 2036119328, 17767448256, 155082786496, 1353856391168, 11820181297536, 103205144171776, 901143678718208, 7868574870153216, 68707438961107456, 599949701428456448

Offset: 0

Klaus Brockhaus, Jul 13 2009

nonn

Seventh binomial transform of A108411. Binomial transform of A162757. Inverse binomial transform of A162759.

Index entries for linear recurrences with constant coefficients, signature (14, -46).

Cf. A108411 (powers of 3 repeated), A162757, A162759.

[ n le 2 select 7*n-6 else 14*Self(n-1)-46*Self(n-2): n in [1..20] ];

LinearRecurrence[{14,-46},{1,8},30] (* Harvey P. Dale, Apr 09 2012 *)

a(n) = ((3+sqrt(3))*(7+sqrt(3))^n+(3-sqrt(3))*(7-sqrt(3))^n)/6.

G.f.: (1-6*x)/(1-14*x+46*x^2).

A188440 Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 4, 1, 4, 4, 1, 6, 12, 8, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 14, 84, 280

Offset: 0

Dennis P. Walsh, Mar 31 2011

A subset S of {1,2,...,n} is antisymmetric if x is an element of S implies n+1-x is not an element of S. In other words, the sum of any two elements of S does not equal n+1. For example, {1,2,5} is an antisymmetric subset of {1,2,3,4,5,6,7}. If n is odd, (n+1)/2 cannot be an element of an antisymmetric subset of {1,2,...,n}. (Note that for n=0, we define {1,...,n} to be the empty set, and thus T(0,0)=1 since the empty set is vacuously antisymmetric.)
We note, for example, that T(100,k) provides the number of possible size-k committees of the U.S. Senate in which no two members are from the same state.
Triangle, read by rows, A013609 rows repeated. - Philippe Deléham, Apr 09 2012
Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012

			Triangle T(n,k) initial values 0 <= k <= floor(n/2), n=0..13:
  1
  1
  1   2
  1   2
  1   4   4
  1   4   4
  1   6  12   8
  1   6  12   8
  1   8  24  32  16
  1   8  24  32  16
  1  10  40  80  80  32
  1  10  40  80  80  32
  1  12  60 160 240 192  64
  1  12  60 160 240 192  64
  ...
For n=7 and k=2, T(7,2)=12 since there are 12 antisymmetric size-2 subsets of {1,2,...,7}:
  {1,2}, {1,3}, {1,5}, {1,6}, {2,3}, {2,5},
  {2,7}, {3,6}, {3,7}, {5,6}, {5,7}, and {6,7}.
(1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins:
  1
  1   0
  1   2   0
  1   2   0   0
  1   4   4   0   0
  1   4   4   0   0   0
  1   6  12   8   0   0   0
  1   6  12   8   0   0   0   0
  1   8  24  32  16   0   0   0   0
  1   8  24  32  16   0   0   0   0   0
  1  10  40  80  80  32   0   0   0   0   0
  1  10  40  80  80  32   0   0   0   0   0   0
  1  12  60 160 240 192  64   0   0   0   0   0   0
  1  12  60 160 240 192  64   0   0   0   0   0   0   0
- _Philippe Deléham_, Apr 09 2012

T. D. Noe, Rows n = 0..100, flattened
Dennis Walsh, Notes on antisymmetric subsets of {1,2,...,n}

Cf. A108411, row sums of triangle T(n,k).

Cf. A000079, A007318, A013109, A152198

seq(seq(binomial(floor(n/2),k)*2^k,k=0..floor(n/2)),n=0..22);

Table[ CoefficientList[(1 + 2*x)^n, x] , {n, 0, 7}, {2}] // Flatten (* Jean-François Alcover, Aug 19 2013, after Philippe Deléham *)

T(n,k) = 2^k*C(floor(n/2),k) where C(*,*) denotes a binomial coefficient.
Sum(T(n,k),k=0..floor(n/2)) = 3^floor(n/2) = A108411(n).
G.f. for columns(k fixed):(2t^2)^k/((1-t)*(1-t^2)^k).
T(n,k) = A152198(n,k)*2^k. - Philippe Deléham, Apr 09 2012
G.f.: (1+x)/(1-x^2-2*y*x^2). - Philippe Deléham, Apr 09 2012
T(n,k) = T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n.- Philippe Deléham, Apr 09 2012

A122007 Expansion of 2x^2(1-2x) / ((3x-1)(3x^2-1)).

