A002409 -id:A002409 - cp's OEIS frontend

A000079 Powers of 2: a(n) = 2^n.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

N. J. A. Sloane

2^0 = 1 is the only odd power of 2.
Number of subsets of an n-set.
There are 2^(n-1) compositions (ordered partitions) of n (see for example Riordan). This is the unlabeled analog of the preferential labelings sequence A000670.
This is also the number of weakly unimodal permutations of 1..n + 1, that is, permutations with exactly one local maximum. E.g., a(4) = 16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - Jon Perry, Jul 27 2003 [Proof: see next line! See also A087783.]
Proof: n must appear somewhere and there are 2^(n-1) possible choices for the subset that precedes it. These must appear in increasing order and the rest must follow n in decreasing order. QED. - N. J. A. Sloane, Oct 26 2003
a(n+1) is the smallest number that is not the sum of any number of (distinct) earlier terms.
Same as Pisot sequences E(1, 2), L(1, 2), P(1, 2), T(1, 2). See A008776 for definitions of Pisot sequences.
With initial 1 omitted, same as Pisot sequences E(2, 4), L(2, 4), P(2, 4), T(2, 4). - David W. Wilson
Not the sum of two or more consecutive numbers. - Lekraj Beedassy, May 14 2004
Least deficient or near-perfect numbers (i.e., n such that sigma(n) = A000203(n) = 2n - 1). - Lekraj Beedassy, Jun 03 2004. [Comment from Max Alekseyev, Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2.]
Almost-perfect numbers referred to as least deficient or slightly defective (Singh 1997) numbers. Does "near-perfect numbers" refer to both almost-perfect numbers (sigma(n) = 2n - 1) and quasi-perfect numbers (sigma(n) = 2n + 1)? There are no known quasi-perfect or least abundant or slightly excessive (Singh 1997) numbers.
The sum of the numbers in the n-th row of Pascal's triangle; the sum of the coefficients of x in the expansion of (x+1)^n.
The Collatz conjecture (the hailstone sequence will eventually reach the number 1, regardless of which positive integer is chosen initially) may be restated as (the hailstone sequence will eventually reach a power of 2, regardless of which positive integer is chosen initially).
The only hailstone sequence which doesn't rebound (except "on the ground"). - Alexandre Wajnberg, Jan 29 2005
With p(n) as the number of integer partitions of n, p(i) is the number of parts of the i-th partition of n, d(i) is the number of different parts of the i-th partition of n, m(i,j) is the multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i = 1..p(n)} (p(i)! / (Product_{j=1..d(i)} m(i,j)!)). - Thomas Wieder, May 18 2005
The number of binary relations on an n-element set that are both symmetric and antisymmetric. Also the number of binary relations on an n-element set that are symmetric, antisymmetric and transitive.
The first differences are the sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
a(n) is the largest number with shortest addition chain involving n additions. - David W. Wilson, Apr 23 2006
Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - Giovanni Teofilatto, Aug 06 2006
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n} -> {1, 2} such that for a fixed x in {1, 2, ..., n} and a fixed y in {1, 2} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
Let P(A) be the power set of an n-element set A. Then a(n) is the number of pairs of elements {x,y} of P(A) for which x = y. - Ross La Haye, Jan 09 2008
a(n) is the number of permutations on [n+1] such that every initial segment is an interval of integers. Example: a(3) counts 1234, 2134, 2314, 2341, 3214, 3241, 3421, 4321. The map "p -> ascents of p" is a bijection from these permutations to subsets of [n]. An ascent of a permutation p is a position i such that p(i) < p(i+1). The permutations shown map to 123, 23, 13, 12, 3, 2, 1 and the empty set respectively. - David Callan, Jul 25 2008
