A027472 -id:A027472 - cp's OEIS frontend

A001788 a(n) = n(n+1)2^(n-2).

0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520, 28160, 67584, 159744, 372736, 860160, 1966080, 4456448, 10027008, 22413312, 49807360, 110100480, 242221056, 530579456, 1157627904, 2516582400, 5452595200, 11777605632, 25367150592, 54492397568, 116769423360, 249644974080, 532575944704

Offset: 0

N. J. A. Sloane

Number of 2-dimensional faces in (n+1)-dimensional hypercube; also number of 4-cycles in the (n+1)-dimensional hypercube. - Henry Bottomley, Apr 14 2000
Also the number of edges in the (n+1)-halved cube graph. - Eric W. Weisstein, Jun 21 2017
From Philippe Deléham, Apr 28 2004: a(n) is the sum, over all nonempty subsets E of {1, 2, ..., n}, of all elements of E. E.g., a(3) = 24: the nonempty subsets are {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} and 1 + 2 + 3 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 2 + 3 = 24.
Equivalently, sum of all nodes (except the last one, equal to n+1) of all integer compositions of n+1. - Olivier Gérard, Oct 22 2011
The inverse binomial transform of a(n-k) for k=-1..4 gives A001844, A000290, A000217(n-1), A002620(n-1), A008805(n-4), A000217 interspersed with 0's. - Michael Somos, Jul 18 2003
Take n points on a finite line. They all move with the same constant speed; they instantaneously change direction when they collide with another; and they fall when they quit the line. a(n-1) is the total number of collisions before falling when the initials directions are the 2^n possible. The mean number of collisions is then n(n-1)/8. E.g., a(1)=0 before any collision is possible. a(2)=1 because there is a collision only if the initials directions are, say, right-left. - Emmanuel Moreau, Feb 11 2006
Also number of pericondensed hexagonal systems with n hexagons. For example, if n=5 then the number of pericondensed hexagonal systems with n hexagons is 24. - Parthasarathy Nambi, Sep 06 2006
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>1, a(n-1) is equal to the number of (n+2)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly two u's. Example: a(2)=6 because we have uuw, uuv, uwu, uvu, wuu and vuu. - Zerinvary Lajos, Dec 29 2007
For n>0 where [0]={}, the empty set, and [n]={1,2,...n} a(n) is the number of ways to separate [n-1] into three non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. - Geoffrey Critzer, Feb 07 2009
Form an array with m(n,0) = m(0,n) = n^2 and m(i,j) = m(i-1,j-1) + m(i-1,j). Then m(1,n) = A001844(n) and m(n,n) = a(n). - J. M. Bergot, Nov 07 2012
The sum of the number of inversions of all sequences of zeros and ones with length n+1. - Evan M. Bailey, Dec 09 2020
a(n) is the number of strings of length n defined on {0,1,2,3} that contain at most one 2, exactly one 3, and have no restriction on the number of 0s and 1s. For example, a(3)=24 since the strings are 321 (6 of this type), 320 (6 of this type), 310 (6 of this type), 300 (3 of this type) and 311 (3 of this type). - Enrique Navarrete, May 04 2025

			The nodes of an integer composition are the partial sums of its elements, seen as relative distances between nodes of a 1-dimensional polygon. For a composition of 7 such as 1+2+1+3, the nodes are 0,1,3,4,7. Their sum (without the last node) is 8. The sum of all nodes of all 2^(7-1)=64 integer compositions of 7 is 672.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
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..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Herbert Izbicki, Über Unterbäume eines Baumes, Monatshefte fur Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
Han Mao Kiah, Alexander Vardy, and Hanwen Yao, Efficient Algorithms for the Bee-Identification Problem, arXiv:2212.09952 [cs.IT], 2022.
Duško Letić, Nenad Cakić, Branko Davidović, Ivana Berković and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations, Vol. 2011 (2011), Article 60.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
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.
Lara Pudwell, Nathan Chenette and Manda Riehl, Statistics on Hypercube Orientations, AMS Special Session on Experimental and Computer Assisted Mathematics, Joint Mathematics Meetings (Denver 2020).
John Riordan and N. J. A. Sloane, Correspondence, 1974.
R. Tosic, D. Masulovic, I. Stojmenovic, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., Vol. 35, No. 2 (1995), pp. 181-187.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Idempotent Number.
Eric Weisstein's World of Mathematics, Halved Cube Graph.
Eric Weisstein's World of Mathematics, Hypercube Graph.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A001787, A001789, A001793 (sum of all nodes of integer compositions, n included).
Cf. A001844, A038207, A290031 (6-cycles).
Row sums of triangle A094305.
Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), this sequence (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

