A006234 -id:A006234 - cp's OEIS frontend

A027471 a(n) = (n-1)*3^(n-2), n > 0.

0, 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, 649539, 2125764, 6908733, 22320522, 71744535, 229582512, 731794257, 2324522934, 7360989291, 23245229340, 73222472421, 230127770466, 721764371007, 2259436291848

Offset: 1

Olivier Gérard

Arithmetic derivative of 3^(n-1): a(n) = A003415(A000244(n-1)). - Reinhard Zumkeller, Feb 26 2002 [Offset corrected by Jianing Song, May 28 2024]
Binomial transform of A053220(n+1) is a(n+2). Binomial transform of A001787 is a(n+1). Binomial transform of A045883(n-1). - Michael Somos, Jul 10 2003
If X_1,X_2,...,X_n are 3-blocks of a (3n+1)-set X then, for n >= 1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). > - Milan Janjic, Nov 18 2007
Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then a(n+1) = the sum of the differences in size (i.e., |y|-|x|) for all (x, y) of S. - Ross La Haye, Nov 19 2007
Number of substrings 00 (or 11, or 22) in all ternary words of length n: a(3) = 6 because we have 000, 001, 002, 100, 200 (with 000 contributing two substrings). - Darrell Minor, Jul 17 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
Milan Janjić, Two Enumerative Functions
Milan Janjić and Boris Petković, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Frank Ellermann, Illustration of binomial transforms
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 715
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 20.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Second column of A027465.
Partial sums of A081038.
Cf. A006234.

List([1..40], n-> (n-1)*3^(n-2)); # Muniru A Asiru, Jul 15 2018

[(n-1)*3^(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

seq((n-1)*3^(n-2), n=1..40); # Muniru A Asiru, Jul 15 2018

Table[(n-1)3^(n-2),{n,30}] (* or *)
LinearRecurrence[{6,-9},{0,1},30] (* Harvey P. Dale, Apr 14 2016 *)
Range[0, 24]! CoefficientList[ Series[x*Exp[3 x], {x, 0, 24}], x] (* Robert G. Wilson v, Aug 03 2018 *)

PARI
```
a(n)=if(n<1, 0, (n-1)*3^(n-2));
```

[3^(n-2)*(n-1) for n in (1..30)] # G. C. Greubel, May 20 2021

From Wolfdieter Lang: (Start)
G.f.: (x/(1-3*x))^2.
E.g.f.: (1 + (3*x-1)*exp(3*x))/9.
a(n) = 3^(n-2)*(n-1) (convolution of A000244, powers of 3, with itself). (End)
a(n) = 6*a(n-1) - 9*a(n-2), n > 2, a(1)=0, a(2)=1. - Barry E. Williams, Jan 13 2000
a(n) = A036290(n-1)/3, for n>0. - Paul Barry, Feb 06 2004 [corrected by Jerzy R Borysowicz, Apr 03 2025]
a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
From Paul Barry, Feb 15 2005: (Start)
a(n) = (1/3)*Sum_{k=0..2n} T(n, k)*k, where T(n, k) is given by A027907.
a(n) = (1/3)*Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j-k).
a(n+1) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k+1). (End)
Sum_{n>=2} 1/a(n) = 3*log(3/2). - Jaume Oliver Lafont, Sep 19 2009
a(n) = 3*a(n-1) + 3^(n-2) (with a(1)=0). - Vincenzo Librandi, Dec 30 2010
Sum_{n>=2} (-1)^n/a(n) = 3*log(4/3). - Amiram Eldar, Oct 28 2020

Edited by Michael Somos, Jul 10 2003

A036290 a(n) = n*3^n.

Original entry on oeis.org

0, 3, 18, 81, 324, 1215, 4374, 15309, 52488, 177147, 590490, 1948617, 6377292, 20726199, 66961566, 215233605, 688747536, 2195382771, 6973568802, 22082967873, 69735688020, 219667417263, 690383311398, 2165293113021, 6778308875544, 21182215236075, 66088511536554, 205891132094649

Offset: 0

N. J. A. Sloane

If X_1,X_2,...,X_n is a partition of a 3n-set X into 3-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
László Németh, The trinomial transform triangle, Journal of Integer Sequences, Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A000244, A006234, A016578, A027471, A083679, A289399 (partial sums).

[n*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2017

A036290 := proc(n) n*3^n ; end proc: # R. J. Mathar, Jun 18 2011

nn=20;a=1/(1-3x);CoefficientList[Series[x D[ a,x] ,{x,0,nn}],x]  (* Geoffrey Critzer, Nov 18 2012 *)
Table[n 3^n, {n, 0, 30}] (* Vincenzo Librandi, Jul 06 2017 *)

a(n)=3^n*n \\ Charles R Greathouse IV, Jun 18 2011

From Paul Barry, Feb 06 2004: (Start)
A trinomial transform. Differentiate (1+x+x^2)^n and set x=1.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (2*n-2*i-j)*n!/(i!*j!*(n-i-j)!). (End)
From Paul Barry, Feb 15 2005: (Start)
a(n) = Sum_{k=0..2*n} T(n, k)*k, where T(n, k) is given by A027907.
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k). (End)
From R. J. Mathar, Jun 19 2011: (Start)
G.f.: 3*x/(3*x-1)^2.
a(n) = 3*A027471(n+1). (End)
Sum_{n>=1} 1/a(n) = log(3/2) = 0.405465108... = A016578. - Franz Vrabec, Jan 07 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = log(4/3) = A083679. - Amiram Eldar, Jul 20 2020
a(n) = 6*a(n-1) - 9*a(n-2). - Wesley Ivan Hurt, Apr 26 2021
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 3*x*exp(3*x).
a(n) = n*A000244(n). (End)

A287768 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.

Original entry on oeis.org

1, 3, -2, 9, -9, 1, 27, -36, 4, 6, 81, -135, 15, 45, -5, 243, -486, 54, 243, -36, -18, 1, 729, -1701, 189, 1134, -189, -189, 7, 21, 2187, -5832, 648, 4860, -864, -1296, 36, 216, 54, -8, 6561, -19683, 2187, 19683, -3645, -7290, 162, 1458, 729, -81, -81, 1, 19683, -65610, 7290, 76545, -14580, -36450, 675, 8100, 6075, -540, -1080, 10, -162, 45

Offset: 1

Gregory Gerard Wojnar, May 31 2017

Let SM_k = Sum( d_(t_1, t_2, t_3)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3) summed over all length 3 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3=k, where SM_k are the averaged k-th power sum symmetric polynomials in 3 data (i.e., SM_k = S_k/3 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(3,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3) form an irregular triangle, with one row for each k value starting with k=1; "irregular" means that the number of terms in successive rows is nondecreasing.

The sum of the positive terms in successive rows appears to be A195350; row sums of negative terms is always 1 less than corresponding sum of positive terms.

			Triangle begins:
    1;
    3,   -2;
    9,   -9,  1;
   27,  -36,  4,   6;
   81, -135, 15,  45,  -5;
  243, -486, 54, 243, -36, -18, 1;
  ...
The first few rows describe:
Row 1: SM_1 = 1 eM_1;
Row 2: SM_2 = 3*(eM_1)^2 - 2*eM_2;
Row 3: SM_3 = 9*(eM_1)^3 - 9*eM_1*eM_2 + 1*eM_3;
Row 4: SM_4 = 27*(eM_1)^4 - 36*(eM_1)^2*eM_2 + 4*eM_1*eM_3 + 6*(eM_2)^2;
Row 5: SM_5 = 81*(eM_1)^5 - 135*(eM_1)^3*eM_2 + 15*(eM_1)^2*eM_3 + 45*eM_1*(eM_2)^2 - 5*eM_2*eM_3.

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=3, p.22, arXiv:1706.08381 [math.GM], 2017.

Row sums of the positive terms appears to be A195350.
First entries of row n is A000244(n).
Second entries of row n, for n>1, is given by -n*3^(n-2).
Third entries of row n, for n>2, is given by n*3^(n-4), A006234.
Fourth entries of row n, for n>3, is given by n*(n-3)*3^(n-3)/2!.
Fifth entries of row n, for n>4, is given by -n*(n-4)*3^(n-5)/1!.
Corresponding sequences for different sized data multisets are:  A028297 (m=2), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).
Cf. A210258.

Java
```
// See Wojnar link.
```

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

Original entry on oeis.org

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

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683

Offset: 0

N. J. A. Sloane, May 03 2000

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008
Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
Cf. A084938, A098399, A110523, A133494, A136158, A193722.

A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023

A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, 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))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023

def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.
Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006
T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007
G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 27 2023: (Start)
T(n, 0) = A000012(n).
T(n, 1) = A016777(n-1).
T(n, 2) = A062741(n-1).
T(n, 3) = 9*A002411(n-2).
T(n, 4) = 27*A001296(n-3).
T(n, 5) = 81*A051836(n-4).
T(n, n) = A133494(n).
T(n, n-1) = A006234(n+2).
T(n, n-2) = A080420(n-2).
T(n, n-3) = A080421(n-3).
T(n, n-4) = A080422(n-4).
T(n, n-5) = A080423(n-5).
T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

More terms from Michel Marcus, Jul 25 2023

A064017 Number of ternary trees (A001764) with n nodes and maximal diameter.

Original entry on oeis.org

1, 3, 12, 45, 162, 567, 1944, 6561, 21870, 72171, 236196, 767637, 2480058, 7971615, 25509168, 81310473, 258280326, 817887699, 2582803260, 8135830269, 25569752274, 80196041223, 251048476872, 784526490225, 2447722649502

Offset: 1

Danail Bonchev (bonchevd(AT)aol.com), Sep 07 2001

A problem important for polymer science because it counts the trees having unbranched branches; they are called "combs".
Equals (1, 3, 9, 27, 81, ...) convolved with (1, 0, 3, 9, 27, 81, ...). Example: a(5) = 162 = (81, 27, 9, 3, 1) dot (1, 0, 3, 9, 27) = 81 + 3*27. - Gary W. Adamson, Jul 31 2010
Floretion Algebra Multiplication Program, FAMP Code: lesforseq[ - 'i + 'j - 'kk' - 'ki' - 'kj' ], vesforseq(n) = 3^n, tesforseq = A006234

			a(5) = 162 because we can write (5+1)*3^(5-2) = 6*3^3 = 6*27.

Harry J. Smith, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A001764, A006234, A014915, A025192, A027261, A079272, A196389.

a:=n->ceil(sum(3^(n-2),j=0..n)): seq(a(n), n=1..26); # Zerinvary Lajos, Jun 05 2008

Join[{1},Table[(n+1)3^(n-2),{n,2,30}]] (* or *) Join[{1}, LinearRecurrence[ {6,-9},{3,12},30]] (* Harvey P. Dale, Feb 07 2012 *)

{ for (n=1, 200, if (n>1, a=(n + 1)*p; p*=3, a=p=1); write("b064017.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 06 2009

a(n)=if(n==1, 1, (n+1)*3^(n-2)); \\ Joerg Arndt, May 06 2013

@CachedFunction
def BB(n, k, x):  # modified cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= k) else 1
    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
    if n == 0: return 1
    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
def A064017(n) : return 3^(n-1)*EulerianPolynomial(1,n-1,1/3) if n != 1 else 1
[A064017(n) for n in (1..25)]  # Peter Luschny, May 04 2013

a(n) = 3*a(n-1) + 3^(n-2).
a(n) = (n+1)*3^(n-2), for n > 1.
From Paul Barry, Sep 05 2003: (Start)
a(n) = (n+2)3^(n-1) + 0^n/3 (offset 0).
a(n) = A025192(n) + A027471(n). (End)
A006234(n+4) - a(n+2) = 3^n. - Creighton Dement, Mar 01 2005
a(n+1) = Sum_{k=0..n} A196389(n,k)*3^k. - Philippe Deléham, Oct 31 2011
G.f.: (1 - 3*x + 3*x^2)*x/(1 - 3*x)^2. - Philippe Deléham, Oct 31 2011
a(n) = 6*a(n-1) - 9*a(n-2), with a(1)=1, a(2)=3, a(3)=12. - Harvey P. Dale, Feb 07 2012
E.g.f.: (exp(3*x)*(1 + 3*x) - 1)/9. - Stefano Spezia, Mar 05 2020
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = 27*log(3/2) - 19/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 17/2 - 27*log(4/3). (End)

A112626 Triangle read by rows: T(n,k) = Sum_{j=0..n} binomial(n, k+j)*2^(n-k-j).

Original entry on oeis.org

1, 3, 1, 9, 5, 1, 27, 19, 7, 1, 81, 65, 33, 9, 1, 243, 211, 131, 51, 11, 1, 729, 665, 473, 233, 73, 13, 1, 2187, 2059, 1611, 939, 379, 99, 15, 1, 6561, 6305, 5281, 3489, 1697, 577, 129, 17, 1, 19683, 19171, 16867, 12259, 6883, 2851, 835, 163, 19, 1, 59049, 58025

Offset: 0

Ross La Haye, Dec 26 2005

T(n, 0) = A000244(n), T(n, 1) = A001047(n), T(n, 2) = A066810(n).
Column 0 is the row sums of A038207 starting at column 0, column 1 is the row sums of A038207 starting at column 1 etc. etc. Helpful suggestions related to Riordan arrays given by Paul Barry.
Riordan array ( 1/(1 - 3*x), x/(1 - 2*x) ). Matrix inverse is a signed version of A209149. - Peter Bala, Jul 17 2013
T(n,k) is the number of strings of length n over an alphabet of 3 letters that contain a given string of length k as a subsequence. - Robert Israel, Jan 14 2020

			Triangle begins as:
    1;
    3,   1;
    9,   5,   1;
   27,  19,   7,   1;
   81,  65,  33,   9,  1;
  243, 211, 131,  51, 11,  1;
  729, 665, 473, 233, 73, 13, 1...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Row sums = n*3^(n-1) + 3^n = A006234(n+3) (Frank Ruskey and class).
Cf. A000244, A001047, A038207, A066810.
Cf. A209149 (unsigned matrix inverse).

Flat(List([0..10], n-> List([0..n], k-> Sum([k..n], j-> Binomial(n, j)*2^(n-j)) ))); # G. C. Greubel, Nov 18 2019

[&+[Binomial(n,j)*2^(n-j): j in [k..n]]: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq( add(binomial(n,j)*2^(n-j), j=k..n), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Flatten[Table[Sum[Binomial[n, k+m]*2^(n-k-m), {m, 0, n}], {n, 0, 10}, {k, 0, n}]]

T(n,k) = sum(j=k,n, binomial(n,j)*2^(n-j)); \\ G. C. Greubel, Nov 18 2019

[[sum(binomial(n,j)*2^(n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

T(n, k) = Sum_{j=0..n} binomial(n, k+j)*2^(n-k-j).
O.g.f. (by columns): x^k /((1-3*x)*(1-2*x)^k). - Frank Ruskey and class
T(n,k) = Sum_{j=k..n} binomial(n,j)*2^(n-j). - Ross La Haye, May 02 2006
Binomial transform (by columns) of A055248.

More terms from Ross La Haye, Dec 31 2006

A079028 a(0) = 1, a(n) = (n + 4)*4^(n-1) for n >= 1.

Original entry on oeis.org

1, 5, 24, 112, 512, 2304, 10240, 45056, 196608, 851968, 3670016, 15728640, 67108864, 285212672, 1207959552, 5100273664, 21474836480, 90194313216, 377957122048, 1580547964928, 6597069766656, 27487790694400, 114349209288704, 474989023199232, 1970324836974592, 8162774324609024

Offset: 0

Benoit Cloitre, Feb 01 2003

a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i) = 5, m(i,j) = i/j.
Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 3*m(i-1,j-1).
4th binomial transform of (1,1,0,0,0,0,...). - Paul Barry, Mar 07 2003
Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the complete graph K_n (see A235113). Example: a(1)=5; indeed, K_1 is the one vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}. - Emeric Deutsch, Jan 13 2014
Row sums of A235113.

F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, A closed formula for the number of convex permutominoes, arXiv:math/0702550 [math.CO], 2007.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A001792, A006234, A081105, A006234, A235113.

Cf. A002697, A034007, A079861, A045891, A087449.

LinearRecurrence[{8, -16}, {1, 5}, 22] (* Jean-François Alcover, Nov 06 2018 *)

[lucas_number2(n, 4, 0)*n/2^10 for n in range(4, 26)] # Zerinvary Lajos, Mar 13 2009

a(n) = 8*a(n-1)-16*a(n-2), a(0) = 1, a(1) = 5. - Paul Barry, Mar 07 2003
G.f.: (1 - 3*x)/(1 - 4*x)^2. - Philippe Deléham, Dec 11 2008
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=0} 1/a(n) = 1024*log(4/3) - 880/3.
Sum_{n>=0} (-1)^n/a(n) = 688/3 - 1024*log(5/4). (End)
E.g.f.: exp(4*x)*(1 + x). - Stefano Spezia, Mar 05 2023

More terms from Stefano Spezia, Mar 05 2023

A080419 Triangle of generalized Chebyshev coefficients.

Original entry on oeis.org

1, 4, 1, 15, 7, 1, 54, 36, 10, 1, 189, 162, 66, 13, 1, 648, 675, 360, 105, 16, 1, 2187, 2673, 1755, 675, 153, 19, 1, 7290, 10206, 7938, 3780, 1134, 210, 22, 1, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1, 78732, 137781, 139968, 91854, 40824, 12474, 2592

Offset: 1

Paul Barry, Feb 19 2003

Second binomial transform of 'pruned' Pascal triangle Binomial(i+1,j+1), (i,j>=0).

			Rows are:
{1},
{4,1},
{15,7,1},
{54,36,10,1},
{189,162,66,13,1},
...
For example, 10 = 7+3*1, 66 = 36+3*10.

Columns include A006234, A080420, A080421, A080422, A080423.

T(n, k) = if (k==1, (n+2)*3^(n-2), if (k==n, 1, if (k < n, T(n-1, k-1) + 3*T(n-1, k), 0)));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 15 2018

T(n,1) = A006234(n+2), T(n,n) = 1, T(n,k) = T(n-1,k-1) + 3*T(n-1,k), T(n,k)=0 for k>n. - corrected by Michel Marcus, Apr 15 2018

As a square array, T1(n, k)= (n+3k)3^n Product{j=1..(k-1), n+j}/(3k(k-1)!) (k>=1, n>=0).

A328887 Array read by antidiagonals: T(n,m) is the number of acyclic edge sets in the complete bipartite graph K_{n,m}.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 15, 8, 1, 1, 16, 54, 54, 16, 1, 1, 32, 189, 328, 189, 32, 1, 1, 64, 648, 1856, 1856, 648, 64, 1, 1, 128, 2187, 9984, 16145, 9984, 2187, 128, 1, 1, 256, 7290, 51712, 129000, 129000, 51712, 7290, 256, 1, 1, 512, 24057, 260096, 968125, 1475856, 968125, 260096, 24057, 512, 1

Offset: 0

Andrew Howroyd, Oct 29 2019

In other words, the number of spanning forests of the complete bipartite graph K_{n,m} with isolated vertices allowed.

			Array begins:
====================================================================
n\m | 0   1    2      3       4         5          6           7
----+---------------------------------------------------------------
  0 | 1   1    1      1       1         1          1           1 ...
  1 | 1   2    4      8      16        32         64         128 ...
  2 | 1   4   15     54     189       648       2187        7290 ...
  3 | 1   8   54    328    1856      9984      51712      260096 ...
  4 | 1  16  189   1856   16145    129000     968125     6925000 ...
  5 | 1  32  648   9984  129000   1475856   15450912   151201728 ...
  6 | 1  64 2187  51712  968125  15450912  219682183  2862173104 ...
  7 | 1 128 7290 260096 6925000 151201728 2862173104 48658878080 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Mathematics Stack Exchange, Spanning forests of bipartite graphs and distinct row/column sums of binary matrices.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.

Column k=2 is A006234.
Main diagonal is A297077.
Cf. A072590, A328888.

\\ here U is A328888 as matrix.
U(n, m=n)={my(M=matrix(n, m), N=matrix(n, m, n, m, n^(m-1) * m^(n-1))); for(n=1, n, for(m=1, m, M[n,m] = N[n,m] + sum(i=1, n-1, sum(j=1, m-1, binomial(n-1, i-1)*binomial(m, j)*N[i,j]*M[n-i, m-j])))); M}
T(n, m=n)={my(M=U(n, m)); matrix(n+1, m+1, n, m, 1 + sum(i=1, n-1, sum(j=1, m-1, binomial(n-1,i)*binomial(m-1,j)*M[i,j])))}
{ my(A=T(7)); for(i=1, #A, print(A[i,])) }

T(n,m) = 1 + Sum_{i=1..n} Sum_{j=1..m} binomial(n,i)*binomial(m,j)*A328888(i,j).

cp's OEIS Frontend

A027471 a(n) = (n-1)*3^(n-2), n > 0.

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A036290 a(n) = n*3^n.

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A287768 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.

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

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A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

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A064017 Number of ternary trees (A001764) with n nodes and maximal diameter.

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