2^(n-1) is the largest number having n divisors (in the sense of A077569); A005179(n) is the smallest. - T. D. Noe, Sep 02 2008
a(n) appears to match the number of divisors of the modified primorials (excluding 2, 3 and 5). Very limited range examined, PARI example shown. - Bill McEachen, Oct 29 2008
Successive k such that phi(k)/k = 1/2, where phi is Euler's totient function. - Artur Jasinski, Nov 07 2008
A classical transform consists (for general a(n)) in swapping a(2n) and a(2n+1); examples for Jacobsthal A001045 and successive differences: A092808, A094359, A140505. a(n) = A000079 leads to 2, 1, 8, 4, 32, 16, ... = A135520. - Paul Curtz, Jan 05 2009
This is also the (L)-sieve transform of {2, 4, 6, 8, ..., 2n, ...} = A005843. (See A152009 for the definition of the (L)-sieve transform.) - John W. Layman, Jan 23 2009
a(n) = a(n-1)-th even natural number (A005843) for n > 1. - Jaroslav Krizek, Apr 25 2009
For n >= 0, a(n) is the number of leaves in a complete binary tree of height n. For n > 0, a(n) is the number of nodes in an n-cube. - K.V.Iyer, May 04 2009
Permutations of n+1 elements where no element is more than one position right of its original place. For example, there are 4 such permutations of three elements: 123, 132, 213, and 312. The 8 such permutations of four elements are 1234, 1243, 1324, 1423, 2134, 2143, 3124, and 4123. - Joerg Arndt, Jun 24 2009
Catalan transform of A099087. - R. J. Mathar, Jun 29 2009
a(n) written in base 2: 1,10,100,1000,10000,..., i.e., (n+1) times 1, n times 0 (A011557(n)). - Jaroslav Krizek, Aug 02 2009
Or, phi(n) is equal to the number of perfect partitions of n. - Juri-Stepan Gerasimov, Oct 10 2009
These are the 2-smooth numbers, positive integers with no prime factors greater than 2. - Michael B. Porter, Oct 04 2009
A064614(a(n)) = A000244(n) and A064614(m) < A000244(n) for m < a(n). - Reinhard Zumkeller, Feb 08 2010
a(n) is the largest number m such that the number of steps of iterations of {r - (largest divisor d < r)} needed to reach 1 starting at r = m is equal to n. Example (a(5) = 32): 32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4; 4 - 2 = 2; 2 - 1 = 1; number 32 has 5 steps and is the largest such number. See A105017, A064097, A175125. - Jaroslav Krizek, Feb 15 2010
a(n) is the smallest proper multiple of a(n-1). - Dominick Cancilla, Aug 09 2010
The powers-of-2 triangle T(n, k), n >= 0 and 0 <= k <= n, begins with: {1}; {2, 4}; {8, 16, 32}; {64, 128, 256, 512}; ... . The first left hand diagonal T(n, 0) = A006125(n + 1), the first right hand diagonal T(n, n) = A036442(n + 1) and the center diagonal T(2*n, n) = A053765(n + 1). Some triangle sums, see A180662, are: Row1(n) = A122743(n), Row2(n) = A181174(n), Fi1(n) = A181175(n), Fi2(2*n) = A181175(2*n) and Fi2(2*n + 1) = 2*A181175(2*n + 1). - Johannes W. Meijer, Oct 10 2010
Records in the number of prime factors. - Juri-Stepan Gerasimov, Mar 12 2011
Row sums of A152538. - Gary W. Adamson, Dec 10 2008
A078719(a(n)) = 1; A006667(a(n)) = 0. - Reinhard Zumkeller, Oct 08 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 2-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Equals A001405 convolved with its right-shifted variant: (1 + 2x + 4x^2 + ...) = (1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + ...) * (1 + x + x^2 + 2x^3 + 3x^4 + 6x^5 + ...). - Gary W. Adamson, Nov 23 2011
The number of odd-sized subsets of an n+1-set. For example, there are 2^3 odd-sized subsets of {1, 2, 3, 4}, namely {1}, {2}, {3}, {4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, and {2, 3, 4}. Also, note that 2^n = Sum_{k=1..floor((n+1)/2)} C(n+1, 2k-1). - Dennis P. Walsh, Dec 15 2011
a(n) is the number of 1's in any row of Pascal's triangle (mod 2) whose row number has exactly n 1's in its binary expansion (see A007318 and A047999). (The result of putting together A001316 and A000120.) - Marcus Jaiclin, Jan 31 2012
