A013620 -id:A013620 - cp's OEIS frontend

A000351 Powers of 5: a(n) = 5^n.

1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 5), L(1, 5), P(1, 5), T(1, 5). Essentially same as Pisot sequences E(5, 25), L(5, 25), P(5, 25), T(5, 25). See A008776 for definitions of Pisot sequences.
a(n) has leading digit 1 if and only if n = A067497 - 1. - Lekraj Beedassy, Jul 09 2002
With interpolated zeros 0, 1, 0, 5, 0, 25, ... (g.f.: x/(1 - 5*x^2)) second inverse binomial transform of Fibonacci(3n)/Fibonacci(3) (A001076). Binomial transform is A085449. - Paul Barry, Mar 14 2004
Sums of rows of the triangles in A013620 and A038220. - Reinhard Zumkeller, May 14 2006
Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4)^n. a(n) is number of compositions of natural numbers into n parts less than 5. a(2) = 25 there are 25 compositions of natural numbers into 2 parts less than 5. - Adi Dani, Jun 22 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(5n) = 5n + sigma(n). In fact we have this theorem: p is a prime if and only if all solutions of the equation sigma(p*x) = p*x + sigma(x) are powers of p. - Jahangeer Kholdi, Nov 23 2013
From Doug Bell, Jun 22 2015: (Start)
Empirical observation: Where n is an odd multiple of 3, let x = (a(n) + 1)/9 and let y be the decimal expansion of x/a(n); then y*(x+1)/x + 1 = y rotated to the left.
Example:
  a(3) = 125;
  x = (125 + 1)/9 = 14;
  y = 112, which is the decimal expansion of 14/125 = 0.112;
  112*(14 + 1)/14 + 1 = 121 = 112 rotated to the left.
(End)
a(n) is the number of n-digit integers that contain only odd digits (A014261). - Bernard Schott, Nov 12 2022
Number of pyramids in the Sierpinski fractal square-based pyramid at the n-th step, while A279511 gives the corresponding number of vertices (see IREM link with drawings). - Bernard Schott, Nov 29 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [math.HO], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 270
IREM Paris-Nord, La pyramide de Sierpinski (in French).
Tanya Khovanova, Recursive Sequences
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
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Box Fractal
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A009969 (even bisection), A013710 (odd bisection), A005054 (first differences), A003463 (partial sums).
Cf. A006495, A006496, A159991, A001021.
Cf. A001076, A008776, A013620, A038220, A067497, A085449, A014261.
Sierpinski fractal square-based pyramid: A020858 (Hausdorff dimension), A279511 (number of vertices), this sequence (number of pyramids).

a000351 = (5 ^)
a000351_list = iterate (* 5) 1  -- Reinhard Zumkeller, Oct 31 2012

[5^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

[ seq(5^n,n=0..30) ];
A000351:=-1/(-1+5*z); # Simon Plouffe in his 1992 dissertation

Table[5^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 06 2006 *)
5^Range[0, 30] (* Harvey P. Dale, Aug 22 2011 *)

makelist(5^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

a(n)=5^n \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return 5**n
print([a(n) for n in range(24)]) # Michael S. Branicky, Nov 12 2022

(List.fill(50)(5: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

a(n) = 5^n.
a(0) = 1; a(n) = 5*a(n-1) for n > 0.
G.f.: 1/(1 - 5*x).
E.g.f.: exp(5*x).
a(n) = A006495(n)^2 + A006496(n)^2.
a(n) = A159991(n) / A001021(n). - Reinhard Zumkeller, May 02 2009
From Bernard Schott, Nov 12 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4.
Sum_{n>=0} (-1)^n/a(n) = 5/6. (End)
a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000045(2*k+1). - Vladimir Kruchinin, Jan 14 2025

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

N. J. A. Sloane

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

/* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

{T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024

A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.

Original entry on oeis.org

1, 3, 2, 9, 12, 4, 27, 54, 36, 8, 81, 216, 216, 96, 16, 243, 810, 1080, 720, 240, 32, 729, 2916, 4860, 4320, 2160, 576, 64, 2187, 10206, 20412, 22680, 15120, 6048, 1344, 128, 6561, 34992, 81648, 108864, 90720, 48384, 16128, 3072, 256

Offset: 0

N. J. A. Sloane

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

Triangle of coefficients in expansion of (3 + 2x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 23 2018

			Triangle begins:
   1;
   3,   2;
   9,  12,   4;
  27,  54,  36,   8;
  81, 216, 216,  96,  16;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A013620, A000079, A000244, A013613, A038221.

a038220 n k = a038220_tabl !! n !! k
a038220_row n = a038220_tabl !! n
a038220_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + 2 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten  (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ Charles R Greathouse IV, Jul 19 2016

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006
G.f.: 1/(1 - 3*x - 2*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 2 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018

A154692 Triangle read by rows: T(n, k) = (2^(n-k)3^k + 2^k3^(n-k))*binomial(n, k).

Original entry on oeis.org

2, 5, 5, 13, 24, 13, 35, 90, 90, 35, 97, 312, 432, 312, 97, 275, 1050, 1800, 1800, 1050, 275, 793, 3492, 7020, 8640, 7020, 3492, 793, 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315, 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

			Triangle begins
     2;
     5,     5;
    13,    24,    13;
    35,    90,    90,     35;
    97,   312,   432,    312,     97;
   275,  1050,  1800,   1800,   1050,    275;
   793,  3492,  7020,   8640,   7020,   3492,   793;
  2315, 11550, 26460,  37800,  37800,  26460, 11550,  2315;
  6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000225, A007482, A013620, A088137, A119309.
Sums include: A010673 (alternating sign row), A020699 (row), A020729 (row).
Related sequences: A007318, A154690,

A154692:= func< n,k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n,k) >;
[A154692(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154692 := proc(n,m)
        (2^(n-m)*3^m+2^m*3^(n-m))*binomial(n,m) ;
end proc:
seq(seq(A154692(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Oct 24 2011

p=2; q=3;
T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n,m];
Table[T[n,m], {n,0,10}, {m,0,n}]//Flatten

from sage.all import *
def A154692(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*binomial(n,k)
print(flatten([[A154692(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Sum_{k=0..n} T(n, k) = A020729(n) = A020699(n+1).
T(n,m) = A013620(n,m) + A013620(m,n). - R. J. Mathar, Oct 24 2011
From G. C. Greubel, Jan 18 2025: (Start)
T(2*n, n) = A119309(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1) + A007482(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A088137(n+1) + A000225(n+1). (End)

A303901 Triangle read by rows of coefficients in expansion of (3-2x)^n, where n is a nonnegative integer.

Original entry on oeis.org

1, 3, -2, 9, -12, 4, 27, -54, 36, -8, 81, -216, 216, -96, 16, 243, -810, 1080, -720, 240, -32, 729, -2916, 4860, -4320, 2160, -576, 64, 2187, -10206, 20412, -22680, 15120, -6048, 1344, -128, 6561, -34992, 81648, -108864, 90720, -48384, 16128, -3072, 256, 19683, -118098, 314928, -489888, 489888, -326592, 145152, -41472, 6912, -512

Offset: 0

Zagros Lalo, May 02 2018

This is a signed version of A038220.
Row n gives coefficients in expansion of (3-2x)^n.
The numbers in rows of triangles in A302747 and A303941 are along skew diagonals pointing top-left and top-right in center-justified triangle of coefficients in expansions of (3-2x)^n (A303901).
This is the lower triangular Riordan matrix (1/(1-3*t), -2*t/(1-3*t)), hence a convolution matrix. See the g.f.s. - Wolfdieter Lang, Jun 28 2018

			Triangle begins:
  n \k 0     1       2       3      4      5       6      7      8    9  ...
  --------------------------------------------------------------------------
  0 |  1
  1 |  3     -2
  2 |  9     -12     4
  3 |  27    -54     36     -8
  4 |  81    -216    216    -96     16
  5 |  243   -810    1080   -720    240    -32
  6 |  729   -2916   4860   -4320   2160   -576    64
  7 |  2187  -10206  20412  -22680  15120  -6048   1344   -128
  8 |  6561  -34992  81648  -108864 90720  -48384  16128  -3072  256
  9 |  19683 -118098 314928 -489888 489888 -326592 145152 -41472 6912 -512

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 394, 396, 398.

Zagros Lalo, Center-justified Triangle
Zagros Lalo, Skew Diagonals in center-justified Triangle

Cf. A013620 (unsigned), A000012 (row sums), A000351 (alternating row sums).

For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(3 - 2 x)^i],x]]]

T(0,0) = 1; T(n,k) = 3*T(n-1,k) - 2*T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.
G.f. of row polynomials: 1 / (1 - 3*t + 2*t*x).
G.f. of column k: (-2*x)^k/(1-3*x)^(k+1), for k >= 0.

Edited - Wolfdieter Lang, Jun 28 2018

A119309 a(n) = binomial(2n,n) 6^n.

Original entry on oeis.org

1, 12, 216, 4320, 90720, 1959552, 43110144, 960740352, 21616657920, 489977579520, 11171488813056, 255928652808192, 5886359014588416, 135839054182809600, 3143703825373593600, 72933928748667371520

Offset: 0

Reinhard Zumkeller, May 14 2006

nonn

Number of lattice paths from (0,0) to (n,n) using three kinds of steps (1,0) and two kinds of steps (0,1). - Joerg Arndt, Jul 01 2011

Central terms of the triangles in A013620 and A038220.

			a(3) = binomial(2*3,3) * (6^3) = 20 * 216 = 4320. - _Indranil Ghosh_, Mar 03 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A000984, A013620, A038220, A059304, A098658.

Table[Binomial[2n,n]*(6^n), {n, 0, 15}] (* Indranil Ghosh, Mar 03 2017 *)

/* same as in A092566 but use */
steps=[[1,0], [1,0], [1,0], [0,1], [0,1]]; /* note repeated entries */
/* Joerg Arndt, Jun 30 2011 */

a(n)=binomial(2*n,n)*6^n \\ Charles R Greathouse IV, Mar 03 2017

import math
f=math.factorial
def C(n,r): return f(n)//f(r)//f(n-r)
def A119309(n): return C(2*n,n)*(6**n) # Indranil Ghosh, Mar 03 2017

a(n) = 6^n * A000984(n).
G.f.: 1/sqrt(1-24*x). - Zerinvary Lajos, Dec 20 2008 [Corrected by Joerg Arndt, Jul 01 2011]
D-finite with recurrence: n*a(n) +12*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = 2^n*A098658(n) = 3^n*A059304(n). - R. J. Mathar, Jan 20 2020
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 24/23 + 24*sqrt(23)*arcsin(1/sqrt(24))/529.
Sum_{n>=0} (-1)^n/a(n) = 24/25 - 24*arcsinh(1/sqrt(24))/125. (End)
E.g.f.: exp(12*x) * BesselI(0,12*x). - Ilya Gutkovskiy, Sep 14 2021

A317498 Triangle read by rows of coefficients in expansions of (-2 + 3*x)^n, where n is nonnegative integer.

Original entry on oeis.org

1, -2, 3, 4, -12, 9, -8, 36, -54, 27, 16, -96, 216, -216, 81, -32, 240, -720, 1080, -810, 243, 64, -576, 2160, -4320, 4860, -2916, 729, -128, 1344, -6048, 15120, -22680, 20412, -10206, 2187, 256, -3072, 16128, -48384, 90720, -108864, 81648, -34992, 6561, -512, 6912, -41472, 145152, -326592, 489888, -489888, 314928, -118098, 19683

Offset: 0

Zagros Lalo, Jul 31 2018

Row n gives coefficients in expansion of (-2 + 3*x)^n.
This is a signed version of A013620.
The coefficients in the expansion of 1/(1-x) are given by the sequence generated by the row sums.
The row sums give A000012 (The simplest sequence of positive numbers: the all 1's sequence).
The numbers in rows of triangles in A302747 and A303941 (Triangle of coefficients of Fermat polynomials) are along first layer skew diagonals pointing top-right and top-left in center-justified triangle of coefficients in expansions of (-2 + 3*x)^n, see links.

			Triangle begins:
     1;
    -2,     3;
     4,   -12,      9;
    -8,    36,    -54,     27;
    16,   -96,    216,   -216,      81;
   -32,   240,   -720,   1080,    -810,     243;
    64,  -576,   2160,  -4320,    4860,   -2916,     729;
  -128,  1344,  -6048,  15120,  -22680,   20412,  -10206,   2187;
   256, -3072,  16128, -48384,   90720, -108864,   81648, -34992,    6561;
  -512,  6912, -41472, 145152, -326592,  489888, -489888, 314928, -118098, 19683;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 394-396.

Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3 x)^n.
Eric Weisstein's World of Mathematics, Fermat Polynomial.

Row sums give A000012.
Cf. A013620 ((2+3*x)^n).
Cf. A302747, A303941.

Flat(List([0..8],n->List([0..n],k->(-2)^(n-k)*3^k/(Factorial(n-k)*Factorial(k))*Factorial(n)))); # Muniru A Asiru, Aug 01 2018

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, -2 t[n - 1, k] + 3 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
t[n_, k_] := t[n, k] = ((-2)^(n - k) 3^k)/((n - k)! k!) n!;Table[t[n, k], {n, 0, 9}, {k, 0, n} ] // Flatten
Table[CoefficientList[(-2 + 3 x)^n, x], {n, 0, 9}] // Flatten

trianglerows(n) = my(v=[]); for(k=0, n-1, v=Vec((-2+3*x)^k + O(x^(k+1))); print(v))
/* Print initial 10 rows of triangle as follows */
trianglerows(10) \\ Felix Fröhlich, Jul 31 2018

T(0,0) = 1; T(n,k) = -2 * T(n-1,k) + 3 * T(n-1,k-1) for k = 0,1,...,n and T(n,k)=0 for n or k < 0.
T(n, k) = ((-2)^(n - k) 3^k)/((n - k)! k!) n! for k = 0,1..n.
G.f.: 1 / (1 + 2*x - 3*x*t).

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