Original entry on oeis.org

0, 2, 2, 12, 24, 90, 234, 756, 2160, 6642, 19602, 59292, 176904, 532170, 1593594, 4785156, 14346720, 43053282, 129133602, 387440172, 1162241784, 3486843450, 10460294154, 31381236756, 94143001680, 282430067922, 847288078002, 2541867422652, 7625595890664

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 11 2006

"Linear Algebra, Examples and Applications" by Alain M. Robert, World Scientific, 2005, p. 58.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

M = {{0, 1, 2}, {1, 2, 0}, {2, 0, 1}} v[1] = {1, 0, 0} v[n_] := v[n] = M.v[n - 1] a1 = Table[v[n][[3]], {n, 1, 50}]

concat(0, Vec(2*x^2*(1-2*x)/((3*x-1)*(3*x^2-1)) + O(x^40))) \\ Colin Barker, Sep 23 2016

Limit a(n+1)/a(n)= 3 as n-> infinity.
a(n)= 3*a(n-1) +3*a(n-2) -9*a(n-3) = 3^(n-2) + (-1)^n*A108411(n-2), n>=2.
From Colin Barker, Sep 23 2016: (Start)
a(n) = 3^(n/2-1)+3^(n-2) for n>1 and even.
a(n) = 3^(n-2)-3^((n-3)/2) for n>1 and odd.
(End)

Definition replaced with generating function by the Assoc. Eds. of the OEIS, Mar 27 2010

A-number in formula corrected - R. J. Mathar, Mar 30 2010

A136201 a(n) = 2a(n-1) + 4a(n-2) - 6a(n-3) - 3a(n-4).

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 18, 53, 124, 328, 780, 1969, 4718, 11648, 28014, 68405, 164824, 400240, 965304, 2337409, 5640122, 13637336, 32914794, 79525973, 191966740, 463636600, 1119239940, 2702647921, 6524535782, 15753313808, 38031163398

Offset: 0

Paul Curtz, Mar 16 2008

Based on a Pell recurrence.

Index entries for linear recurrences with constant coefficients, signature (2,4,-6,-3).
G. C. Greubel, Table of n, a(n) for n = 0..1000

a:=proc(n) options operator, arrow: expand((1/8)*(1+sqrt(2))^n+(1/8)*(1-sqrt(2))^n+(1/24)*3^((1/2)*n)*(-3-sqrt(3)-3*(-1)^n+(-1)^n*sqrt(3))) end proc: seq(a(n),n=0..30); # Emeric Deutsch, Mar 31 2008

LinearRecurrence[{2, 4, -6, -3}, {0, 0, 0, 1}, 50] (* G. C. Greubel, Feb 23 2017 *)
CoefficientList[Series[x^3/(1-2 x-4 x^2+6 x^3+3 x^4),{x,0,50}],x] (* Harvey P. Dale, Apr 21 2022 *)

x='x+O('x^50); Vec(x^3/(3*x^4 + 6*x^3 - 4*x^2 - 2*x + 1)) \\ G. C. Greubel, Feb 23 2017

A137255(n+1) - 2*A137255(n), same recurrence.
a(n) = (-A108411(n) + A001333(n))/4. - R. J. Mathar, Apr 01 2008
a(n) = (1/8)*(1+sqrt(2))^n + (1/8)*(1-sqrt(2))^n + (1/24)*3^(n/2)*(-3 - sqrt(3) - 3(-1)^n + (-1)^n*sqrt(3)). - Emeric Deutsch, Mar 31 2008
G.f.: x^3/(3*x^4 + 6*x^3 - 4*x^2 - 2*x + 1). - Alexander R. Povolotsky, Mar 31 2008

A141516 The main diagonal of the array of A141425 and its higher order differences.

Original entry on oeis.org

1, 2, 1, -7, -23, -1, 7, -103, -251, -133, -149, -1387, -3143, -3001, -4913, -19663, -42611, -55693, -101549, -291667, -612863, -960001, -1831433, -4460023, -9185771, -15980053, -31162949, -69500347, -141392183, -261261001

Offset: 0

Paul Curtz, Aug 11 2008

The sequence A141425 and higher order differences in subsequent rows starts (see A141533):
1, 2, 4,  5,  7, 8, 1, 2, 4,  5,  7, 8, 1, 2, 4,...
1, 2, 1,  2,  1,-7, 1, 2, 1,  2,  1,-7, 1, 2, 1, 2,...
1,-1, 1, -1, -8, 8, 1,-1, 1, -1, -8, 8, 1, -1,..
-2, 2,-2, -7, 16,-7,-2, 2,-2, -7, 16,-7, -2,..
4,-4,-5, 23,-23, 5, 4,-4,-5, 23,-23, 5, 4,..
-8,-1,28,-46, 28,-1,-8,-1,28,-46, 28,-1,..
Reading downwards the main diagonal of this array defines the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,6)

