A059304 -id:A059304 - cp's OEIS frontend

A001018 Powers of 8: a(n) = 8^n.

1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656, 2251799813685248, 18014398509481984, 144115188075855872, 1152921504606846976, 9223372036854775808, 73786976294838206464, 590295810358705651712, 4722366482869645213696

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 8), L(1, 8), P(1, 8), T(1, 8). Essentially same as Pisot sequences E(8, 64), L(8, 64), P(8, 64), T(8, 64). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1..2n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1..2n} -> {1,2,3} such that for fixed y_1,y_2,...,y_n in {1,2,3} we have f(X_i)<>{y_i}, (i=1..n). - Milan Janjic, May 24 2007
This is the auto-convolution (convolution square) of A059304. - R. J. Mathar, May 25 2009
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 8-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) is equal to the determinant of a 3 X 3 matrix with rows 2^(n+2), 2^(n+1), 2^n; 2^(n+3), 2^(n+4), 2(n+3); 2^n, 2^(n+1), 2^(n+2) when it is divided by 144. - J. M. Bergot, May 07 2014
a(n) gives the number of small squares in the n-th iteration of the Sierpinski carpet fractal. Equivalently, the number of vertices in the n-Sierpinski carpet graph. - Allan Bickle, Nov 27 2022

			For n=1, the 1st order Sierpinski carpet graph is an 8-cycle.

K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2017; p. 15.
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
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 273
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Caroline Nunn, A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021). See table at p. 9.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Sierpiński Carpet
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A013730, A103333, A013731, A067417, A083233, A055274.
Cf. A008588, A016969, A157176.
Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001019 (powers of 9), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49), A165800 (powers of 50), A159991 (powers of 60).
Cf. A032766 (floor(3*n/2)).
Cf. A271939 (number of edges in the n-Sierpinski carpet graph).

a001018 = (8 ^)
a001018_list = iterate (* 8) 1  -- Reinhard Zumkeller, Apr 29 2015

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

seq(8^n, n=0..23); # Nathaniel Johnston, Jun 26 2011
A001018 := n -> 8^n; # M. F. Hasler, Apr 19 2015

Table[8^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(8^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=8^n \\ Charles R Greathouse IV, May 10 2014

print([8**n for n in range(25)]) # Michael S. Branicky, Dec 29 2021

a(n) = 8^n.
a(0) = 1; a(n) = 8*a(n-1) for n > 0.
G.f.: 1/(1-8*x).
E.g.f.: exp(8*x).
Sum_{n>=0} 1/a(n) = 8/7. - Gary W. Adamson, Aug 29 2008
a(n) = A157176(A008588(n)); a(n+1) = A157176(A016969(n)). - Reinhard Zumkeller, Feb 24 2009
From Stefano Spezia, Dec 28 2021: (Start)
a(n) = (-1)^n*(1 + sqrt(-3))^(3*n) (see Nunn, p. 9).
a(n) = (-1)^n*Sum_{k=0..floor(3*n/2)} (-3)^k*binomial(3*n, 2*k) (see Nunn, p. 9). (End)

A038207 Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 8, 12, 6, 1, 16, 32, 24, 8, 1, 32, 80, 80, 40, 10, 1, 64, 192, 240, 160, 60, 12, 1, 128, 448, 672, 560, 280, 84, 14, 1, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 1024, 5120, 11520, 15360, 13440, 8064, 3360, 960, 180, 20, 1

Offset: 0

N. J. A. Sloane

This infinite matrix is the square of the Pascal matrix (A007318) whose rows are [ 1,0,... ], [ 1,1,0,... ], [ 1,2,1,0,... ], ...
As an upper right triangle, table rows give number of points, edges, faces, cubes,
4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009
Number of different partial sums of 1+[1,1,2]+[2,2,3]+[3,3,4]+[4,4,5]+... with entries that are zero removed. - Jon Perry, Jan 01 2004
Row sums are powers of 3 (A000244), antidiagonal sums are Pell numbers (A000129). - Gerald McGarvey, May 17 2005
Riordan array (1/(1-2x), x/(1-2x)). - Paul Barry, Jul 28 2005
T(n,k) is the number of elements of the Coxeter group B_n with descent set contained in {s_k}, 0<=k<=n-1. For T(n,n), we interpret this as the number of elements of B_n with empty descent set (since s_n does not exist). - Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006
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 T(n,k) = the number of elements (x,y) of S for which y has exactly k more elements than x. - Ross La Haye, Oct 12 2007
T(n,k) is number of paths in the first quadrant going from (0,0) to (n,k) using only steps B=(1,0) colored blue, R=(1,0) colored red and U=(1,1). Example: T(3,2)=6 because we have BUU, RUU, UBU, URU, UUB and UUR. - Emeric Deutsch, Nov 04 2007
T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (0,1), and two kinds of step (1,0). - Joerg Arndt, Jul 01 2011
T(i,j) is the number of i-permutations of {1,2,3} containing j 1's. Example: T(2,1)=4 because we have 12, 13, 21 and 31; T(3,2)=6 because we have 112, 113, 121, 131, 211 and 311. - Zerinvary Lajos, Dec 21 2007
Triangle of coefficients in expansion of (2+x)^n. - N-E. Fahssi, Apr 13 2008
Sum of diagonals are Jacobsthal-numbers: A001045. - Mark Dols, Aug 31 2009
Triangle T(n,k), read by rows, given by [2,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009
Eigensequence of the triangle = A004211: (1, 3, 11, 49, 257, 1539, ...). - Gary W. Adamson, Feb 07 2010
f-vectors ("face"-vectors) for n-dimensional cubes [see e.g., Hoare]. (This is a restatement of Bottomley's above.) - Tom Copeland, Oct 19 2012
With P = Pascal matrix, the sequence of matrices I, A007318, A038207, A027465, A038231, A038243, A038255, A027466 ... = P^0, P^1, P^2, ... are related by Copeland's formula below to the evolution at integral time steps n= 0, 1, 2, ... of an exponential distribution exp(-x*z) governed by the Fokker-Planck equation as given in the Dattoli et al. ref. below. - Tom Copeland, Oct 26 2012
The matrix elements of the inverse are T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 12 2013
Unsigned diagonals of A133156 are rows of this array. - Tom Copeland, Oct 11 2014
Omitting the first row, this is the production matrix for A039683, where an equivalent differential operator can be found. - Tom Copeland, Oct 11 2016
T(n,k) is the number of functions f:[n]->[3] with exactly k elements mapped to 3. Note that there are C(n,k) ways to choose the k elements mapped to 3, and there are 2^(n-k) ways to map the other (n-k) elements to {1,2}. Hence, by summing T(n,k) as k runs from 0 to n, we obtain 3^n = Sum_{k=0..n} T(n,k). - Dennis P. Walsh, Sep 26 2017
Since this array is the square of the Pascal lower triangular matrix, the row polynomials of this array are obtained as the umbral composition of the row polynomials P_n(x) of the Pascal matrix with themselves. E.g., P_3(P.(x)) = 1 P_3(x) + 3 P_2(x) + 3 P_1(x) + 1 = (x^3 + 3 x^2 + 3 x + 1) + 3 (x^2 + 2 x + 1) + 3 (x + 1) + 1 = x^3 + 6 x^2 + 12 x + 8. - Tom Copeland, Nov 12 2018
T(n,k) is the number of 2-compositions of n+1 with some zeros allowed that have k zeros; see the Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
Also the convolution triangle of A000079. - Peter Luschny, Oct 09 2022

