A036290 -id:A036290 - 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

A006234 a(n) = n*3^(n-4).

Original entry on oeis.org

1, 4, 15, 54, 189, 648, 2187, 7290, 24057, 78732, 255879, 826686, 2657205, 8503056, 27103491, 86093442, 272629233, 860934420, 2711943423, 8523250758, 26732013741, 83682825624, 261508830075, 815907549834, 2541865828329

Offset: 3

N. J. A. Sloane

For n >= 1 a(n) is also the determinant of the n-3 X n-3 matrix with 4's on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
a(n+3) = det(M(n)) where M(n) is the n X n matrix with m(i,i) = 4, m(i,j) = i/j for i != j. - Benoit Cloitre, Feb 01 2003
Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 2*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003
a(n+3) is the number of words of length n on {A, B, C, D} with no D appearing anywhere to the right of an A. - Rob Pratt, Aug 04 2004
Number of spanning trees in the book graph of order n-2, i.e., S_{n-2} X P_2 (S_k = the star graph on k nodes) (conjectured). This conjecture is true - see Doslic (2013). - N. J. A. Sloane, Dec 28 2013
Conjecture: a(n+2) is the total number of parts used in the compositions of n if the parts can be runs of any length from 1 to n, and contain any integers from 1 to n. (The number of such compositions is given by A000244(n-1).) - Gregory L. Simay, May 27 2017
a(n+3) is the number of words of length n defined on 4 letters where one of the letters is used at most once. - Enrique Navarrete, Mar 14 2024

			For n=3, the total number of parts is (3+2)3^(3+2-4)=(5)(3)=15 (each part indicated by "[]"): [3]; [2,1]; [1,2]; [2],[1]; [1],[2]; [1,1,1]; [1,1],[1]; [1],[1,1]; [1],[1],[1]. Note that these 15 parts are arranged into 9 = A000244(3-1)compositions. - _Gregory L. Simay_, May 27 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Book Graph.
Eric Weisstein's World of Mathematics, Spanning Tree.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Binomial transform of A001792.

Cf. A000244, A036290, A050914, A112626.

[ n*3^(n-4): n in [3..30] ]; // Vincenzo Librandi, Aug 19 2011

Table[n 3^(n-4), {n, 3, 30}] (* or *)
CoefficientList[Series[(1-2 x)/(1-3 x)^2, {x,0,30}], x] (* Michael De Vlieger, May 28 2017 *)
LinearRecurrence[{6,-9},{1,4},30] (* Harvey P. Dale, Aug 17 2020 *)

a(n)=n*3^(n-4) \\ Charles R Greathouse IV, Sep 24 2015

[n*3^(n-4) for n in range(3,31)] # G. C. Greubel, Dec 27 2023

G.f.: (1-2*x)/(1-3*x)^2. - Simon Plouffe in his 1992 dissertation.
a(n+3) = Sum_{k=0..n} A112626(n, k). - Ross La Haye, Jan 11 2006
G.f.: Hypergeometric2F1([1,4],[3],3*x). - R. J. Mathar, Aug 09 2015
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = 81*log(3/2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 81*log(4/3). (End)
E.g.f.: x*(exp(3*x) - 3*x - 1)/27. - Stefano Spezia, Mar 04 2023
E.g.f. (with offset 0): exp(3*x)*(1+x). - Enrique Navarrete, Mar 14 2024

A124647 a(n) = (2n + 1)*3^n.

Original entry on oeis.org

1, 9, 45, 189, 729, 2673, 9477, 32805, 111537, 373977, 1240029, 4074381, 13286025, 43046721, 138706101, 444816117, 1420541793, 4519905705, 14334558093, 45328197213, 142958160441, 449795187729, 1412147682405, 4424729404869, 13839047287569, 43211719081593, 134718888901437

Offset: 0

Gary W. Adamson, Dec 22 2006

1 - 1/9 + 1/45 - 1/189 + ... = Pi/(2*sqrt(3)) = A093766. [Jolley eq 271].
If X_1,X_2,...,X_n are 3-blocks of a (4n+1)-set X then, for n>=1, a(n) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007
Sum_{k>=0} 1/a(k) = log(2+sqrt(3))*sqrt(3)/2 =  1.1405189944... - Jaume Oliver Lafont, Nov 30 2009

			a(3) = 189 = 7*(3^3).

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 50

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 14.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A000010, A000244, A018804, A036290, A081038, A093766.

