A291000 -id:A291000 - cp's OEIS frontend

A058396 Expansion of ((1-x)/(1-2*x))^3.

1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, 8192000, 17891328, 38928384, 84410368, 182452224, 393216000, 845152256, 1811939328, 3875536896, 8271167488, 17616076800

Offset: 0

Henry Bottomley, Nov 24 2000

If X_1,X_2,...,X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
Equals row sums of triangle A152230. - Gary W. Adamson, Nov 29 2008
a(n) is the number of weak compositions of n with exactly 2 parts equal to 0. - Milan Janjic, Jun 27 2010
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^3; see A291000. - Clark Kimberling, Aug 24 2017

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A045623, A001793, A152230. A diagonal of A058395.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^3)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^3,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

CoefficientList[ Series[(1 - x)^3/(1 - 2x)^3, {x, 0, 28}], x] (* Robert G. Wilson v, Jun 28 2005 *)
Join[{1},LinearRecurrence[{6,-12,8},{3,9,25},40]] (* Harvey P. Dale, Oct 17 2011 *)

Vec((1-x)^3/(1-2*x)^3+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = (n+2)*(n+7)*2^(n-4) for n > 0.
a(n) = Sum_{k=0..floor((n+2)/2)} C(n+2, 2k)*k(k+1)/2. - Paul Barry, May 15 2003
Binomial transform of quarter squares A002620 (without leading zeros). - Paul Barry, May 27 2003
a(n) = Sum_{k=0..n} C(n, k)*floor((k+2)^2/4). - Paul Barry, May 27 2003
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 3. - Harvey P. Dale, Oct 17 2011
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 145189/525 - 1984*log(2)/5.
Sum_{n>=0} (-1)^n/a(n) = 30103/175 - 2112*log(3/2)/5. (End)
E.g.f.: (1 + exp(2*x)*(7 + 10*x + 2*x^2))/8. - Stefano Spezia, Feb 01 2025

A034007 First differences of A045891.

Original entry on oeis.org

1, 0, 2, 4, 9, 20, 44, 96, 208, 448, 960, 2048, 4352, 9216, 19456, 40960, 86016, 180224, 376832, 786432, 1638400, 3407872, 7077888, 14680064, 30408704, 62914560, 130023424, 268435456, 553648128, 1140850688, 2348810240

Offset: 0

Wolfdieter Lang

Let M_n be the n X n matrix m_(i,j) = 4 + abs(i-j) then det(M_n) = (-1)^(n+1)*a(n+2). - Benoit Cloitre, May 28 2002
Number of ordered pairs of (possibly empty) ordered partitions, each not beginning with 1. - Christian G. Bower, Jan 23 2004
If X_1, X_2, ..., X_n are 2-blocks of a (2n+4)-set X then, for n>=1, a(n+3) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S^2)^2; see A291000. - Clark Kimberling, Aug 24 2017
Conjecture 1: For compositions of n+k-1, a(n) is the number of runs of 1 of length k. Example: Among the compositions of 4+2-1 = 5, there are a(4) = 4 runs of two 1's: 3,[1,1]; [1,1],3; 1,2,[1,1] and [1,1],2,1. - Gregory L. Simay, Feb 18 2018
Conjecture 2: More generally, let R(n,m,k) = the number of runs of k m's in all compositions of n. Then R(n,m,k) = A045623(n-m*k) - 2*A045623(n-m*(k+1)) + A045623(n-m*(k+2)). For example, R(7,1,1) = A045623(6) - 2*A045623(5) + A045623(4) = 144 - 2*64 + 28 = 44 = a(7). - Gregory L. Simay, Feb 20 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Frank Ellermann, Illustration of binomial transforms.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A045623, A045891, A291000.
Convolution of A034008 with itself.
Columns of A091613 converge to this sequence.

[1,0,2] cat [(n+5)*2^(n-4): n in [3..30]]; // G. C. Greubel, Sep 27 2022

Join[{1,0,2,a=4},Table[a=(2*(n+7)*a)/(n+6),{n,2,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)

a(n)=if(n<3,[1,0,2][n+1],(n+5)*2^(n-4)) \\ Charles R Greathouse IV, Jun 01 2011

[1,0,2]+[(n+5)*2^(n-4) for n in range(3,30)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k = 0..n-3} (k+4)*binomial(n-3,k) for n >= 3. - N. J. A. Sloane, Jan 30 2008
a(n) = (n+5)*2^(n-4), n >= 3; a(0)=1, a(1)=0, a(2)=2.
G.f.: ((1-x)^2/(1-2*x))^2.
a(n) = Sum_{k=0..n} (k+1)*C(n-3,n-k). - Peter Luschny, Apr 20 2015
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=2} 1/a(n) = 512*log(2) - 74327/210.
Sum_{n>=2} (-1)^n/a(n) = 14579/70 - 512*log(3/2). (End)
E.g.f.: (1/16)*(11 - 12*x + 2*x^2 + (5+2*x)*exp(2*x)). - G. C. Greubel, Sep 27 2022