List([0..30], n-> n*(n+1)*2^(n-2)); # G. C. Greubel, Aug 27 2019

a001788 n = if n < 2 then n else n * (n + 1) * 2 ^ (n - 2)
a001788_list = zipWith (*) a000217_list $ 1 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014

[n*(n+1)*2^(n-2): n in [0..30]]; // G. C. Greubel, Aug 27 2019

A001788 := n->n*(n+1)*2^(n-2);
A001788:=-1/(2*z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero

CoefficientList[Series[x/(1-2x)^3, {x,0,30}], x]
Table[n*(n+1)*2^(n-2), {n,0,30}]
With[{n = 30}, Join[{0}, Times @@@ Thread[{Accumulate[Range[n]], 2^Range[0, n - 1]}]]] (* Harvey P. Dale, Jul 16 2013 *)
LinearRecurrence[{6, -12, 8}, {0, 1, 6}, 30] (* Harvey P. Dale, Jul 16 2013 *)

PARI
```
a(n)=if(n<0,0,2^n*n*(n+1)/4)
```

A001788_upto(n)=Vec(x/(1-2*x)^3+O(x^n),-n) \\ for illustration. - M. F. Hasler, Oct 05 2024

[n if n < 2 else n * (n + 1) * 2**(n - 2) for n in range(28)] # Zerinvary Lajos, Mar 10 2009

G.f.: x/(1-2*x)^3.
E.g.f.: x*(1 + x)*exp(2*x).
a(n) = 2*a(n-1) + n*2^(n-1) = 2*a(n-1) + A001787(n).
a(n) = A038207(n+1,2).
a(n) = A055252(n, 2).
a(n) = Sum_{i=1..n} i^2 * binomial(n, i): binomial transform of A000290. - Yong Kong, Dec 26 2000
a(n) = Sum_{j=0..n} binomial(n+1,j)*(n+1-j)^2. - Zerinvary Lajos, Aug 22 2006
If the leading 0 is deleted, the binomial transform of A001844: (1, 5, 13, 25, 41, ...); = double binomial transform of [1, 4, 4, 0, 0, 0, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = Sum_{1<=i<=k<=n} (-1)^(i+1)*i^2*binomial(n+1,k+i)*binomial(n+1,k-i). - Mircea Merca, Apr 09 2012
a(0)=0, a(1)=1, a(2)=6, a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). - Harvey P. Dale, Jul 16 2013
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(3/2) - 4. (End)

A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

Original entry on oeis.org

1, -1, 2, 1, -8, 9, -1, 24, -81, 64, 1, -64, 486, -1024, 625, -1, 160, -2430, 10240, -15625, 7776, 1, -384, 10935, -81920, 234375, -279936, 117649, -1, 896, -45927, 573440, -2734375, 5878656, -5764801, 2097152, 1, -2048, 183708, -3670016, 27343750, -94058496, 161414428, -134217728, 43046721