			Triangle begins with T(0,0):
   1;
   2,  1;
   4,  4,  1;
   8, 12,  6,  1;
  16, 32, 24,  8,  1;
  32, 80, 80, 40, 10,  1;
  ... -  corrected by _Clark Kimberling_, Aug 05 2011
Seen as an array read by descending antidiagonals:
[0] 1, 2,  4,   8,    16,    32,    64,     128,     256, ...     [A000079]
[1] 1, 4,  12,  32,   80,    192,   448,    1024,    2304, ...    [A001787]
[2] 1, 6,  24,  80,   240,   672,   1792,   4608,    11520, ...   [A001788]
[3] 1, 8,  40,  160,  560,   1792,  5376,   15360,   42240, ...   [A001789]
[4] 1, 10, 60,  280,  1120,  4032,  13440,  42240,   126720, ...  [A003472]
[5] 1, 12, 84,  448,  2016,  8064,  29568,  101376,  329472, ...  [A054849]
[6] 1, 14, 112, 672,  3360,  14784, 59136,  219648,  768768, ...  [A002409]
[7] 1, 16, 144, 960,  5280,  25344, 109824, 439296,  1647360, ... [A054851]
[8] 1, 18, 180, 1320, 7920,  41184, 192192, 823680,  3294720, ... [A140325]
[9] 1, 20, 220, 1760, 11440, 64064, 320320, 1464320, 6223360, ... [A140354]

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 155.
H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

T. D. Noe, Rows n=0..100 of triangle, flattened
Peter Bala, A note on the diagonals of a proper Riordan Array
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Jhon J. Bravo, Jose L. Herrera, and José L. Ramírez, Combinatorial Interpretation of Generalized Pell Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.2.1.
John Cartan, Starmaze: Cartan's Triangle.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras.
B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), 109-121.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular Structure 376 (1996), 495-505.
G. Dattoli, A. Mancho, M. Quattromini and A. Torre, Exponential operators, generalized polynomials and evolution problems, Radiation Physics and Chemistry 61 (2001), 99-108. [From _Tom Copeland_, Oct 25 2012]
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
Russell Jay Hendel, A Method for Uniformly Proving a Family of Identities, arXiv:2107.03549 [math.CO], 2021.
Graham Hoare, Hypercubes and Chebyshev, Math. Gaz. 74 (470) (1990), 375-377.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Katarzyna Kril and Wojciech Mlotkowski, Permutations of Type B with Fixed Number of Descents and Minus Signs, The Electronic Journal of Combinatorics, Vol. 26(1) (2019), Article P1.27.
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.
Thomas Selig and Haoyue Zhu, Complete non-ambiguous trees and associated permutations: connections through the Abelian sandpile model, arXiv:2303.15756 [math.CO], 2023, see p. 27.
Wikipedia, Hypercube.

See A065109 for a signed version.
Columns 0-10 are A000079, A001787, A001788(n-1), A001789, A003472, A054849, A002409(n-6), A054851, A140325(n-8), A140354(n-9), A172242.
T(2n,n) gives A059304.
Cf. A007318, A013609, A013610, A062715, A099089, A004211, A135387.
Cf. A001861, A008277, A027465, A027466, A038231, A038243, A038255, A039683, A112857, A118801, A132440, A133156, A133314, A167374, A218272, A238385, A323939.