A139761 a(n) = Sum_{k >= 0} binomial(n,5*k+4).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3004, 6385, 13380, 27370, 54740, 107883, 211585, 416405, 826045, 1652090, 3321891, 6690150, 13455325, 26985675, 53971350, 107746282, 214978335, 429124630, 857417220, 1714834440, 3431847189

Offset: 0

N. J. A. Sloane, Jun 13 2008

Sequence is identical to its fifth differences. - Paul Curtz, Jun 18 2008
{A139398, A133476, A139714, A139748, A139761} is the difference analog of the hyperbolic functions of order 5, {h_1(x), h_2(x), h_3(x), h_4(x), h_5 (x)}. For a definition see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jun 28 2017
This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^5; see A291000.  - Clark Kimberling, Aug 24 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2).

Cf. A049016, A133476, A139398, A139714, A139748, A139761.

I:=[0,0,0,0,1]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+2*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 21 2015

a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[2, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2015

CoefficientList[Series[x^4/((1-2x)(x^4-2x^3+4x^2-3x+1)), {x,0,40}], x] (* Vincenzo Librandi, Dec 21 2015 *)
LinearRecurrence[{5,-10,10,-5,2}, {0,0,0,0,1}, 35] (* Jean-François Alcover, Feb 14 2018 *)

a(n) = sum(k=0, n\5, binomial(n,5*k+4)); \\ Michel Marcus, Dec 21 2015

my(x='x+O('x^100)); concat([0,0,0,0], Vec(-x^4/((2*x-1)*(x^4-2*x^3 +4*x^2-3*x+1)))) \\ Altug Alkan, Dec 21 2015

def A139761(n): return sum(binomial(n,5*k+4) for k in range(1+n//5))
[A139761(n) for n in range(41)] # G. C. Greubel, Jan 23 2023

a(n) = A049016(n-4). - R. J. Mathar, Nov 08 2010
From Paul Curtz, Jun 18 2008: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5).
a(n) = A139398(n+1) - A139398(n). (End)
G.f.: x^4/((1-2*x)*(1-3*x+4*x^2-2*x^3+x^4)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
a(n) = round((2/5)*(2^(n-1) + phi^n*cos(Pi*(n-8)/5))), where phi is the golden ratio, round(x) is the integer nearest to x. - Vladimir Shevelev, Jun 28 2017
a(n+m) = a(n)*H_1(m) + H_4(n)*H_2(m) + H_3(n)*H_3(m) + H_2(n)*H_4(m) + H_1(n)*a(m), where H_1=A139398, H_2=A133476, H_3=A139714, H_4=A139748. - Vladimir Shevelev, Jun 28 2017

A062109 Expansion of ((1-x)/(1-2*x))^4.

Original entry on oeis.org

1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, 15859712, 36175872, 82051072, 185139200, 415760384, 929562624, 2069889024, 4591714304, 10150215680, 22364028928, 49123688448, 107592286208, 235015241728, 512040632320

Offset: 0

Henry Bottomley, May 30 2001

If X_1,X_2,...,X_n are 2-blocks of a (2n+4)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007
If the offset here is set to zero, the binomial transform of A006918. - R. J. Mathar, Jun 29 2009
a(n) is the number of weak compositions of n with exactly 3 parts equal to 0. - Milan Janjic, Jun 27 2010
Binomial transform of A002623. - Carl Najafi, Jan 22 2013
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^4; see A291000. - Clark Kimberling, Aug 24 2017

Harry J. Smith, Table of n, a(n) for n = 0..200
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^4, x,n+1),x,n),n=0..30); # Muniru A Asiru, Jul 01 2018

CoefficientList[Series[(1 - x)^4/(1 - 2 x)^4, {x, 0, 26}], x] (* Michael De Vlieger, Jul 01 2018 *)
LinearRecurrence[{8,-24,32,-16},{1,4,14,44,129},30] (* Harvey P. Dale, Sep 02 2022 *)

a(n)=if(n<1,n==0,(n+5)*(n^2+13*n+18)*2^n/96)

a(n) = (n+5)*(n^2 + 13*n + 18)*2^(n-5)/3, with a(0)=1.
a(n) = A055809(n-5)*2^(n-4).
a(n) = 2*a(n-1) + A058396(n) - A058396(n-1).
a(n) = Sum_{kA058396(n).
a(n) = A062110(4, n).
G.f.: (1-x)^4/(1-2*x)^4.

A169792 Expansion of ((1-x)/(1-2x))^5.

Original entry on oeis.org

1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, 23336960, 55623680, 131563520, 309002240, 721092608, 1672806400, 3859415040, 8859156480, 20240138240, 46038777856, 104291368960, 235342397440, 529153392640

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 4 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^5; see A291000. - Clark Kimberling, Aug 24 2017

Muniru A Asiru, Table of n, a(n) for n = 0..500
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

Concatenation([1],List([1..30],n->2^n*(n+4)*(n^3+26*n^2+171*n+186)/768)); # Muniru A Asiru, Aug 22 2018

seq(coeff(series(((1-x)/(1-2*x))^5, x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 22 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^5, {x, 0, 28}], x] (* Michael De Vlieger, Oct 15 2018 *)

Vec(((1-x)/(1-2*x))^5+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5), n >= 6. - Vincenzo Librandi, Mar 14 2011

a(n) = 2^n*(n+4)*(n^3 + 26*n^2 + 171*n + 186)/768, n > 0. - R. J. Mathar, Mar 14 2011

A290995 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 8, 36, 120, 330, 792, 1716, 3432, 6436, 11456, 19584, 32640, 54264, 93024, 170544, 341088, 735472, 1653632, 3749760, 8386560, 18289440, 38724480, 79594560, 159189120, 311058496, 597137408, 1133991936, 2147450880, 4089171840

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A291000 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8).

Cf. A000012, A033453, A289780, A291000.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), this sequence (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0,0] cat Coefficients(R!( x^7/((1-x)^8 - x^8) )); // G. C. Greubel, Apr 11 2023

z = 60; s = x/(1 - x); p = 1 - s^8;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290995 *)

concat(vector(7), Vec(x^7 / ((1 - 2*x)*(1 - 2*x + 2*x^2)*(1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4)) + O(x^50))) \\ Colin Barker, Aug 22 2017

def A290995_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^7/((1-x)^8 - x^8) ).list()
A290995_list(60) # G. C. Greubel, Apr 11 2023

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) for n >= 9.
G.f.: x^7 / ((1 - 2*x)*(1 - 2*x + 2*x^2)*(1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4)). - Colin Barker, Aug 22 2017
G.f.: x^7/((1-x)^8 - x^8). - G. C. Greubel, Apr 11 2023

A120926 Number of isolated 0's in all ternary words of length n on {0,1,2}.

Original entry on oeis.org

1, 4, 16, 60, 216, 756, 2592, 8748, 29160, 96228, 314928, 1023516, 3306744, 10628820, 34012224, 108413964, 344373768, 1090516932, 3443737680, 10847773692, 34093003032, 106928054964, 334731302496, 1046035320300, 3263630199336, 10167463313316, 31632108085872

Offset: 1

Emeric Deutsch, Jul 16 2006

nonn

This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - 2 S); see A291000. - Clark Kimberling, Aug 24 2017

			a(2) = 4 because in the 9 ternary words of length 2, namely 00, 01, 02, 10, 11, 12, 20, 21 and 22, we have altogether 4 isolated 0's.

Index entries for linear recurrences with constant coefficients, signature (6, -9).

Cf. A120924.

Maple
```
1,seq(4*(n+1)*3^n/27,n=2..28);
```

a(n) = (4/27)*(n+1)*3^n for n >= 2.
G.f.: z*(1-z)^2/(1-3*z)^2.
a(n) = Sum_{k=0..ceiling(n/2)} k*A120924(n,k).

A290998 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^4.

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 43, 92, 205, 462, 1035, 2301, 5099, 11303, 25088, 55728, 123800, 274969, 610628, 1355970, 3011157, 6686979, 14850196, 32978725, 73237462, 162641499, 361184653, 802098203, 1781254927, 3955712256, 8784625824, 19508406192, 43323176177

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
See A291000 for a guide to related sequences.
For n >= 1, a(n-1) is the number of ways to split [n] into an unspecified number of intervals and then choose 3 blocks (i.e., subintervals) from each interval. For example, for n=9, a(8)=205 since the number of ways to split [9] into intervals and then select 3 blocks from each interval is C(9,3) + C(6,3)*C(3,3) + C(5,3)*C(4,3) + C(4,3)*C(5,3) + C(3,3)*C(6,3) + C(3,3)*C(3,3)*C(3,3) for a total of 205 ways. - Enrique Navarrete, Dec 23 2023
a(n-1) is also the number of compositions of n using parts of size at least 3 where there are binomial(i,3) types of i, n>=1, i>=3 (see example). - Enrique Navarrete, Dec 25 2023