Offset: 1

Wolfdieter Lang, Oct 02 2002

Coefficients of the Sidi polynomials (-1)^(n-1)*D_{n-1,1,n-1}(x), for n >=1, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980].
The row polynomials p(n, x) := Sum_{m=0..n-1} a(n, m)x^m, n >= 1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x.
The row polynomials p(n, y) := Sum_{m=0..n-1} a(n, m)*y^m, n >= 1, are also obtained from ((d^m/dx^m)((exp(x)-1)^m)/m)/exp(x) after replacement of exp(x) by y. Here (d^m/dx^m)f(x), m >= 1, denotes m-fold differentiation of f(x) with respect to x.
b(k,m,n) := (Sum_{p=0..m-1} (a(m, p)*((p+1)*k)^n))/(m-1)!, n >= 0, has g.f. 1/Product_{p=1..m} (1 - k*p*x) for k = 1, 2,... and m = 1, 2,...
The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n).
The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)*A036071(n-5), (6^5)*A036084(n-6), (7^6)*A036226(n-7), (8^7)*A053107(n-8) for m=0..7.
Right edge of triangle is A000169. - Michel Marcus, May 17 2013

			The triangle T(n, m)  begins:
  n\m 0     1      2        3        4         5         6          7       8
  1:  1
  2: -1     2
  3:  1    -8      9
  4: -1    24    -81       64
  5:  1   -64    486    -1024      625
  6: -1   160  -2430    10240   -15625      7776
  7:  1  -384  10935   -81920   234375   -279936    117649
  8: -1   896 -45927   573440 -2734375   5878656  -5764801    2097152
  9:  1 -2048 183708 -3670016 27343750 -94058496 161414428 -134217728 4304672
  [Reformatted by _Wolfdieter Lang_, Oct 12 2022]
-----------------------------------------------------------------------------
p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.

Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Doron S. Lubinsky and Herbert Stahl, Some Explicit Biorthogonal Polynomials, (IN) Approximation Theory XI, (C.K. Chui, M. Neamtu, L. Schumaker, eds.), Nashboro Press, Nashville, 2005, pp. 279-285.
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.

Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013, A258773.

# Assuming offset 0.
seq(seq((-1)^(n-k)*binomial(n, k)*(k+1)^n, k=0..n), n=0..8);
# Alternative:
egf := x -> 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y):
ser := x -> series(egf(x), x, 12):
row := n -> seq(coeff(n!*coeff(ser(x), x, n), y, k), k=0..n):
seq(print(row(n)), n = 0..8); # Peter Luschny, Oct 21 2022

p[n_, x_] := p[n, x] = Nest[ x*D[#, x]& , (x-1)^n, n]/(n*x); a[n_, m_] := Coefficient[ p[n, x], x, m]; Table[a[n, m], {n, 1, 9}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 03 2013 *)

tabl(nn) = {for (n=1, nn, for (m=0, n-1, print1((-1)^(n-m-1)*binomial(n-1, m)*(m+1)^(n-1), ", ");); print(););} \\ Michel Marcus, May 17 2013

T(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0.
G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0.
E.g.f.: -LambertW(-x*y*exp(-x))/((1+LambertW(-x*y*exp(-x)))*x*y). - Vladeta Jovovic, Feb 13 2008 [corrected for offset 0 <= m <= n. For offset n >= 1 take the integral over x. - Wolfdieter Lang, Oct 12 2022]
T(n, k) = S(n, k+1) / n where S(, ) is triangle in A258773. - Michael Somos, May 13 2018
E.g.f. of column k, with offset n >= 0: exp(-(k + 1)*x)*((k + 1)*x)^k/k!. - Wolfdieter Lang, Oct 20 2022
E.g.f: 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y) assuming offset = 0. - Peter Luschny, Oct 21 2022

A027465 Cube of lower triangular normalized binomial matrix.

