A058396 -id:A058396 - cp's OEIS frontend

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

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

A176027 Binomial transform of A005563.

Original entry on oeis.org

0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, 479232, 1089536, 2457600, 5505024, 12255232, 27131904, 59768832, 131072000, 286261248, 622854144, 1350565888, 2919235584, 6291456000, 13522436096

Offset: 0

Paul Curtz, Dec 06 2010

The numbers appear on the diagonal of a table T(n,k), where the left column contains the elements of A005563, and further columns are recursively T(n,k) = T(n,k-1)+T(n-1,k-1):
....0....-1.....0.....0.....0.....0.....0.....0.....0.....0.
....3.....3.....2.....2.....2.....2.....2.....2.....2.....2.
....8....11....14....16....18....20....22....24....26....28.
...15....23....34....48....64....82...102...124...148...174.
...24....39....62....96...144...208...290...392...516...664.
...35....59....98...160...256...400...608...898..1290..1806.
...48....83...142...240...400...656..1056..1664..2562..3852.
...63...111...194...336...576...976..1632..2688..4352..6914.
...80...143...254...448...784..1360..2336..3968..6656.11008.
...99...179...322...576..1024..1808..3168..5504..9472.16128.
..120...219...398...720..1296..2320..4128..7296.12800.22272.
The second column is A142463, the third A060626, the fourth essentially A035008 and the fifth essentially A016802. Transposing the array gives A005563 and its higher order differences in the individual rows.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A001792, A001793, A005563, A058396, A084266, A127276.

[2^(n-2)*n*(5+n) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

LinearRecurrence[{6,-12,8},{0,3,14},30] (* Harvey P. Dale, Oct 19 2015 *)

a(n)=n*(n+5)<<(n-2) \\ Charles R Greathouse IV, Sep 21 2017

G.f.: x*(-3+4*x)/(2*x-1)^3. - R. J. Mathar, Dec 11 2010
a(n) = 2^(n-2)*n*(5+n). - R. J. Mathar, Dec 11 2010
a(n) = A127276(n) - A127276(n+1).
a(n+1)-a(n) = A084266(n+1).
a(n+2) = 16*A058396(n) for n > 0.
a(n) = 2*a(n-1) + A001792(n).
a(n) = A001793(n) - 2^(n-1) for n > 0. - Brad Clardy, Mar 02 2012
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+3) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Aug 13 2022: (Start)
Sum_{n>=1} 1/a(n) = 1322/75 - 124*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = 132*log(3/2)/5 - 782/75. (End)

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 6, 3, 4, 3, 1, 0, 6, 6, 6, 4, 1, 10, 6, 9, 10, 9, 5, 1, 0, 10, 12, 15, 16, 13, 6, 1, 15, 10, 16, 21, 25, 25, 18, 7, 1, 0, 15, 20, 28, 36, 41, 38, 24, 8, 1, 21, 15, 25, 36, 49, 61, 66, 56, 31, 9, 1, 0, 21, 30, 45, 64, 85, 102, 104, 80, 39, 10, 1, 28, 21, 36, 55, 81, 113, 146, 168, 160, 111, 48, 11, 1

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n, 0) = T(n, 3) with T(2n, 0) = T(n, m) for some other value of m would change the generating function to the coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^m. This would produce A058393, A058394, A057884 (and effectively A007318).

			The array T(n, k) starts:
[0] 1, 0,  3,   0,   6,   0,  10,    0,   15,    0, ...
[1] 1, 1,  3,   3,   6,   6,  10,   10,   15,   15, ...
[2] 1, 2,  4,   6,   9,  12,  16,   20,   25,   30, ...
[3] 1, 3,  6,  10,  15,  21,  28,   36,   45,   55, ...
[4] 1, 4,  9,  16,  25,  36,  49,   64,   81,  100, ...
[5] 1, 5, 13,  25,  41,  61,  85,  113,  145,  181, ...
[6] 1, 6, 18,  38,  66, 102, 146,  198,  258,  326, ...
[7] 1, 7, 24,  56, 104, 168, 248,  344,  456,  584, ...
[8] 1, 8, 31,  80, 160, 272, 416,  592,  800, 1040, ...
[9] 1, 9, 39, 111, 240, 432, 688, 1008, 1392, 1840, ...