			From _Enrique Navarrete_, Dec 25 2023: (Start)
Since there are binomial(3,3) = 1 type of 3, binomial(4,3) = 4 types of 4, binomial(5,3) = 10 types of 5, binomial(6,3) = 20 types of 6, and binomial(9,3) = 84 types of 9, we can write 9 in the following ways:
 9 in 84 ways;
 6+3 in 20 ways;
 5+4 in 40 ways;
 4+5 in 40 ways;
 3+6 in 20 ways;
 3+3+3 in 1 way, for a total of 205 ways. (End)

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-1).

Cf. A000012, A033453, A289780, A290997, A291000.

I:=[0,0,1,4]; [n le 4 select I[n] else 4*Self(n-1) -6*Self(n-2) +5*Self(n-3) -Self(n-4): n in [1..41]]; // G. C. Greubel, Apr 25 2023

z = 60; s = x/(1 - x); p = 1 - s^3 - s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* this sequence *)
LinearRecurrence[{4,-6,5,-1}, {0,0,1,4}, 41] (* G. C. Greubel, Apr 25 2023 *)

concat(vector(2), Vec(x^2 / (1 - 4*x + 6*x^2 - 5*x^3 + x^4) + O(x^50))) \\ Colin Barker, Aug 22 2017

def A290998_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2/(1-4*x+6*x^2-5*x^3+x^4) ).list()
A290998_list(40) # G. C. Greubel, Apr 25 2023

a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - a(n-4) for n >= 5.
G.f.: x^2 / (1 - 4*x + 6*x^2 - 5*x^3 + x^4). - Colin Barker, Aug 22 2017
G.f.: 1/(x*(1-Sum_{k>=3} binomial(k,3)*x^k)) - 1/x. - Enrique Navarrete, Dec 26 2023

A169793 Expansion of ((1-x)/(1-2*x))^6.

Original entry on oeis.org

1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, 68889600, 170065920, 416071680, 1009582080, 2431254528, 5814222848, 13815054336, 32629850112, 76640681984, 179080003584, 416412598272, 963876225024

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 5 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^6; see A291000. - Clark Kimberling, Aug 24 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. for ((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^6)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^6,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^6, {x, 0, 27}], x] (* Michael De Vlieger, Oct 15 2018 *)

x='x+O('x^30); Vec(((1-x)/(1-2*x))^6) \\ G. C. Greubel, Oct 16 2018

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

For n > 0, a(n) = 2^(n-9)*(n+7)*(n^4 + 38*n^3 + 419*n^2 + 1342*n + 1080)/15. - Bruno Berselli, Aug 07 2011

A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2Pell(n+1)+2Pell(n)-2^n, with Pell = A000129.

Original entry on oeis.org

1, 4, 10, 26, 66, 166, 414, 1026, 2530, 6214, 15214, 37154, 90546, 220294, 535230, 1298946, 3149506, 7630726, 18476494, 44714786, 108168210, 261575494, 632367774, 1528408194, 3693378466, 8923553734, 21557263150, 52071634466

Offset: 0

Johannes W. Meijer, Aug 06 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
The sequence above corresponds to 24 A[5] vectors with decimal values 23, 29, 53, 83, 86, 89, 92, 113, 116, 149, 209, 212, 275, 278, 281, 284, 305, 308, 338, 344, 368, 401, 404 and 464. These vectors lead for the side squares to A000079 and for the corner squares to 2*A094723 (a(n)=2*Pell(n+1)-2^n).
From Clark Kimberling, Aug 23 2017 (Start)
p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3.
Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453). See A291000 for a guide to related sequences. (End)

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

Cf. A175654, A175655 (central square).

Cf. A000129 (Pell(n)), A078057 (Pell(n)+Pell(n+1)), A094723 (Pell(n+2)-2^n).

I:=[1,4,10]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013

nmax:=27; m:=5; A[5]:= [0,0,0,0,1,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{4,-3,-2},{1,4,10},30] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[Series[(1 - 3 x^2) / (1 - 4 x + 3 x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)

Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017

G.f.: ( 1-3*x^2 ) / ( (2*x-1)*(x^2+2*x-1) ).
a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=10.
Limit_{n->oo} a(n+1)/a(n) = 1+sqrt(2).
a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - Colin Barker, Aug 29 2017

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A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2Pell(n+1)+2Pell(n)-2^n, with Pell = A000129.