Original entry on oeis.org

1, 3, 1, 9, 6, 1, 27, 27, 9, 1, 81, 108, 54, 12, 1, 243, 405, 270, 90, 15, 1, 729, 1458, 1215, 540, 135, 18, 1, 2187, 5103, 5103, 2835, 945, 189, 21, 1, 6561, 17496, 20412, 13608, 5670, 1512, 252, 24, 1, 19683, 59049, 78732, 61236, 30618, 10206, 2268

Offset: 0

Olivier Gérard, N. J. A. Sloane

Rows of A013610 reversed. - Michael Somos, Feb 14 2002
Row sums are powers of 4 (A000302), antidiagonal sums are A006190 (a(n) = 3*a(n-1) + a(n-2)). - Gerald McGarvey, May 17 2005
Triangle of coefficients in expansion of (3+x)^n.
Also: Pure Galton board of scheme (3,1). Also: Multiplicity (number) of pairs of n-dimensional binary vectors with dot product (overlap) k. There are 2^n = A000079(n) binary vectors of length n and 2^(2n) = 4^n = A000302(n) different pairs to form dot products k = Sum_{i=1..n} v[i]*u[i] between these, 0 <= k <= n. (Since dot products are symmetric, there are only 2^n*(2^n-1)/2 different non-ordered pairs, actually.) - R. J. Mathar, Mar 17 2006
Mirror image of A013610. - Zerinvary Lajos, Nov 25 2007
T(i,j) is the number of i-permutations of 4 objects a,b,c,d, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007
The antidiagonals of the sequence formatted as a square array (see Examples section) and summed with alternating signs gives a bisection of Fibonacci sequence, A001906. Example: 81-(27-1)=55. Similar rule applied to rows gives A000079. - Mark Dols, Sep 01 2009
Triangle T(n,k), read by rows, given by (3,0,0,0,0,0,0,0,...)DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011
T(n,k) = binomial(n,k)*3^(n-k), the number of subsets of [2n] with exactly k symmetric pairs, where elements i and j of [2n] form a symmetric pair if i+j=2n+1. Equivalently, if n couples attend a (ticketed) event that offers door prizes, then the number of possible prize distributions that have exactly k couples as dual winners is T(n,k). - Dennis P. Walsh, Feb 02 2012
T(n,k) is the number of ordered pairs (A,B) of subsets of {1,2,...,n} such that the intersection of A and B contains exactly k elements. For example, T(2,1) = 6 because we have ({1},{1}); ({1},{1,2}); ({2},{2}); ({2},{1,2}); ({1,2},{1}); ({1,2},{2}). Sum_{k=0..n} T(n,k)*k = A002697(n) (see comment there by Ross La Haye). - Geoffrey Critzer, Sep 04 2013
Also the convolution triangle of A000244. - Peter Luschny, Oct 09 2022

			Example: n = 3 offers 2^3 = 8 different binary vectors (0,0,0), (0,0,1), ..., (1,1,0), (1,1,1). a(3,2) = 9 of the 2^4 = 64 pairs have overlap k = 2: (0,1,1)*(0,1,1) = (1,0,1)*(1,0,1) = (1,1,0)*(1,1,0) = (1,1,1)*(1,1,0) = (1,1,1)*(1,0,1) = (1,1,1)*(0,1,1) = (0,1,1)*(1,1,1) = (1,0,1)*(1,1,1) = (1,1,0)*(1,1,1) = 2.
For example, T(2,1)=6 since there are 6 subsets of {1,2,3,4} that have exactly 1 symmetric pair, namely, {1,4}, {2,3}, {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}.
The present sequence formatted as a triangular array:
     1
     3     1
     9     6     1
    27    27     9     1
    81   108    54    12    1
   243   405   270    90   15    1
   729  1458  1215   540  135   18   1
  2187  5103  5103  2835  945  189  21  1
  6561 17496 20412 13608 5670 1512 252 24 1
  ...
A013610 formatted as a triangular array:
  1
  1  3
  1  6   9
  1  9  27   27
  1 12  54  108   81
  1 15  90  270  405   243
  1 18 135  540 1215  1458   729
  1 21 189  945 2835  5103  5103  2187
  1 24 252 1512 5670 13608 20412 17496 6561
   ...
A099097 formatted as a square array:
      1     0     0    0   0 0 0 0 0 0 0 ...
      3     1     0    0   0 0 0 0 0 0 ...
      9     6     1    0   0 0 0 0 0 ...
     27    27     9    1   0 0 0 0 ...
     81   108    54   12   1 0 0 ...
    243   405   270   90  15 1 ...
    729  1458  1215  540 135 ...
   2187  5103  5103 2835 ...
   6561 17496 20412 ...
  19683 59049 ...
  59049 ...