Flat(List([0..15], n->List([0..n], k->Binomial(n, k)*2^(n-k)))); # Stefano Spezia, Nov 21 2018

a038207 n = a038207_list !! n
a038207_list = concat $ iterate ([2,1] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

a038207' n k = a038207_tabl !! n !! k
a038207_row n = a038207_tabl !! n
a038207_tabl = iterate f [1] where
   f row = zipWith (+) ([0] ++ row) (map (* 2) row ++ [0])
-- Reinhard Zumkeller, Feb 27 2013

/* As triangle */ [[(&+[Binomial(n,i)*Binomial(i,k): i in [k..n]]): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Nov 16 2018

for i from 0 to 12 do seq(binomial(i, j)*2^(i-j), j = 0 .. i) end do; # yields sequence in triangular form - Emeric Deutsch, Nov 04 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 2^(n-1)); # Peter Luschny, Oct 09 2022

Table[CoefficientList[Expand[(y + x + x^2)^n], y] /. x -> 1, {n, 0,10}] // TableForm (* Geoffrey Critzer, Nov 20 2011 *)
Table[Binomial[n,k]2^(n-k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, May 22 2020 *)

{T(n, k) = polcoeff((x+2)^n, k)}; /* Michael Somos, Apr 27 2000 */

def A038207_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in range(dim): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+2*M[n-1,k]
    return M
A038207_triangle(9)  # Peter Luschny, Sep 20 2012

T(n, k) = Sum_{i=0..n} binomial(n,i)*binomial(i,k).
T(n, k) = (-1)^k*A065109(n,k).
G.f.: 1/(1-2*z-t*z). - Emeric Deutsch, Nov 04 2007
Rows of the triangle are generated by taking successive iterates of (A135387)^n * [1, 0, 0, 0, ...]. - Gary W. Adamson, Dec 09 2007
From the formalism of A133314, the e.g.f. for the row polynomials of A038207 is exp(x*t)*exp(2x). The e.g.f. for the row polynomials of the inverse matrix is exp(x*t)*exp(-2x). p iterates of the matrix give the matrix with e.g.f. exp(x*t)*exp(p*2x). The results generalize for 2 replaced by any number. - Tom Copeland, Aug 18 2008
Sum_{k=0..n} T(n,k)*x^k = (2+x)^n. - Philippe Deléham, Dec 15 2009
n-th row is obtained by taking pairwise sums of triangle A112857 terms starting from the right. - Gary W. Adamson, Feb 06 2012
T(n,n) = 1 and T(n,k) = T(n-1,k-1) + 2*T(n-1,k) for kJon Perry, Oct 11 2012
The e.g.f. for the n-th row is given by umbral composition of the normalized Laguerre polynomials A021009 as p(n,x) = L(n, -L(.,-x))/n! = 2^n L(n, -x/2)/n!. E.g., L(2,x) = 2 -4*x +x^2, so p(2,x)= (1/2)*L(2, -L(.,-x)) = (1/2)*(2*L(0,-x) + 4*L(1,-x) + L(2,-x)) = (1/2)*(2 + 4*(1+x) + (2+4*x+x^2)) = 4 + 4*x + x^2/2. - Tom Copeland, Oct 20 2012
From Tom Copeland, Oct 26 2012: (Start)
From the formalism of A132440 and A218272:
Let P and P^T be the Pascal matrix and its transpose and H= P^2= A038207.
Then with D the derivative operator,
   exp(x*z/(1-2*z))/(1-2*z)= exp(2*z D_z z) e^(x*z)= exp(2*D_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} z^n * 2^n Lag_n(-x/2)= exp[z*EF(.,x)], an o.g.f. for the f-vectors (rows) of A038207 where EF(n,x) is an e.g.f. for the n-th f-vector. (Lag_n(x) are the un-normalized Laguerre polynomials.)
Conversely,
   exp(z*(2+x))= exp(2D_x) exp(x*z)= exp(2x) exp(x*z)
   = (1 x x^2 x^3 ...) H^T (1 z z^2/2! z^3/3! ...)^T
   = (1 z z^2/2! z^3/3! ...) H (1 x x^2 x^3 ...)^T
   = exp(z*OF(.,x)), an e.g.f for the f-vectors of A038207 where
     OF(n,x)= (2+x)^n is an o.g.f. for the n-th f-vector.
(End)
G.f.: R(0)/2, where R(k) = 1 + 1/(1 - (2*k+1+ (1+y))*x/((2*k+2+ (1+y))*x + 1/R(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
A038207 = exp[M*B(.,2)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). B(n,2) = A001861(n). - Tom Copeland, Apr 17 2014
T = (A007318)^2 = A112857*|A167374| = |A118801|*|A167374| = |A118801*A167374| = |P*A167374*P^(-1)*A167374| = |P*NpdP*A167374|. Cf. A118801. - Tom Copeland, Nov 17 2016
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial 2^n*Sum_{k = 0..n} binomial(n,k)*x^k/k!. For example, the e.g.f. for the third subdiagonal is exp(x)*(8 + 24*x + 12*x^2 + 4*x^3/3) = 8 + 32*x + 80*x^2/2! + 160*x^3/3! + .... - Peter Bala, Mar 05 2017
T(3*k+2,k) = T(3*k+2,k+1), T(2*k+1,k) = 2*T(2*k+1,k+1). - Yuchun Ji, May 26 2020
From Robert A. Russell, Aug 05 2020: (Start)
G.f. for column k: x^k / (1-2*x)^(k+1).
E.g.f. for column k: exp(2*x) * x^k / k!. (End)
Also the array A(n, k) read by descending antidiagonals, where A(n, k) = (-1)^n*Sum_{j= 0..n+k} binomial(n + k, j)*hypergeom([-n, j+1], [1], 1). - Peter Luschny, Nov 09 2021

A013609 Triangle of coefficients in expansion of (1+2*x)^n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024

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 two 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
Also sum of rows in A046816. - Lior Manor, Apr 24 2004
Also square array of unsigned coefficients of Chebyshev polynomials of second kind. - Philippe Deléham, Aug 12 2005
The rows give the number of k-simplices in the n-cube. For example, 1, 6, 12, 8 shows that the 3-cube has 1 volume, 6 faces, 12 edges and 8 vertices. - Joshua Zucker, Jun 05 2006
Triangle whose (i, j)-th entry is binomial(i, j)*2^j.
With offset [1,1] the triangle with doubled numbers, 2*a(n,m), enumerates sequences of length m with nonzero integer entries n_i satisfying sum(|n_i|) <= n. Example n=4, m=2: [1,3], [3,1], [2,2] each in 2^2=4 signed versions: 2*a(4,2) = 2*6 = 12. The Sum over m (row sums of 2*a(n,m)) gives 2*3^(n-1), n >= 1. See the W. Lang comment and a K. A. Meissner reference under A024023. - Wolfdieter Lang, Jan 21 2008
n-th row of the triangle = leftmost column of nonzero terms of X^n, where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (2,2,2,...) in the subdiagonal. - Gary W. Adamson, Jul 19 2008
Numerators of a matrix square-root of Pascal's triangle A007318, where the denominators for the n-th row are set to 2^n. - Gerald McGarvey, Aug 20 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
The triangle sums (see A180662 for their definitions) link the Pell-Jacobsthal triangle, whose mirror image is A038207, with twenty-four different sequences; see the crossrefs.
This triangle may very well be called the Pell-Jacobsthal triangle in view of the fact that A000129 (Kn21) are the Pell numbers and A001045 (Kn11) the Jacobsthal numbers.
(End)
T(n,k) equals the number of n-length words on {0,1,2} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 2-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
T(n,k) is the number of chains 0=x_0Geoffrey Critzer, Oct 01 2022
Excluding the initial 1, T(n,k) is the number of k-faces of a regular n-cross polytope. See A038207 for n-cube and A135278 for n-simplex. - Mohammed Yaseen, Jan 14 2023

			Triangle begins:
  1;
  1,  2;
  1,  4,   4;
  1,  6,  12,    8;
  1,  8,  24,   32,   16;
  1, 10,  40,   80,   80,    32;
  1, 12,  60,  160,  240,   192,    64;
  1, 14,  84,  280,  560,   672,   448,    128;
  1, 16, 112,  448, 1120,  1792,  1792,   1024,    256;
  1, 18, 144,  672, 2016,  4032,  5376,   4608,   2304,    512;
  1, 20, 180,  960, 3360,  8064, 13440,  15360,  11520,   5120,  1024;
  1, 22, 220, 1320, 5280, 14784, 29568,  42240,  42240,  28160, 11264,  2048;
  1, 24, 264, 1760, 7920, 25344, 59136, 101376, 126720, 112640, 67584, 24576, 4096;
From _Peter Bala_, Apr 20 2012: (Start)
The triangle can be written as the matrix product A038207*(signed version of A013609).
  |.1................||.1..................|
  |.2...1............||-1...2..............|
  |.4...4...1........||.1..-4...4..........|
  |.8..12...6...1....||-1...6...-12...8....|
  |16..32..24...8...1||.1..-8....24.-32..16|
  |..................||....................|
(End)

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
G. Hotz, Zur Reduktion von Schaltkreispolynomen im Hinblick auf eine Verwendung in Rechenautomaten, El. Datenverarbeitung, Folge 5 (1960), pp. 21-27.

T. D. Noe, Rows n=0..50 of triangle, flattened
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
John Cartan, Cartan's triangle shows the relationship to the n-cube.
R. Ehrenborg and M. Readdy, Characterization of the factorial functions of Eulerian binomial and Sheffer posets
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients of k rooks on the 2xn board, all heights 2.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

Cf. A007318, A013610, etc.
Appears in A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
Triangle sums (see the comments): A000244 (Row1); A000012 (Row2); A001045 (Kn11); A026644 (Kn12); 4*A011377 (Kn13); A000129 (Kn21); A094706 (Kn22); A099625 (Kn23); A001653 (Kn3); A007583 (Kn4); A046717 (Fi1); A007051 (Fi2); A077949 (Ca1); A008998 (Ca2); A180675 (Ca3); A092467 (Ca4); A052942 (Gi1); A008999 (Gi2); A180676 (Gi3); A180677 (Gi4); A140413 (Ze1); A180678 (Ze2); A097117 (Ze3); A055588 (Ze4).
(End)
Cf. A024023, A038207, A046816, A105728, A115068, A135278, A171977, A180662.
T(2n,n) gives A059304.

a013609 n = a013609_list !! n
a013609_list = concat $ iterate ([1,2] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

a013609 n k = a013609_tabl !! n !! k
a013609_row n = a013609_tabl !! n
a013609_tabl = iterate (\row -> zipWith (+) ([0] ++ row) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 22 2013, Feb 27 2013

[2^k*Binomial(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2021

bin2:=proc(n,k) option remember; if k<0 or k>n then 0 elif k=0 then 1 else 2*bin2(n-1,k-1)+bin2(n-1,k); fi; end; # N. J. A. Sloane, Jun 01 2009

Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* Jean-François Alcover, May 17 2011 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 3], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)

a(n,k):=coeff(expand((1+2*x)^n),x^k);
create_list(a(n,k),n,0,6,k,0,n); /* Emanuele Munarini, Nov 21 2012 */

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

flatten([[2^k*binomial(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 17 2021

G.f.: 1 / (1 - x*(1+2*y)).
T(n,k) = 2^k*binomial(n,k).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k). - Jon Perry, Nov 22 2005
Row sums are 3^n = A000244(n). - Joerg Arndt, Jul 01 2011
T(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i). - Mircea Merca, Apr 28 2012
E.g.f.: exp(2*y*x + x). - Geoffrey Critzer, Nov 12 2012
Riordan array (x/(1 - x), 2*x/(1 - x)). Exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(1 + 6*x + 12*x^2/2! + 8*x^3/3!) = 1 + 8*x + 40*x^2/2! + 160*x^3/3! + 560*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1 - x)). - Peter Bala, Dec 21 2014
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 3^j. - Kolosov Petro, Jan 28 2019
T(n,k) = 2*(n+1-k)*T(n,k-1)/k, T(n,0) = 1. - Alexander R. Povolotsky, Oct 08 2023
For n >= 1, GCD(T(n,1), ..., T(n,n)) = GCD(T(n,1),T(n,n)) = GCD(2*n,2^n) = A171977(n). - Pontus von Brömssen, Nov 01 2024

A036442 a(n) = 2^((n-1)*(n+2)/2).

Original entry on oeis.org

1, 4, 32, 512, 16384, 1048576, 134217728, 34359738368, 17592186044416, 18014398509481984, 36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, 20282409603651670423947251286016, 664613997892457936451903530140172288

Offset: 1

Abdallah Rayhan (rayhan(AT)engr.uvic.ca)

Number of redundant paths for a fault-tolerant ATM switch.
Hankel transform (see A001906 for definition ) of A001850, A006139, A084601; also Hankel transform of the sequence 1, 0, 4, 0, 24, 0, 160, 0, 1120, ... (A059304 with interpolated zeros). - Philippe Deléham, Jul 03 2005
Hankel transform of A109980. Unsigned version of A127945. - Philippe Deléham, Dec 11 2008
a(n) = the multiplicative Wiener index of the wheel graph with n+3 vertices. The multiplicative Wiener index of a connected simple graph G is defined as the product of the distances between all pairs of distinct vertices of G. The wheel graph with n+3 vertices has (n+3)(n+2)/2 pairs of distinct vertices, of which 2(n+2) are adjacent; each of the remaining (n+2)(n-1)/2 pairs are at distance 2; consequently, the multiplicative Wiener index is 2^((n-1)(n+2)/2) = a(n). - Emeric Deutsch, Aug 17 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..80
P. Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161, 2013 and J. Int. Seq. 17 (2014) # 14.5.1
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
C. Lo and C. Chiu, A Fault-Tolerant Architecture for ATM Networks, 20th IEEE Conf. Local Computer Networks, 1995, pp. 29-36
Index to divisibility sequences

I:=[1]; [n le 1 select I[n] else Self(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Oct 24 2012

Table[2^((n-1) * (n+2)/2), {n, 1, 30}] (* Vincenzo Librandi, Oct 24 2012 *)

A036442[n]:=2^((n-1)*(n+2)/2)$
makelist(A036442[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

a(n)=2^((n-1)*(n+2)/2) \\ Charles R Greathouse IV, Oct 24 2012

a(1) = 1, a(n) = a(n-1) * 2^n. - Vincenzo Librandi, Oct 24 2012

A004981 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 1).

Original entry on oeis.org

1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, 7049900, 52169260, 388898120, 2916735900, 21987701400, 166478310600, 1265235160560, 9647418099270, 73774373700300, 565603531702300, 4346216612028200, 33465867912617140, 258165266754475080, 1994913424920943800

Offset: 0

Joe Keane (jgk(AT)jgk.org)

The convolution of this sequence with itself yields A059304. - T. D. Noe, Jun 11 2002

Conjecture: a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 4) and all positive integers n. Cf. A004982 and A298799. - Peter Bala, Dec 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, arXiv preprint arXiv:1108.2993 [hep-ph], 2011.

Cf. A002897, A004982, A059304, A127776, A298799.

List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+1)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [2^n*&*[4*k+1: k in [0..n-1]]/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019

A004981 := n -> (-8)^n*binomial(-1/4, n):
seq(A004981(n), n=0..25); # Peter Luschny, Oct 23 2018

CoefficientList[Series[(1-8x)^(-1/4), {x, 0, 25}], x] (* Vincenzo Librandi, Mar 16 2014 *)
Table[8^n*Pochhammer[1/4, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)

a(n):=if n=0 then 1 else (sum(m*sum(binomial(-m+2*k-1,k-1) *2^(n+m-k)*binomial(2*n-k-1,n-1),k,m,n),m,1,n))/(n); /* Vladimir Kruchinin, Dec 26 2011 */

PARI
```
a(n)=if(n<0,0,prod(k=1,n,(8*k-6)/k))
```

{a(n)=if(n<0, 0, polcoeff( (1-8*x+x*O(x^n))^(-1/4), n))} /* Michael Somos, Jan 31 2007 */

[8^n*rising_factorial(1/4, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

a(n) ~ Gamma(1/4)^-1*n^(-3/4)*2^(3*n)*{1 - 3/32*n^-1 - ...}
G.f.: (1-8*x)^(-1/4).
A002897(n) = Sum_{k=0..n} a(k)^2*a(n-k)^2. - Michael Somos, Jan 31 2007
a(n) = (Sum_{m=1..n} m*Sum_{k=m..n} binomial(-m+2*k-1,k-1)*2^(n+m-k)*binomial(2*n-k-1,n-1))/n, n>0, a(0)=1. - Vladimir Kruchinin, Dec 26 2011
D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1). - R. J. Mathar, Mar 14 2014
From Karol A. Penson, Dec 19 2015: (Start)
a(n) = (-8)^n*binomial(-1/4,n).
E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([1/4], [1], 8*x).
Representation as n-th moment of a positive function on (0, 8): a(n)=int(x^n*(sqrt(2)/(16*Pi*(x/8)^(3/4)*(1-x/8)^(1/4))), x=0..8), n=0, 1, ... . This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End)

More terms from James Sellers, May 01 2000

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

Original entry on oeis.org

1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725

Offset: 0

Paul D. Hanna, Jun 10 2003

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - N-E. Fahssi, Mar 30 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Sums of squares of coefficients of (1+2*x)^n. - Joerg Arndt, Jul 06 2011
The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007
Partial sums of A085363. - J. M. Bergot, Jun 12 2013
Diagonal of rational functions 1/(1 - x - y - 3*x*y), 1/(1 - x - y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Tewodros Amdeberhan, In search of multiple expressions for a sequence
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Curtis Greene, Posets of shuffles, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).
Christopher Huffaker, Nathan McCue, Cameron N. Miller, and Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], 2015.
Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See p. 3.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Cf. A001850, A059231, A059304, A246923 (a(n)^2).

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*4^k)); # Muniru A Asiru, Jul 29 2018

[3^n*Evaluate(LegendrePolynomial(n), 5/3) : n in [0..40]]; // G. C. Greubel, May 30 2023

seq(simplify(hypergeom([-n,1/2], [1], -8)),n=0..19); # Peter Luschny, Apr 26 2016

Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *)
Table[Hypergeometric2F1[-n, -n, 1, 4], {n,0,30}] (* Vladimir Reshetnikov, Nov 29 2013 *)
CoefficientList[Series[1/Sqrt[1-10x+9x^2],{x,0,30}],x] (* Harvey P. Dale, Mar 08 2016 *)
Table[3^n*LegendreP[n, 5/3], {n, 0, 40}] (* G. C. Greubel, May 30 2023 *)

{a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0,n, binomial(n, k)^2 * 4^k))}; /* Michael Somos, Oct 08 2003 */

{a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* Michael Somos, Oct 08 2003 */

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

a(n)={local(v=Vec((1+2*x)^n));sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2+imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */

[3^n*gen_legendre_P(n, 0, 5/3) for n in range(41)] # G. C. Greubel, May 30 2023

G.f.: 1 / sqrt(1 - 10*x + 9*x^2).
From Vladeta Jovovic, Aug 20 2003: (Start)
Binomial transform of A059304.
G.f.: Sum_{k >= 0} binomial(2*k,k)*(2*x)^k/(1-x)^(k+1).
E.g.f.: exp(5*x)*BesselI(0, 4*x). (End)
a(n) = Sum_{k = 0..n} Sum_{j = 0..n-k} C(n,j)*C(n-j,k)*C(2*n-2*j,n-j). - Paul Barry, May 19 2006
a(n) = Sum_{k = 0..n} 4^k*C(n,k)^2. - heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010
a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Sep 11 2012
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - R. J. Mathar, Nov 26 2012
a(n) = hypergeom([-n, -n], [1], 4). - Vladimir Reshetnikov, Nov 29 2013
a(n) = hypergeom([-n, 1/2], [1], -8). - Peter Luschny, Apr 26 2016
From Michael Somos, Jun 01 2017: (Start)
a(n) = -3 * 9^n * a(-1-n) for all n in Z.
0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End)
From Peter Bala, Nov 13 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 5*x^2 + 29*x^3 + 185*x^4 + 1257*x^5 + ... is the g.f. of A059231.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all positive integers n and r and all primes p. (End)
a(n) = 3^n * LegendreP(n, 5/3). - G. C. Greubel, May 30 2023
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A098410 Expansion of 1/(sqrt(1-4x)sqrt(1-8*x)).

Original entry on oeis.org

1, 6, 38, 252, 1734, 12276, 88796, 652728, 4856902, 36478404, 275975028, 2099978568, 16054486044, 123213933576, 948713646072, 7325088811632, 56692748053062, 439689331938276, 3416328042565124, 26587566855421608, 207218159714453044, 1617124976299315224, 12634892752595949192

Offset: 0

Paul Barry, Sep 07 2004

Convolution of A000984(n) and 2^n*A000984(n). Convolution of A000984(n) and A059304. 4th binomial transform of A000984.
Largest coefficient of (1 + 6*x + x^2)^n. - Philippe Deléham, Oct 02 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 6 colors. - N-E. Fahssi, Mar 31 2008
Self-convolution of a(n)/4^n gives A126646. - Vladimir Reshetnikov, Oct 10 2016
Diagonal of rational function 1/(1 - (x^2 + 6*x*y + y^2)). - Gheorghe Coserea, Aug 03 2018

			G.f. = 1 + 6*x + 38*x^2 + 252*x^3 + 1734*x^4 + 12276*x^5 + 88796*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011 (the sequence b_{6,n}).
Rigoberto Flórez, Leandro Junes and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column 6 of A292627. Cf. A025230, A104454 (binomial transf.)

Table[SeriesCoefficient[1/(Sqrt[1-4*x]*Sqrt[1-8*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
a[ n_] := If[n < 0, 0, 4^n Hypergeometric2F1[-n, 1/2, 1, -1]]; (* Michael Somos, May 06 2017 *)
a[ n_] := SeriesCoefficient[ D[ InverseJacobiSD[2 x, -1] / 2, x], {x, 0, 2 n}]; (* Michael Somos, May 06 2017 *)

x='x+O('x^66); Vec(1/sqrt(1-12*x+32*x^2)) \\ Joerg Arndt, May 11 2013

{a(n) = sum(k=0, n, 8^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 6^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

a = lambda n: 4^n*hypergeometric([-n, 1/2], [1], -1)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

G.f.: 1/sqrt(1 - 12*x + 32*x^2).
E.g.f.: exp(6*x)*BesselI(0, 2*x).
a(n) = Sum_{k=0..n} 2^k*binomial(2*k, k)*binomial(2*(n-k), n-k).
a(n) = Sum_{k=0..n} 4^(n-k)*binomial(n,k)*binomial(2k,k). - Paul Barry, Mar 08 2005
D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) - 32*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ 2^(3*n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 15 2012
a(n) = 4^n*hypergeometric([-n, 1/2], [1], -1). - Peter Luschny, May 19 2015
a(n) = Sum_{k=0..n} 8^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 6^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
From Peter Bala, Jan 10 2022: (Start)
3*x + x^2*exp(Sum_{n >= 1} a(n)*x^n/n) = 3*x + x^2 + 6*x^3 + 37*x^4 + 234*x^5 + 1514*x^6 + ... is the o.g.f. of A025230.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
a(n) = (1/Pi) * Integral_{x = -1..1} (4 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (8 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)

A103488 a(n) = 2^(n^2-1).

Original entry on oeis.org

1, 8, 256, 32768, 16777216, 34359738368, 281474976710656, 9223372036854775808, 1208925819614629174706176, 633825300114114700748351602688, 1329227995784915872903807060280344576, 11150372599265311570767859136324180752990208

Offset: 1

Zerinvary Lajos, Jul 30 2006

From the classic story: number of grains of wheat on last square of n X n chessboard.
Hankel transform of A059304 and of A084771. - Philippe Deléham, Dec 02 2007
The number of compositions of n^2. - Alois P. Heinz, Feb 22 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..50
Peter Kagey and William Keehn, Counting Tilings of the n X m Grid, Cylinder, and Torus, J. Int. Seq. (2024) Vol. 27, Art. No. 24.6.1. See section A, pp. 6, 10.

Cf. A000079, A000290, A005563, A011782.

[2^(n^2-1): n in [1..15]]; // Vincenzo Librandi, Feb 25 2014

Maple
```
[seq(2^(n^2-1),n=1..14)];
```

Table[2^(n^2 - 1), {n, 1, 20}] (* Vincenzo Librandi, Feb 25 2014 *)

a(n) = 2^(n^2-1); \\ Joerg Arndt, Feb 23 2014

a(n) = A000079(A005563(n-1)). - Michel Marcus, Feb 25 2014

a(n) = A011782(A000290(n)). - Alois P. Heinz, Feb 22 2020

A053175 Catalan-Larcombe-French sequence.

Original entry on oeis.org

1, 8, 80, 896, 10816, 137728, 1823744, 24862720, 346498048, 4911669248, 70560071680, 1024576061440, 15008466534400, 221460239482880, 3287994183188480, 49074667327062016, 735814252604162048

Offset: 0

Peter J Larcombe, Nov 12 2001

These numbers were proposed as 'Catalan' numbers by an associate of Catalan. They appear as coefficients in the series expansion of an elliptic integral of the first kind. Defining f(x; c) = 1 /(1 - c^2*sin^2(x))^(1/2), consider the function I(c) obtained by integrating f(x; c) with respect to x between 0 and Pi/2. I(c) is transformed and written as a power series in c (through an intermediate variable) which acts as a generating function for the sequence.

Conjecture: Let P(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,n. Then P(n)/2^(n*(n+3)) is a positive odd integer. - Zhi-Wei Sun, Aug 14 2013

			G.f. = 1 + 8*x + 80*x^2 + 896*x^3 + 10816*x^4 + 137728*x^5 + 1823774*x^6 + ...

P. J. Larcombe, D. R. French and E. J. Fennessey, The asymptotic behavior of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Utilitas Mathematica, 60 (2001), 67-77.
P. J. Larcombe, D. R. French and C. A. Woodham, A note on the asymptotic behavior of a prime factor decomposition of the general Catalan-Larcombe-French number, Congressus Numerantium, 156 (2002), 17-25.

T. D. Noe, Table of n, a(n) for n=0..200
E. Catalan, Sur les Nombres de Segner, Rend. Circ. Mat. Pal., 1 (1887), 190-201. [From _Peter Luschny_, Jun 26 2009]
Lane Clark, An asymptotic expansion for the Catalan-Larcombe-French sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.1.
A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congressus Numerantium, 169 (2004), 79-99.
A. F. Jarvis, P. J. Larcombe and D. R. French, Applications of the a.g.m. of Gauss: some new properties of the Catalan-Larcombe-French sequence, Congressus Numerantium, 161 (2003), 151-162.
A. F. Jarvis, P. J. Larcombe and D. R. French, Power series identities generated by two recent integer sequences, Bulletin ICA, 43 (2005), 85-95.
A. F. Jarvis, P. J. Larcombe and D. R. French, On Small Prime Divisibility of the Catalan-Larcombe-French sequence, Indian Journal of Mathematics, 47 (2005), 159-181.
A. F. Jarvis, P. J. Larcombe and D. R. French, A short proof of the 2-adic valuation of the Catalan-Larcombe-French number, Indian Journal of Mathematics, 48 (2006), 135-138.
F. Jarvis, H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.
Xiao-Juan Ji, Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015 and JIS vol 19 (2016) # 16.3.4
P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.
Peter J. Larcombe, Daniel R. French, On the “Other” Catalan Numbers: A Historical Formulation Re-Examined, Congressus Numerantium, 143 (2000), 33-64.
P. J. Larcombe and D. R. French, On the integrality of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Congressus Numerantium, 148 (2001), 65-91.
P. J. Larcombe and D. R. French, A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic, Congressus Numerantium, 166 (2004), 161-172.
Guo-Shuai Mao, Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers, arXiv:1511.06222 [math.NT], 2015.
Brian Yi Sun, Baoyindureng Wu, Two-log-convexity of the Catalan-Larcombe-French sequence, arXiv:1602.04909 [math.CO], 2016. Also Journal of Inequalities and Applications, 2015, 2015:404; DOI: 10.1186/s13660-015-0920-0.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
N. M. Temme, Examples of 3_F_2-polynomials, Asymptotic Methods for Integrals, Chapter 13, pp. 167-179 (2014).
Yang Wen, On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers, American Journal of Mathematical and Computer Modelling, 2017; 2(4): 95-98.
E. X. W. Xia and O. X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, 20 (2013), #P3.

Cf. A065409, A002894, A081085.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 8 else (8*(3*n^2 -3*n+1)*a(n-1)-128*(n-1)^2*a(n-2))/n^2 fi end; # Peter Luschny, Jun 26 2009

a[ n_] := SeriesCoefficient[ EllipticK[ (8 x /(1 - 8 x))^2] / ((1 - 8 x) Pi/2), {x, 0, n}]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 8 x] BesselI[ 0, 4 x]^2, {x, 0, n}]]; (* Michael Somos, Aug 01 2011 *)
Table[(-8)^n Sqrt[Pi] HypergeometricPFQRegularized[{1/2, -n, -n}, {1, 1/2 - n}, -1]/n!, {n, 0, 20}] (* Vladimir Reshetnikov, May 21 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 16*x + x * O(x^n)), n))}; /* Michael Somos, Feb 12 2003 */

{a(n) = if( n<0, 0, polcoeff( sum( k=0, n, binomial( 2*k ,k)^2 * (2*x - 16*x^2)^k, x * O(x^n)), n))}; /* Michael Somos, Mar 04 2003 */

G.f.: 1 / AGM(1, 1 - 16*x) = 2 * EllipticK(8*x / (1-8*x)) / ((1-8*x)*Pi), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. Cf. A081085, A089602. - Michael Somos, Mar 04 2003 and Vladeta Jovovic, Dec 30 2003
E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - Vladeta Jovovic, Aug 20 2003
a(n)*n^2 = a(n-1)*8*(3*n^2 - 3*n + 1) - a(n-2)*128*(n-1)^2. - Michael Somos, Apr 01 2003
Exponential convolution of A059304 with itself: Sum(2^n*binomial(2*n, n)*x^n/n!, n=0..infinity)^2 = (BesselI(0, 4*x)*exp(4*x))^2 = hypergeom([1/2], [1], 8*x)^2. - Vladeta Jovovic, Sep 09 2003
a(n) ~ 2^(4n+1)/(Pi*n). - Vaclav Kotesovec, Oct 09 2012
a(n) = 2^n*Sum_{k=0..n} C(n,k)*C(2*k,k)*C(2(n-k),n-k), where C(n,k)=n!/(k!*(n-k)!). This formula has been proved via the Zeilberger algorithm (both sides of the equality satisfy the same recurrence relation). a(n)/2^n also has another expression: Sum_{k=0..floor(n/2)} C(n,2*k)*C(2*k,k)^2*4^(n-2*k). - Zhi-Wei Sun, Mar 21 2013
a(n) = (-1)^n*Sum_{k=0..n}C(2*k,k)*C(2(n-k),n-k)*C(k,n-k)*(-4)^k. I have proved this new formula via the Zeilberger algorithm. - Zhi-Wei Sun, Nov 19 2014

A067001 Triangle T(n,k) = d(n-k,n), 0 <= k <= n, where d(l,m) = Sum_{k=l..m} 2^k * binomial(2m-2k, m-k) * binomial(m+k, m) * binomial(k, l).

Original entry on oeis.org

1, 4, 6, 24, 60, 42, 160, 560, 688, 308, 1120, 5040, 8760, 7080, 2310, 8064, 44352, 99456, 114576, 68712, 17556, 59136, 384384, 1055040, 1572480, 1351840, 642824, 134596, 439296, 3294720, 10695168, 19536000, 21778560, 14912064, 5864640, 1038312

Offset: 0

N. J. A. Sloane, Feb 16 2002

For an explanation on how this triangular array is related to the Boros-Moll polynomial P_n(x) and the theory in Comtet (1967), see my comments in A223549. For example, the bivariate o.g.f. below follows from the theory in Comtet (1967). - Petros Hadjicostas, May 24 2020

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) starts:
     1;
     4,    6;
    24,   60,   42;
   160,  560,  688,  308;
  1120, 5040, 8760, 7080, 2310;
  ...

Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87.
V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
V. H. Moll, Combinatorial sequences arising from a rational integral, Onl. J. Anal. Combin. no 2 (2007) #4.

Column k=0 gives A059304.
Row sums give A002458.
Main diagonal gives A004982.
Cf. A223549, A223550.

d := proc(l,m) local k; add(2^k*binomial(2*m-2*k,m-k)*binomial(m+k,m)*binomial(k,l),k=l..m); end:
T:= (n, k)-> d(n-k, n):
seq(seq(T(n, k), k=0..n), n=0..10);

T[n_, k_] := SeriesCoefficient[Sqrt[(1+y)/(1 - 8x (1+y))/(1 + y Sqrt[1 - 8x (1+y)])], {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 05 2020 *)

d(l, m) = sum(kk=l, m, 2^kk*binomial(2*m-2*kk,m-kk)*binomial(m+kk,m)*binomial(kk,l));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(d(n-k, n), ", ");); print(););} \\ Michel Marcus, Jul 18 2015

From Petros Hadjicostas, May 24 2020: (Start)
T(n,k) = 2^(2*n)*A223549(n,n-k)/A223550(n,n-k).
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = sqrt((1 + y)/(1 - 8*x*(1 + y))/(1 + y*sqrt(1 - 8*x*(1 + y)))). (End)

cp's OEIS Frontend

A001018 Powers of 8: a(n) = 8^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Python

Formula

A038207 Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

A013609 Triangle of coefficients in expansion of (1+2*x)^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A036442 a(n) = 2^((n-1)*(n+2)/2).

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

Maxima

PARI

Formula

A004981 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

A098410 Expansion of 1/(sqrt(1-4x)sqrt(1-8*x)).