Rows are A000217 with zeros, A008805, A002620, A000217, A000290, A001844, A005899.
Columns are A000012, A001477, A016028.
Diagonals include A058396, A049611, A001793, A001788, A055580, A055581, A055582.
The triangle A055252 also appears in half of the array.

gf := n -> (1 + x)^n / (1 - x^2)^3: ser := n -> series(gf(n), x, 20):
seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..9)), n = 0..9); # Peter Luschny, Apr 12 2023

T[0, k_] := If[OddQ[k], 0, (k+2)(k+4)/8];
T[n_, k_] := T[n, k] = If[k == 0, 1, T[n-1, k-1] + T[n-1, k]];
Table[T[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 13 2023 *)

T(n, k) = T(n-1, k-1) + T(n, k-1) with T(0, k) = 1, T(2*n, 0) = T(n, 3) and T(2*n + 1, 0) = 0. Coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^3.

A169794 Expansion of ((1-x)/(1-2*x))^7.

Original entry on oeis.org

1, 7, 35, 147, 553, 1925, 6321, 19825, 59906, 175504, 500864, 1397536, 3823680, 10282496, 27230464, 71129856, 183518720, 468213760, 1182433280, 2958376960, 7338426368, 18059821056, 44120473600, 107055742976, 258122317824, 618683957248, 1474700509184

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 6 parts equal to 0. - Milan Janjic, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 (14, -84, 280, -560, 672, -448, 128).

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))^7)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^7,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^7, {x, 0, 26}], x] (* Michael De Vlieger, Oct 15 2018 *)

x='x+O('x^30); Vec(((1-x)/(1-2*x))^7) \\ G. C. Greubel, Oct 16 2018

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

For n > 0, a(n) = 2^(n-11)*(n+3)*(n+6)*(n^4 + 54*n^3 + 931*n^2 + 5454*n + 5080)/45. - Bruno Berselli, Aug 07 2011

A169795 Expansion of ((1-x)/(1-2x))^8.

Original entry on oeis.org

1, 8, 44, 200, 806, 2984, 10364, 34232, 108545, 332688, 990736, 2878144, 8182432, 22823680, 62595328, 169090048, 450568960, 1185832960, 3085885440, 7947714560, 20275478528, 51272351744, 128605356032, 320145981440, 791358537728, 1943278714880, 4742573981696

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 7 parts equal to 0. - Milan Janjic, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 (16, -112, 448, -1120, 1792, -1792, 1024, -256).

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

CoefficientList[Series[((1-x)/(1-2x))^8,{x,0,30}],x] (* Harvey P. Dale, Nov 24 2016 *)

G.f.: ((1-x)/(1-2*x))^8.

For n > 0, a(n) = 2^(n-12)*(n+9) * (n^6 + 75*n^5 + 1999*n^4 + 23169*n^3 + 115768*n^2 + 232284*n + 142800)/315. - Bruno Berselli, Aug 07 2011

A169796 Expansion of ((1-x)/(1-2x))^9.

Original entry on oeis.org

1, 9, 54, 264, 1134, 4446, 16272, 56412, 187137, 598417, 1854882, 5597172, 16498632, 47638512, 135048672, 376592064, 1034663040, 2804590080, 7509232640, 19880294400, 52088352768, 135173578752, 347680161792, 886900948992, 2245014454272, 5641949085696

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 8 parts equal to 0. - Milan Janjic, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 (18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512).

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

CoefficientList[Series[((1 - x)/(1 - 2 x))^9, {x, 0, 25}], x] (* Michael De Vlieger, Oct 15 2018 *)

G.f.: ((1-x)/(1-2*x))^9.

For n > 0, a(n) = 2^(n-16)*(n+8)*(n^7 + 100*n^6 + 3778*n^5 + 68056*n^4 + 606961*n^3 + 2543284*n^2 + 4524300*n + 2575440)/315. - Bruno Berselli, Aug 07 2011

A382615 Expansion of 1/(1 - x/(1 - x)^3)^3.

Original entry on oeis.org

1, 3, 15, 64, 261, 1032, 3982, 15066, 56094, 206068, 748452, 2691966, 9600233, 33982197, 119495229, 417724302, 1452550371, 5026878774, 17321417650, 59450099958, 203306331429, 692955932103, 2354664287943, 7978488379398, 26963061909228, 90897971951727

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-57,139,-195,174,-102,39,-9,1).