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.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).

Cf. A000244, A007318, A013610, A013610, A099097, A027471, A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.

a027465 n k = a027465_tabl !! n !! k
a027465_row n = a027465_tabl !! n
a027465_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 1) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013

for i from 0 to 12 do seq(binomial(i, j)*3^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Nov 25 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 3^(n-1)); # Peter Luschny, Oct 09 2022

t[n_, k_] := Binomial[n, k]*3^(n-k); Table[t[n, n-k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 19 2012 *)

{T(n, k) = polcoeff( (3 + x)^n, k)}; /* Michael Somos, Feb 14 2002 */

Numerators of lower triangle of (b^2)[ i, j ] where b[ i, j ] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
Triangle whose (i, j)-th entry is binomial(i, j)*3^(i-j).
a(n, m) = 4^(n-1)*Sum_{j=m..n} b(n, j)*b(j, m) = 3^(n-m)*binomial(n-1, m-1), n >= m >= 1; a(n, m) := 0, n < m. G.f. for m-th column: (x/(1-3*x))^m (m-fold convolution of A000244, powers of 3). - Wolfdieter Lang, Feb 2006
G.f.: 1 / (1 - x(3+y)).
a(n,k) = 3*a(n-1,k) + a(n-1,k-1) - R. J. Mathar, Mar 17 2006
From the formalism of A133314, the e.g.f. for the row polynomials of A027465 is exp(x*t)*exp(3x). The e.g.f. for the row polynomials of the inverse matrix is exp(x*t)*exp(-3x). p iterates of the matrix give the matrix with e.g.f. exp(x*t)*exp(p*3x). The results generalize for 3 replaced by any number. - Tom Copeland, Aug 18 2008
T(n,k) = A164942(n,k)*(-1)^k. - Philippe Deléham, Oct 09 2011
Let P and P^T be the Pascal matrix and its transpose and H = P^3 = A027465. Then from the formalism of A132440 and A218272,
  exp[x*z/(1-3z)]/(1-3z) = exp(3z D_z z) e^(x*z)= exp(3D_x x D_x) e^(z*x)
  = (1 z z^2 z^3 ...) H (1 x x^2/2! x^3/3! ...)^T
  = (1 x x^2/2! x^3/3! ...) H^T (1 z z^2 z^3 ...)^T = Sum_{n>=0} (3z)^n L_n(-x/3), where D is the derivative operator and L_n(x) are the regular (not normalized) Laguerre polynomials. - Tom Copeland, Oct 26 2012
E.g.f. for column k: x^k/k! * exp(3x). - Geoffrey Critzer, Sep 04 2013

A038845 3-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 12, 96, 640, 3840, 21504, 114688, 589824, 2949120, 14417920, 69206016, 327155712, 1526726656, 7046430720, 32212254720, 146028888064, 657129996288, 2937757630464, 13056700579840, 57724360458240, 253987186016256, 1112705767309312, 4855443348258816

Offset: 0

Wolfdieter Lang

Also convolution of A002802 with A000984 (central binomial coefficients).
With a different offset, number of n-permutations of 5 objects u, v, w, z, x with repetition allowed, containing exactly two u's. - Zerinvary Lajos, Dec 29 2007
Also convolution of A000302 with A002697, also convolution of A002457 with itself. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Adam Ehrenberg, Joseph T. Iosue, Abhinav Deshpande, Dominik Hangleiter, and Alexey V. Gorshkov, The Second Moment of Hafnians in Gaussian Boson Sampling, arXiv:2403.13878 [quant-ph], 2024. See p. 30.
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A000302, A000984, A002457, A002697, A002802, A038231, A052780.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), this sequence (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

List([0..30], n-> 4^n*Binomial(n+2,n) ); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+2, 2): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq((n+2)*(n+1)*4^n/2, n=0..30); # Zerinvary Lajos, Apr 25 2007

Table[4^n*Binomial[n+2,n], {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)

a(n)=(n+2)*(n+1)<<(2*n-1) \\ Charles R Greathouse IV, Aug 21 2015

[4^(n-2)*binomial(n,2) for n in range(2, 30)] # Zerinvary Lajos, Mar 11 2009

a(n) = (n+2)*(n+1)*2^(2*n-1).
G.f.: 1/(1-4*x)^3.
a(n) = Sum_{u+v+w+x+y+z=n} f(u)*f(v)*f(w)*f(x)*f(y)*f(z) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
a(n) = binomial(n+2,n) * 4^n. - Rui Duarte, Oct 08 2011
E.g.f.: (1 + 8*x + 8*x^2)*exp(4*x). - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 8 - 24*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 40*log(5/4) - 8. (End)

A036216 Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 12, 90, 540, 2835, 13608, 61236, 262440, 1082565, 4330260, 16888014, 64481508, 241805655, 892820880, 3252418920, 11708708112, 41712272649, 147219785820, 515269250370, 1789882659180, 6175095174171, 21171754882872

Offset: 0

Wolfdieter Lang

With three leading zeros, 3rd binomial transform of (0,0,0,1,0,0,0,0,...). - Paul Barry, Mar 07 2003

Number of n-permutations (n=4) of 4 objects u, v, w, z, with repetition allowed, containing exactly three u's. - Zerinvary Lajos, May 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), this sequence (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

[3^n* Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

seq(3^n*binomial(n+3, 3), n=0..30)]; # Zerinvary Lajos, Dec 21 2006

CoefficientList[Series[1/(1-3x)^4,{x,0,30}],x] (* or *) LinearRecurrence[ {12,-54,108,-81},{1,12,90,540},30] (* Harvey P. Dale, Jul 27 2017 *)

a(n) = 3^n*binomial(n+3, 3) \\ Charles R Greathouse IV, Oct 03 2016

[3^n*binomial(n+3,3) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+3, 3).
a(n) = A027465(n+4, 4).
G.f.: 1/(1 - 3*x)^4.
With three leading zeros, a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4), a(0) = a(1) = a(2) = 0, a(3) = 1. - Paul Barry, Mar 07 2003
With three leading zeros, C(n, 3)*3^(n-3) is the second binomial transform of C(n, 3). - Paul Barry, Jul 24 2003
E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(3*x). - Franck Maminirina Ramaharo, Nov 23 2018
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 36*log(3/2) - 27/2.
Sum_{n>=0} (-1)^n/a(n) = 144*log(4/3) - 81/2. (End)

A081139 9th binomial transform of (0,0,1,0,0,0,...).

Original entry on oeis.org

0, 0, 1, 27, 486, 7290, 98415, 1240029, 14880348, 172186884, 1937102445, 21308126895, 230127770466, 2447722649502, 25701087819771, 266895911974545, 2745215094595320, 28001193964872264, 283512088894331673

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A001019 (powers of 9).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (27,-243,729).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), this sequence (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

Cf. A001019.

[9^n* Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

LinearRecurrence[{27,-243,729},{0,0,1},30] (* Harvey P. Dale, Jan 30 2018 *)

a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 9^(n-2)*binomial(n, 2).
G.f.: x^2/(1-9*x)^3.
E.g.f.: (x^2/2)*exp(9*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 18 - 144*log(9/8).
Sum_{n>=2} (-1)^n/a(n) = 180*log(10/9) - 18. (End)

A013610 Triangle of coefficients in expansion of (1+3*x)^n.

Original entry on oeis.org

1, 1, 3, 1, 6, 9, 1, 9, 27, 27, 1, 12, 54, 108, 81, 1, 15, 90, 270, 405, 243, 1, 18, 135, 540, 1215, 1458, 729, 1, 21, 189, 945, 2835, 5103, 5103, 2187, 1, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 1, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and three kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. - Joerg Arndt, Jul 01 2011
Rows of A027465 reversed. - Michael Somos, Feb 14 2002
T(n,k) equals the number of n-length words on {0,1,2,3} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 3-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference.  - Brian Hopkins, Aug 16 2020

			Triangle begins
  1;
  1,    3;
  1,    6,    9;
  1,    9,   27,   27;
  1,   12,   54,  108,   81;
  1,   15,   90,  270,  405,  243;
  1,   18,  135,  540, 1215, 1458,  729;
  1,   21,  189,  945, 2835, 5103, 5103, 2187;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients on the 3xn board (all heights 3) with k rooks
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.

Cf. A007318, A013609, A027465, etc.

Diagonals of the triangle: A000244 (k=n), A027471 (k=n-1), A027472 (k=n-2), A036216 (k=n-3), A036217 (k=n-4), A036219 (k=n-5), A036220 (k=n-6), A036221 (k=n-7), A036222 (k=n-8), A036223 (k=n-9), A172362 (k=n-10).

a013610 n k = a013610_tabl !! n !! k
a013610_row n = a013610_tabl !! n
a013610_tabl = iterate (\row ->
   zipWith (+) (map (* 1) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013

[3^k*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 19 2021

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+3*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 25 2015

t[n_, k_] := Binomial[n, k]*3^(n-k); Table[t[n, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2013 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 4], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)
T[0, 0] := 1; T[n_, k_]/;0<=k<=n := T[n, k] = 3T[n-1, k-1]+T[n-1, k]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Oliver Seipel, Jan 26 2025 *)

{T(n, k) = polcoeff((1 + 3*x)^n, k)}; /* Michael Somos, Feb 14 2002 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,1], [1,1]]; /* note triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

flatten([[3^k*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 19 2021

G.f.: 1 / (1 - x*(1+3*y)).
Row sums are 4^n. - Joerg Arndt, Jul 01 2011
T(n,k) = 3^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*2^(n-i). - Mircea Merca, Apr 28 2012
From Peter Bala, Dec 22 2014: (Start)
Riordan array ( 1/(1 - x), 3*x/(1 - x) ).
exp(3*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(3*x)*(1 + 9*x + 27*x^2/2! + 27*x^3/3!) = 1 + 12*x + 90*x^2/2! + 540*x^3/3! + 2835*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 3*x/(1 - x) ). (End)
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 4^j. - Kolosov Petro, Jan 28 2019
T(0,0)=1, T(n,k)=3*T(n-1,k-1)+T(n-1,k) for 0<=k<=n, T(n,k)=0 for k<0 or k>n. - Oliver Seipel, Feb 10 2025

A081135 5th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 15, 150, 1250, 9375, 65625, 437500, 2812500, 17578125, 107421875, 644531250, 3808593750, 22216796875, 128173828125, 732421875000, 4150390625000, 23345947265625, 130462646484375, 724792480468750

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, three-fold convolution of A000351 (powers of 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), this sequence (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[5^(n-2)*Binomial(n, 2): n in [0..30]]; // Vincenzo Librandi, Aug 06 2013

seq(n*(n-1)*5^(n-2)/2, n=0..30); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[x^2/(1-5x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{15,-75,125},{0,0,1},30] (* Harvey P. Dale, Sep 13 2017 *)

[5^(n-2)*binomial(n,2) for n in range(0, 30)] # Zerinvary Lajos, Mar 12 2009

a(n) = 15*a(n-1) - 75*a(n-2) + 125*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 5^(n-2)*binomial(n, 2).
G.f.: x^2/(1-5*x)^3.
E.g.f.: (x^2/2)*exp(5*x). - G. C. Greubel, May 14 2021
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=2} 1/a(n) = 10 - 40*log(5/4).
Sum_{n>=2} (-1)^n/a(n) = 60*log(6/5) - 10. (End)

A081136 6th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 18, 216, 2160, 19440, 163296, 1306368, 10077696, 75582720, 554273280, 3990767616, 28298170368, 198087192576, 1371372871680, 9403699691520, 63945157902336, 431629815840768, 2894458765049856, 19296391766999040

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, three-fold convolution of A000400 (powers of 6).

Number of n-permutations of 7 objects: p, u, v, w, z, x, y with repetition allowed, containing exactly two u's. - Zerinvary Lajos, May 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (18,-108,216).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), this sequence (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[6^n*Binomial(n+2,2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

seq(binomial(n, 2)*6^(n-2), n=0..19); # Zerinvary Lajos, May 23 2008

nn=20;Range[0,nn]!CoefficientList[Series[x^2/2! Exp[6x],{x,0,nn}],x] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{18,-108,216},{0,0,1},30] (* Harvey P. Dale, Apr 20 2022 *)

[6^(n-2)*binomial(n,2) for n in range(0, 21)] # Zerinvary Lajos, Mar 13 2009

a(n) = 18*a(n-1) -108*a(n-2) +216*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 6^(n-2)*C(n, 2).
G.f.: x^2/(1-6*x)^3.
E.g.f.: exp(6*x) * x^2/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=2} 1/a(n) = 12 - 60*log(6/5).
Sum_{n>=2} (-1)^n/a(n) = 84*log(7/6) - 12. (End)

A027474 a(n) = 7^(n-2) * C(n,2).

Original entry on oeis.org

1, 21, 294, 3430, 36015, 352947, 3294172, 29647548, 259416045, 2219448385, 18643366434, 154231485954, 1259557135291, 10173346092735, 81386768741880, 645668365352248, 5084638377148953, 39779817891812397, 309398583602985310

Offset: 2

Olivier Gérard

7th binomial transform of (0,0,1,0,0,0,........). Starting at 1, the three-fold convolution of A000420 (powers of 7). - Paul Barry, Mar 08 2003

Vincenzo Librandi, Table of n, a(n) for n = 2..400
Index entries for linear recurrences with constant coefficients, signature (21,-147,343).

Third column of A027466.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), this sequence (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[7^(n-2)* Binomial(n, 2): n in [2..20]]; /* Vincenzo Librandi, Oct 12 2011 */

seq(binomial(n, 2)*7^(n-2), n=2..30); # Zerinvary Lajos, Jun 12 2008

Table[7^(n-2) Binomial[n,2], {n,2,20}] (* Harvey P. Dale, Sep 25 2011 *)

a(n)=7^(n-2)*n*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

[7^(n-2)*binomial(n,2) for n in range(2, 21)] # Zerinvary Lajos, Mar 13 2009

From Paul Barry, Mar 08 2003: (Start)
G.f.: x^2 / (1-7*x)^3.
a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3), a(0) = a(1) = 0, a(2) = 1. (End)
Numerators of sequence a[3,n] in (a[i,j])^3 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
E.g.f.: (x^2/2)*exp(7*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 14 - 84*log(7/6).
Sum_{n>=2} (-1)^n/a(n) = 112*log(8/7) - 14. (End)

Edited by Ralf Stephan, Dec 30 2004

cp's OEIS Frontend

A001788 a(n) = n*(n+1)*2^(n-2).

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A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

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A027465 Cube of lower triangular normalized binomial matrix.

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A038845 3-fold convolution of A000302 (powers of 4).

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A036216 Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3).

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A081139 9th binomial transform of (0,0,1,0,0,0,...).

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A001788 a(n) = n(n+1)2^(n-2).