[ (2*n+1)*3^n: n in [0..23] ]; // Klaus Brockhaus, Sep 23 2009

Table[3^n*(2*n+1), {n,0,30}] (* G. C. Greubel, May 01 2021 *)

[3^n*(2*n+1) for n in (0..30)] # G. C. Greubel, May 01 2021

G.f.: (1+3*x)/(1-3*x)^2. - Jaume Oliver Lafont, Mar 07 2009
a(n) = 6*a(n-1) - 9*a(n-2) for n > 1; a(0) = 1, a(1) = 9. - Klaus Brockhaus, Sep 23 2009
a(n) = 9*A081038(n-1) for n > 0. - Klaus Brockhaus, Sep 23 2009
a(n) = Sum_{i=1..2*3^n-1} gcd(i,2*3^n) = A018804(2*3^n) -2*3^n. This is an application of the multiplicative property of the gcd sum-function A018804. So we get: 2*3^0 * phi(3^n) + ... + 2*3^(n-1) * phi(3^1) + 2*3^n * phi(3^0)+3^0 * phi(2*3^n) + ... + 3^n * phi(2*3^0) - gcd(2*3^n,2*3^n) = a(n), where phi=A000010 is Euler's totient. A general formula is Sum_{i=1..2*p^n-1} gcd(i,2*p^n) = n*3*p^n * n - 3*n*p^(n-1) + p^n, for p an odd prime. This sequence correspondes to p=3. - Jeffrey R. Goodwin, Nov 10 2011
E.g.f.: exp(3*x)*(1 + 6*x). - Stefano Spezia, May 07 2023

More terms from Klaus Brockhaus, Sep 23 2009

A083679 Decimal expansion of log(4/3).

Original entry on oeis.org

2, 8, 7, 6, 8, 2, 0, 7, 2, 4, 5, 1, 7, 8, 0, 9, 2, 7, 4, 3, 9, 2, 1, 9, 0, 0, 5, 9, 9, 3, 8, 2, 7, 4, 3, 1, 5, 0, 3, 5, 0, 9, 7, 1, 0, 8, 9, 7, 7, 6, 1, 0, 5, 6, 5, 0, 6, 6, 6, 5, 6, 8, 5, 3, 4, 9, 2, 9, 2, 9, 5, 0, 7, 2, 0, 7, 8, 0, 4, 6, 4, 3, 3, 8, 1, 1, 0, 8, 9, 9, 1, 7, 9, 1, 0, 5, 2, 8, 6, 2, 9, 6, 0, 3

Offset: 0

Benoit Cloitre, Jun 15 2003

			log(4/3) = 0.2876820724517809274392190059938274315035097108977610565....

Index entries for transcendental numbers

Cf. A016578, A018215, A036290.

RealDigits[Log[4/3],10,120][[1]] (* Harvey P. Dale, Feb 04 2015 *)

log(4/3) \\ Charles R Greathouse IV, May 15 2019

Limit of a special sum: log(4/3) = Sum_{k>=1} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1).
Asymptotically: log(4/3) = Sum_{k=1..n} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1) + log(2)/2^(n+1) + o(1/2^n).
From Amiram Eldar, Aug 07 2020: (Start)
Equals 2 * arctanh(1/7).
Equals Sum_{n>=1} 1/(n * 4^n) = Sum_{n>=1} 1/A018215(n).
Equals Sum_{n>=1} (-1)^(n+1)/(n * 3^n) = Sum_{n>=1} (-1)^(n+1)/A036290(n).
Equals Integral_{x=0..oo} 1/(3*exp(x) + 1) dx. (End)
log(4/3) = 2*Sum_{n >= 1} 1/(n*P(n, 7)*P(n-1, 7)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(4/3) = 0.28768207245178092743921(31...), correct to 23 decimal places. - Peter Bala, Mar 18 2024
Equals Sum_{n >= 1} (-1)^(n+1) * 7/(n*binomial(2*n, n)*12^n). The n-th term of the series is O(7*sqrt(Pi/n)*1/48^n). - Peter Bala, Mar 04 2025
Equals Integral_{x=0..1} (x^(1/3) - 1)/log(x) dx. - Kritsada Moomuang, May 27 2025

A050914 a(n) = n*3^n + 1.

Original entry on oeis.org

1, 4, 19, 82, 325, 1216, 4375, 15310, 52489, 177148, 590491, 1948618, 6377293, 20726200, 66961567, 215233606, 688747537, 2195382772, 6973568803, 22082967874, 69735688021, 219667417264, 690383311399, 2165293113022, 6778308875545, 21182215236076, 66088511536555, 205891132094650

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Cf. A002064, A050915.

Equals A036290(n) + 1.

[ n*3^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n*3^n+1,{n,0,30}] (* or *) LinearRecurrence[{7,-15,9},{1,4,19},30] (* Harvey P. Dale, Nov 07 2012 *)

a(n)=n*3^n+1 \\ Charles R Greathouse IV, Oct 07 2015

From Colin Barker, Oct 14 2012: (Start)
a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3).
G.f.: -(6*x^2 - 3*x + 1)/((x-1)*(3*x-1)^2). (End)
E.g.f.: exp(x)*(3*x*exp(2*x) + 1). - Elmo R. Oliveira, Sep 09 2024

A171607 Expressible as A*B^A in a nontrivial way.

Original entry on oeis.org

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592

Offset: 1

Robert Munafo, Dec 12 2009

nonn

			8=2*2^2. 24=3*2^3. 375=3*5^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. Munafo, Generalized Cullen and Woodall Numbers

Cf. A171606. Union of the "KN^K" sequences A001105, A117642, A141046, ... or of the "NK^N" sequences A036289, A036290, A018215, A036291, ... but omitting the trivial initial terms.

is(n)=if(n<8, return(0)); for(a=2,logint(n\2,2), if(n%a==0 && ispower(n/a,a), return(1))); 0 \\ Charles R Greathouse IV, Feb 19 2017

list(lim)=my(v=List()); if(lim<8,return([])); for(a=2,logint(lim\2,2), for(b=2,sqrtnint(lim\a,a), listput(v,a*b^a))); Set(v) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 2n^2 - O(n^(5/3)). - Charles R Greathouse IV, Feb 19 2017

A120906 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having k drops (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21.

Original entry on oeis.org

1, 3, 6, 3, 10, 16, 1, 15, 51, 15, 21, 126, 90, 6, 28, 266, 357, 77, 1, 36, 504, 1107, 504, 36, 45, 882, 2907, 2304, 414, 9, 55, 1452, 6765, 8350, 2850, 210, 1, 66, 2277, 14355, 25653, 14355, 2277, 66, 78, 3432, 28314, 69576, 58278, 16236, 1221, 12, 91, 5005

Offset: 0

Emeric Deutsch, Jul 15 2006

Row n has 1+floor(2n/3) terms. Row sums are the powers of 3 (A000244). T(n,0)=A000217(n+1) (the triangular numbers). Sum(k*T(n,k),k>=0)=(n-1)*3^(n-1)=A036290(n-1).

			T(5,3) = 6 because we have 1/02/1/0, 2/02/1/0, 2/1/01/0, 2/1/02/0, 2/12/1/0 and 2/1/02/1, the middle points of the drops being indicated by /.
Triangle starts:
1;
3;
6,    3;
10,  16,  1;
15,  51, 15;
21, 126, 90, 6;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000244, A000217, A036290.

G:=1/((1-z)^3-3*t*z^2+2*t*z^3-t^2*z^3): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..floor(2*n/3)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, expand(
      add(b(n-1, j)*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..15);  # Alois P. Heinz, May 19 2014

sol=Solve[{a==v z^2,b==v z^2,c==v(z^2+a z)},{a,b,c}];f[z_,u_]:=1/(1-3z-a-b-c)/.sol/.v->u-1;nn=10;Map[Select[#,#>0&]&,Level[CoefficientList[Series[f[z,u],{z,0,nn}],{z,u}],{2}]]//Grid (* Geoffrey Critzer, May 19 2014 *)

G.f.: G(t,z) = 1/[(1-z)^3-3tz^2+2tz^3-t^2*z^3].

A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).

Original entry on oeis.org

1, 2, 3, 4, 18, 16, 8, 81, 192, 125, 16, 324, 1536, 2500, 1296, 32, 1215, 10240, 31250, 38880, 16807, 64, 4374, 61440, 312500, 699840, 705894, 262144, 128, 15309, 344064, 2734375, 9797760, 17294403, 14680064, 4782969

Offset: 0

Peter Luschny, Jan 14 2009

Formatted as a square array:
1st row is A000079(n). Subsets of an n-set.
2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.
2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.
1st column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).
Alternating sum of rows in the triangle, Sum_{k=0..n} (-1)^(n-k) * T(n,k) = n! = A000142(n).
This triangle gives the coefficient of Sidi's polynomials D_{n,2,n}(-z)/(-z), for n >= 0. See [Sidi 1980]. - Wolfdieter Lang, Oct 27 2022

			Triangle begins as:
   1;
   2,    3;
   4,   18,    16;
   8,   81,   192,    125;
  16,  324,  1536,   2500,   1296;
  32, 1215, 10240,  31250,  38880,  16807;
  64, 4374, 61440, 312500, 699840, 705894, 262144;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874. Eq. (4.10), p. 862.

Cf. A000079, A000272, A036290, A066274, A075513.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*(k+2)^n ))); # G. C. Greubel, May 09 2019

[[Binomial(n,k)*(k+2)^n: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019

T := proc(n,k) binomial(n,k)*(k+2)^n end;

Table[Binomial[n, k]*(k+2)^n, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 09 2019 *)

{T(n, k) = binomial(n,k)*(k+2)^n}; \\ G. C. Greubel, May 09 2019

[[binomial(n,k)*(k+2)^n for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019

T(n,k) = binomial(n,k)*(k+2)^n, where n >= 0, and k >= 0.
From Wolfdieter Lang, Oct 20 2022: (Start)
O.g.f. of column k: (-x)^k*(k + 2)^k/(1 - (k + 2)*x)^(k+1), for k >= 0. See |A075513| with offset 0.
E.g.f. of column k: exp((k+2)*x)*((k+2)*x)^k/k!, for k >= 0. (End)
E.g.f. of triangle (of row polynomials in y): exp(2*x)*substitute(z = x*y*exp(x), LambertW(-z)^2/(-z)*2*(1 + LambertW(-z)))). - Wolfdieter Lang, Oct 24 2022

A158749 a(n) = n*9^n.

Original entry on oeis.org

0, 9, 162, 2187, 26244, 295245, 3188646, 33480783, 344373768, 3486784401, 34867844010, 345191655699, 3389154437772, 33044255768277, 320275094369454, 3088366981419735, 29648323021629456, 283512088894331673, 2701703435345984178, 25666182635786849691, 243153309181138576020

Offset: 0

Zerinvary Lajos, Mar 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001019, A018215, A036289, A036290, A036291, A036292, A036293, A036294.

Cf. A038299, A126431.

[n*9^n: n in [0..20]]; // Vincenzo Librandi, Feb 25 2014

Table[n 9^n, {n, 0, 20}] (* Vincenzo Librandi, Feb 25 2014 *)

a(n) = n*9^n; \\ Joerg Arndt, Feb 23 2014

a(n) = n*9^n.
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) = A038299(n,1).
G.f.: 9*x/(1-9*x)^2. (End)
a(n) = A001019(n)*n. - Omar E. Pol, Mar 26 2009
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(10/9). (End)
E.g.f.: 9*x*exp(9*x). - Elmo R. Oliveira, Sep 09 2024

A217536 Square array read by antidiagonals, where the top row is the nonnegative integers and the other numbers are the sum of the neighbors in the preceding row.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 3, 6, 10, 14, 4, 9, 18, 32, 46, 5, 12, 27, 55, 101, 147, 6, 15, 36, 81, 168, 315, 462, 7, 18, 45, 108, 244, 513, 975, 1437, 8, 21, 54, 135, 324, 736, 1564, 3001, 4438, 9, 24, 63, 162, 405, 973, 2222, 4761, 9199, 13637, 10, 27, 72, 189, 486, 1215, 2924, 6710, 14472, 28109, 41746

Offset: 0

WG Zeist, Oct 06 2012

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the nonnegative integers). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three).
Replacing the top row with A000012 (the all 1's sequence) and constructing the rest of the array the same way produces A062105. Similarly, replacing the top row with A000007 (a(n) = 0^n) produces A020474. - WG Zeist, Aug 24 2024
For any array constructed with this method, regardless of the sequence chosen for the top row, the sequence in the first column of the array can be computed from the sequence in the top row as follows: let a(0), a(1), a(2), ... be the terms in the top row, and b(0), b(1), b(2), ... the terms in the first column. Then b(n) = Sum_{k=0..n} A064189(n,k) * a(k). The inverse operation, to compute the top row from the first column, is given by a(n) = Sum_{k=0..n} A104562(n,k) * b(k). - WG Zeist, Aug 26 2024

			The array starts:
  0  1  2  3
  1  3  6  9
  4  10 18 27
  14 32 55 81

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Main diagonal gives A036290. First column gives A330796.

Cf. A020474, A062105, A064189, A104562.

A:= proc(n, k) option remember; `if`(k<0, 0,
     `if`(n=0, k, add(A(n-1, k+i), i=-1..1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 24 2024

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=n), m, n >= 0. - M. F. Hasler, Oct 09 2012

Offset 0 from Alois P. Heinz, Aug 24 2024

cp's OEIS Frontend

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