Cf. A000217, A058396, A290918.
Cf. A052529, A382616.
Cf. A290917.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x/(1 - x)^3)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq;// Vincenzo Librandi, Apr 02 2025

Table[Sum[Binomial[k+2,2]*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Apr 02 2025 *)

a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(n+2*k-1, n-k));

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(n+2*k-1,n-k).

a(n) = 12*a(n-1) - 57*a(n-2) + 139*a(n-3) - 195*a(n-4) + 174*a(n-5) - 102*a(n-6) + 39*a(n-7) - 9*a(n-8) + a(n-9) for n > 9.

A109434 Irregular triangle read by rows: row n contains the numbers from 2^n up to (n+3)*2^(n-2), inclusive, read along their common subdiagonal of A109433.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 5, 8, 11, 12, 16, 24, 27, 28, 32, 51, 60, 63, 64, 64, 107, 131, 140, 143, 144, 128, 222, 282, 307, 316, 319, 320, 256, 457, 601, 666, 691, 700, 703, 704, 512, 935, 1270, 1432, 1498, 1523, 1532, 1535, 1536, 1024, 1904, 2665, 3057, 3224, 3290

Offset: 0

Robert G. Wilson v, Jun 28 2005

Evolution of 2^n into 2^(n-2)(n+3) as exhibited by A109433.

The forward differences of the last row of the table approache A058396.

			Triangle begins
   0  0
   1  1
   2  2
   4  5
   8 11 12
  16 24 27 28
  32 51 60 63 64

Cf. A109433, A109436.

T[n_, m_] := Length[ Select[ StringPosition[ #, ToString[(10^m - 1)/9]] & /@ Table[ ToString[ FromDigits[ IntegerDigits[i, 2]]], {i, 2^n, 2^(n + 1) - 1}], # != {} &]]; Join[{0, 0, 1, 1, 2}, Flatten[ Table[ T[n + i, i], {n, 0, 9}, {i, n + 1}]]]

New definition from R. J. Mathar, Nov 27 2007

A370478 G.f. satisfies A(x) = ( 1 + x * (A(x)^(1/3) / (1-x))^(3/2) )^3.

Original entry on oeis.org

1, 3, 12, 46, 174, 654, 2451, 9177, 34368, 128826, 483531, 1817673, 6844294, 25815660, 97539435, 369154485, 1399419360, 5313440610, 20205330660, 76946898744, 293443125804, 1120565939780, 4284550682478, 16402204879386, 62864294076480, 241205747620740

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A058396, A370480.

Cf. A071724, A370477.

my(N=30, x='x+O('x^N)); Vec((1+x*((1-sqrt(1-4*x))/(2*x))^3)^3)

a(n, r=3, s=3/2, t=3/2, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: B(x)^3 where B(x) is the g.f. of A071724.

a(n) = 3 * Sum_{k=0..n} binomial(3*k/2+3,k) * binomial(n+k/2-1,n-k)/(3*k/2+3).

A370480 G.f. satisfies A(x) = ( 1 + x * (A(x)^(1/3) / (1-x))^2 )^3.

Original entry on oeis.org

1, 3, 15, 73, 360, 1800, 9112, 46632, 240936, 1255336, 6589080, 34811784, 184990568, 988156872, 5303039256, 28579068520, 154605138984, 839272725864, 4570409517848, 24961191298248, 136688674353000, 750355591919240, 4128471397725336, 22762905189252264

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A058396, A370478.

Cf. A006319, A370479.

my(N=30, x='x+O('x^N)); Vec((1+x*((1-x-sqrt(1-6*x+x^2))/(2*x))^2)^3)

a(n, r=3, s=2, t=2, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: B(x)^3 where B(x) is the g.f. of A006319.

a(n) = 3 * Sum_{k=0..n} binomial(2*k+3,k) * binomial(n+k-1,n-k)/(2*k+3).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A176027 Binomial transform of A005563.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A169794 Expansion of ((1-x)/(1-2*x))^7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A169795 Expansion of ((1-x)/(1-2x))^8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A169796 Expansion of ((1-x)/(1-2x))^9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A382615 Expansion of 1/(1 - x/(1 - x)^3)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI