A080419 -id:A080419 - cp's OEIS frontend

A136158 Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

1, 1, 1, 3, 4, 1, 9, 15, 7, 1, 27, 54, 36, 10, 1, 81, 189, 162, 66, 13, 1, 243, 648, 675, 360, 105, 16, 1, 729, 2187, 2673, 1755, 675, 153, 19, 1, 2187, 7290, 10206, 7938, 3780, 1134, 210, 22, 1, 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1

Offset: 0

Gary W. Adamson, Dec 16 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 17 2007

Equals A080419 when first column is removed (here). - Georg Fischer, Jul 25 2023

			First few rows of the triangle:
    1;
    1,    1;
    3,    4,    1;
    9,   15,    7,    1;
   27,   54,   36,   10,   1;
   81,  189,  162,   66,  13,   1;
  243,  648,  675,  360, 105,  16,  1;
  729, 2187, 2673, 1755, 675, 153, 19, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A000012, A001519, A003688, A006234, A016777, A062741.
Cf. A080419, A080421, A080422, A080423, A081294, A133494, A136157.
Absolute value of A164948.

A136158:= func< n,k | n eq 0 select 1 else 3^(n-k-1)*(n+2*k)* Binomial(n, k)/n >;
[A136158(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2023; Dec 27 2023

A136158[n_,k_]:= If[n==0, 1, 3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A136158[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2023; Dec 27 2023 *)

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); 3*T(n-1,k) + T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 25 2023

def A136158(n,k): return 1 if (n==0) else 3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A136158(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 22 2023; Dec 27 2023

Sum_{k=0..n} T(n, k) = A081294(n).
Given A136157 = M, an infinite lower triangular bidiagonal matrix with (3, 3, 3, ...) in the main diagonal, (1, 1, 1, ...) in the subdiagonal and the rest zeros; rows of A136157 are generated from M^n * [1, 1, 0, 0, 0, ...], given a(0) = 1.
T(n, k) = A038763(n,n-k). - Philippe Deléham, Dec 17 2007
T(n, k) = 3*T(n-1, k) + T(n-1, k-1) for n > 1, T(0,0) = T(1,1) = T(1,0) = 1. - Philippe Deléham, Oct 30 2013
Sum_{k=0..n} T(n, k)*x^k = (1+x)*(3+x)^(n-1), n >= 1. - Philippe Deléham, Oct 30 2013
G.f.: (1-2*x)/(1-3*x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 22 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = A006234(n+2).
T(n, 2) = A080420(n-2).
T(n, 3) = A080421(n-3).
T(n, 4) = A080422(n-4).
T(n, 5) = A080423(n-5).
T(n, n) = A000012(n).
T(n, n-1) = A016777(n-1).
T(n, n-2) = A062741(n-1).
Sum_{k=0..n} (-1)^k * T(n, k) = 0^n = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A003688(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001519(n). (End)
From G. C. Greubel, Dec 27 2023: (Start)
T(n, k) = 3^(n-k-1)*(n+2*k)*binomial(n,k)/n, for n > 0, with T(0, 0) = 1.
T(n, k) = (-1)^k * A164948(n, k). (End)

More terms from Philippe Deléham, Dec 17 2007

A080420 a(n) = (n+1)(n+6)3^n/6.

Original entry on oeis.org

1, 7, 36, 162, 675, 2673, 10206, 37908, 137781, 492075, 1732104, 6022998, 20726199, 70681653, 239148450, 803538792, 2683245609, 8910671247, 29443957164, 96855122250, 317297380491, 1035574967097, 3368233731366, 10920608743932, 35303692060125, 113819103201843

Offset: 0

Paul Barry, Feb 19 2003

a(n-1) is the number of words of length n defined on 5 letters that have exactly one a and no b's or exactly two b's and no a's. For example, for n=3, a(2) = 36 since the words are (number of permutations in parentheses): acc (3), add (3), aee (3), acd (6), ace (6), ade (6), bbc (3), bbd (3), bbe (3). - Enrique Navarrete, Jun 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

T(n,2) in triangle A080419.

Cf. A136158, A288834.

[(n+1)*(n+6)*3^n/6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)
Table[(n+1)(n+6)3^n/6,{n,0,30}] (* or *) LinearRecurrence[{9,-27,27},{1,7,36},30] (* Harvey P. Dale, Apr 02 2019 *)

[(n+1)*(n+6)*3^n/6 for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: (1-2*x)/(1-3*x)^3.
From G. C. Greubel, Dec 22 2023: (Start)
a(n) = (n+6)*A288834(n)/2, for n >= 1.
a(n) = A136158(n+2, 2).
E.g.f.: (1/2)*(2 + 8*x + 3*x^2)*exp(3*x). (End)
From Amiram Eldar, Jan 11 2024: (Start)
Sum_{n>=0} 1/a(n) = 17721/50 - 4356*log(3/2)/5.
Sum_{n>=0} (-1)^n/a(n) = 4392*log(4/3)/5 - 12591/50. (End)

A080421 a(n) = (n+1)(n+2)(n+9)*3^n/18.

Original entry on oeis.org

1, 10, 66, 360, 1755, 7938, 34020, 139968, 557685, 2165130, 8227494, 30705480, 112842639, 409209570, 1466777160, 5203870272, 18294856425, 63795240522, 220829678730, 759344158440, 2595329855811, 8821564534530, 29832927334956, 100419390748800, 336561864306525

Offset: 0

Paul Barry, Feb 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

T(n,3) in triangle A080419.

Cf. A036068, A136158.

[(n+1)*(n+2)*(n+9)*3^n/18: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

Table[((n+1)(n+2)(n+9)3^n)/18,{n,0,30}] (* or *) LinearRecurrence[ {12,-54,108,-81},{1,10,66,360},30] (* Harvey P. Dale, Mar 21 2012 *)
CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

[(n+1)*(n+2)*(n+9)*3^(n-2)/2 for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: (1-2*x)/(1-3*x)^4.
a(n) = A006503(n+1)*3^(n-1).
a(n) = 12*a(n-1)-54*a(n-2)+108*a(n-3)-81*a(n-4). - Harvey P. Dale, Mar 21 2012
From G. C. Greubel, Dec 22 2023: (Start)
a(n) = (n+9)*A036068(n-1).
a(n) = A136158(n+3, 3).
E.g.f.: (1/2)*(2 + 14*x + 15*x^2 + 3*x^3)*exp(3*x). (End)
From Amiram Eldar, Jan 11 2024: (Start)
Sum_{n>=0} 1/a(n) = 44172*log(3/2)/7 - 20050659/7840.
Sum_{n>=0} (-1)^n/a(n) = 44496*log(4/3)/7 - 14329629/7840. (End)

A080422 a(n) = (n+1)(n+2)(n+3)(n+12)3^n/72.

Original entry on oeis.org

1, 13, 105, 675, 3780, 19278, 91854, 415530, 1804275, 7577955, 30961359, 123589557, 483611310, 1860043500, 7046907660, 26344593252, 97328636181, 355781149065, 1288173125925, 4623863536215, 16466920464456, 58222325927898, 204499905118650, 713919106104750

Offset: 0

Paul Barry, Feb 19 2003

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,243).

T(n, 4) in triangle A080419.

Cf. A136158.

[(n+1)*(n+2)*(n+3)*(n+12)*3^n/72: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

Table[(n + 1) (n + 2) (n + 3) (n + 12) 3^n/72, {n, 0, 30}] (* or *) LinearRecurrence[ {15, -90, 270, -405, 243}, {1, 13, 105, 675, 3780}, 30] (* Harvey P. Dale, Oct 22 2011 *)
CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^5, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

[(n+1)*(n+2)*(n+3)*(n+12)*3^(n-3)/8 for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: (1-2*x)/(1-3*x)^5.
For n>4, a(n)=15*a(n-1)-90*a(n-2)+270*a(n-3)-405*a(n-4)+243*a(n-5). - Harvey P. Dale, Oct 22 2011
From G. C. Greubel, Dec 22 2023: (Start)
a(n) = A136158(n+4,4).
E.g.f.: (1/8)*(8 + 80*x + 144*x^2 + 72*x^3 + 9*x^4)*exp(3*x). (End)
From Amiram Eldar, Jan 11 2024: (Start)
Sum_{n>=0} 1/a(n) = 662816499/42350 - 2122848*log(3/2)/55.
Sum_{n>=0} (-1)^n/a(n) = 2135808*log(4/3)/55 - 94614897/8470. (End)

A080423 a(n) = (n+1)(n+2)(n+3)(n+4)(n+15)*3^n/360.

Original entry on oeis.org

1, 16, 153, 1134, 7182, 40824, 214326, 1058508, 4979799, 22517352, 98513415, 419129802, 1741000716, 7083045648, 28296044604, 111232727064, 431026817373, 1648861601184, 6234757929477, 23328137324646, 86451332438394, 317576323243080, 1157228874847890, 4185605730648420

Offset: 0

Paul Barry, Feb 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-135,540,-1215,1458,-729).

T(n, 5) of triangle A080419.

Cf. A136158.

[(n+1)*(n+2)*(n+3)*(n+4)*(n+15)*3^n/360: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^6, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)
Table[3^n*(n+15)*Binomial[n+4,4]/15, {n,0,30}] (* G. C. Greubel, Dec 22 2023 *)

[3^n*(n+15)*binomial(n+4,4)/15 for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: (1-2*x)/(1-3*x)^6.
From G. C. Greubel, Dec 22 2023: (Start)
a(n) = A136158(n+5, 5).
E.g.f.: (1/40)*(40 + 520*x + 1320*x^2 + 1080*x^3 + 315*x^4 + 27*x^5)*exp(3*x). (End)
From Amiram Eldar, Jan 11 2024: (Start)
Sum_{n>=0} 1/a(n) = 215084880*log(3/2)/1001 - 99766344351/1145144.
Sum_{n>=0} (-1)^n/a(n) = 216218880*log(4/3)/1001 - 498108421095/8016008. (End)

A136159 A Chebyshev polynomial triangle of the first kind defined by T(n+1,x) = 3x*T(n,x) - T(n-1,x).

Original entry on oeis.org

1, 1, 3, -1, 9, -4, 27, -15, 1, 81, -54, 7, 243, -189, 36, -1, 729, -648, 162, -10, 2187, -2187, 675, -66, 1, 6561, -7290, 2673, -360, 13, 19683, -24057, 10206, -1755, 105, -1, 59049, -78732, 37908, -7938, 675, -16

Offset: 0

Gary W. Adamson, Dec 16 2007

Row sums (unsigned) give A003688, (starting 1, 1, 4, 13, 43, 142, 469, ...).

			First few rows of the polynomials are:
1;
x;
3x^2 - 1;
9x^3 - 4x;
27x^4 - 15x^2 + 1;
81x^5 - 54x^3 + 7x;
243x^6 - 189x^4 + 36x^2 - 1;
729x^7 - 648x^5 + 162x^3 - 10x;
...

Cf. A136158, A003688.

Cf. A000244, A006234, A080419, A080420, A080421, A080422, A080423. [Philippe Deléham, Sep 12 2009]

P(n) = if (n==0, 1, if (n==1, x, 3*x*P(n-1) - P(n-2)));
row(n) = select(x->x!=0, Vec(P(n))); \\ Michel Marcus, Apr 15 2018

T(0,x) = 1, T(1,x) = x, T(n+1,x) = 3x*T(n,x) - T(n-1,x).
G.f: (l - tx)/(1 - 3tx + t^2).
Given triangle A136158, shift down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row.

Corrected and extended by Philippe Deléham, Sep 12 2009

Keyword tabf set by Michel Marcus, Apr 15 2018

cp's OEIS Frontend

A136158 Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A080420 a(n) = (n+1)*(n+6)*3^n/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A080421 a(n) = (n+1)*(n+2)*(n+9)*3^n/18.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A080422 a(n) = (n+1)*(n+2)*(n+3)*(n+12)*3^n/72.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A080423 a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(n+15)*3^n/360.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A136159 A Chebyshev polynomial triangle of the first kind defined by T(n+1,x) = 3x*T(n,x) - T(n-1,x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A080420 a(n) = (n+1)(n+6)3^n/6.

A080421 a(n) = (n+1)(n+2)(n+9)*3^n/18.

A080422 a(n) = (n+1)(n+2)(n+3)(n+12)3^n/72.

A080423 a(n) = (n+1)(n+2)(n+3)(n+4)(n+15)*3^n/360.