A108411 := proc(n) 3^floor(n/2) ; end proc:
A141516 := proc(n) if n = 0 then 1; else (-3*(-1)^n-2^n+3*(-1)^(floor((n-1)/2))*A108411(n))/2 ; end if; end proc: # R. J. Mathar, Mar 08 2011

LinearRecurrence[{1,-1,3,6},{1,2,1,-7,-23},30] (* Harvey P. Dale, Nov 23 2022 *)

a(n) = ( -3*(-1)^n -2^n +3*(-1)^(floor((n-1)/2))*A108411(n) )/2, n>0. - R. J. Mathar, Mar 08 2011
a(2n)+a(2n+1)= -A002023(n-1) = -3*A081294(n), n>0.
a(4n)+a(4n+1)+a(4n+2)+a(4n+3) = -120*16^(n-1), n>0.
a(4n+2)+a(4n+3)+a(4n+4)+a(4n+5) = -30*A001025(n).
G.f. x*(-2+x+6*x^2+21*x^3) / ( (2*x-1)*(1+x)*(3*x^2+1) ). - R. J. Mathar, Mar 08 2011

A155084 A Catalan transform of [x^n](1/(1-2x-2x^2)) (A002605).

Original entry on oeis.org

1, 2, 8, 32, 132, 552, 2328, 9872, 42020, 179336, 766888, 3284272, 14081224, 60426576, 259490736, 1114965792, 4792924356, 20611174920, 88662405768, 381494338032, 1641837542232, 7067257125744, 30425523536592

Offset: 0

Paul Barry, Jan 19 2009

Hankel transform is 4^n.

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [_N. J. A. Sloane_, Oct 08 2012]

Cf. A000108, A002605, A101850, A039599, A108411.

G.f.: 1/(1-2x*c(x)-2(x*c(x))^2), where c(x) is the g.f. of A000108.
G.f.: 1/(1-2x-4x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction).
a(n) = Sum_{k=0..n} (k/(2n-k))*binomial(2n-k, n-k)*A002605(k), a(0) = 1.
a(n) = Sum_{0<=k<=n} A039599(n,k)*A108411(k). [Philippe Deléham, Nov 15 2009]
Apparently 3*n*a(n) +6*(3-4*n)*a(n-1) +4*(11*n-18)*a(n-2) +8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Oct 25 2012

A162759 a(n) = 16a(n-1)-61a(n-2) for n > 1; a(0) = 1, a(1) = 9.

Original entry on oeis.org

1, 9, 83, 779, 7401, 70897, 682891, 6601539, 63968273, 620798489, 6030711171, 58622670907, 570089353081, 5545446723969, 53951697045563, 524954902566899, 5108224921291041, 49709349684075817, 483747874746459571

Offset: 0

Klaus Brockhaus, Jul 13 2009

nonn

Eighth binomial transform of A108411. Binomial transform of A162758.

Index entries for linear recurrences with constant coefficients, signature (16, -61).

Cf. A108411 (powers of 3 repeated), A162758.

[ n le 2 select 8*n-7 else 16*Self(n-1)-61*Self(n-2): n in [1..19] ];

LinearRecurrence[{16,-61},{1,9},30] (* Harvey P. Dale, Jan 18 2014 *)

a(n) = ((3+sqrt(3))*(8+sqrt(3))^n+(3-sqrt(3))*(8-sqrt(3))^n)/6.

G.f.: (1-7*x)/(1-16*x+61*x^2).

cp's OEIS Frontend

A208329 Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.

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A162558 a(n) = ((3+sqrt(3))*(5+sqrt(3))^n + (3-sqrt(3))*(5-sqrt(3))^n)/6.

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A162757 a(n) = 12*a(n-1)-33*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

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A162758 a(n) = 14*a(n-1)-46*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

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A188440 Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.

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A122007 Expansion of 2*x^2*(1-2*x) / ((3*x-1)*(3*x^2-1)).

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A136201 a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 3*a(n-4).

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A141516 The main diagonal of the array of A141425 and its higher order differences.

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A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.

A162757 a(n) = 12a(n-1)-33a(n-2) for n > 1; a(0) = 1, a(1) = 7.

A162758 a(n) = 14a(n-1)-46a(n-2) for n > 1; a(0) = 1, a(1) = 8.

A122007 Expansion of 2x^2(1-2x) / ((3x-1)(3x^2-1)).

A136201 a(n) = 2a(n-1) + 4a(n-2) - 6a(n-3) - 3a(n-4).

A162759 a(n) = 16a(n-1)-61a(n-2) for n > 1; a(0) = 1, a(1) = 9.