A204455(k) = 1 if and only if k is in this sequence. - Wolfdieter Lang, Feb 04 2012
For n>=1 apparently the number of distinct finite languages over a unary alphabet, whose minimum regular expression has alphabetic width n (verified up to n=17), see the Gruber/Lee/Shallit link. - Hermann Gruber, May 09 2012
First differences of A000225. - Omar E. Pol, Feb 19 2013
This is the lexicographically earliest sequence which contains no arithmetic progression of length 3. - Daniel E. Frohardt, Apr 03 2013
a(n-2) is the number of bipartitions of {1..n} (i.e., set partitions into two parts) such that 1 and 2 are not in the same subset. - Jon Perry, May 19 2013
Numbers n such that the n-th cyclotomic polynomial has a root mod 2; numbers n such that the n-th cyclotomic polynomial has an even number of odd coefficients. - Eric M. Schmidt, Jul 31 2013
More is known now about non-power-of-2 "Almost Perfect Numbers" as described in Dagal. - Jonathan Vos Post, Sep 01 2013
Number of symmetric Ferrers diagrams that fit into an n X n box. - Graham H. Hawkes, Oct 18 2013
Numbers n such that sigma(2n) = 2n + sigma(n). - Jahangeer Kholdi, Nov 23 2013
a(1), ..., a(floor(n/2)) are all values of permanent on set of square (0,1)-matrices of order n>=2 with row and column sums 2. - Vladimir Shevelev, Nov 26 2013
Numbers whose base-2 expansion has exactly one bit set to 1, and thus has base-2 sum of digits equal to one. - Stanislav Sykora, Nov 29 2013
A072219(a(n)) = 1. - Reinhard Zumkeller, Feb 20 2014
a(n) is the largest number k such that (k^n-2)/(k-2) is an integer (for n > 1); (k^a(n)+1)/(k+1) is never an integer (for k > 1 and n > 0). - Derek Orr, May 22 2014
If x = A083420(n), y = a(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
The mini-sequence b(n) = least number k > 0 such that 2^k ends in n identical digits is given by {1, 18, 39}. The repeating digits are {2, 4, 8} respectively. Note that these are consecutive powers of 2 (2^1, 2^2, 2^3), and these are the only powers of 2 (2^k, k > 0) that are only one digit. Further, this sequence is finite. The number of n-digit endings for a power of 2 with n or more digits id 4*5^(n-1). Thus, for b(4) to exist, one only needs to check exponents up to 4*5^3 = 500. Since b(4) does not exist, it is clear that no other number will exist. - Derek Orr, Jun 14 2014
The least number k > 0 such that 2^k ends in n consecutive decreasing digits is a 3-number sequence given by {1, 5, 25}. The consecutive decreasing digits are {2, 32, 432}. There are 100 different 3-digit endings for 2^k. There are no k-values such that 2^k ends in '987', '876', '765', '654', '543', '321', or '210'. The k-values for which 2^k ends in '432' are given by 25 mod 100. For k = 25 + 100*x, the digit immediately before the run of '432' is {4, 6, 8, 0, 2, 4, 6, 8, 0, 2, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus, we see the digit before '432' will never be a 5. So, this sequence is complete. - Derek Orr, Jul 03 2014
a(n) is the number of permutations of length n avoiding both 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Numbers n such that sigma(n) = sigma(2n) - phi(4n). - Farideh Firoozbakht, Aug 14 2014
This is a B_2 sequence: for i < j, differences a(j) - a(i) are all distinct. Here 2*a(n) < a(n+1) + 1, so a(n) - a(0) < a(n+1) - a(n). - Thomas Ordowski, Sep 23 2014
a(n) counts n-walks (closed) on the graph G(1-vertex; 1-loop, 1-loop). - David Neil McGrath, Dec 11 2014
a(n-1) counts walks (closed) on the graph G(1-vertex; 1-loop, 2-loop, 3-loop, 4-loop, ...). - David Neil McGrath, Jan 01 2015
b(0) = 4; b(n+1) is the smallest number not in the sequence such that b(n+1) - Prod_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n) = a(n) for n > 2. - Derek Orr, Jan 15 2015
a(n) counts the permutations of length n+2 whose first element is 2 such that the permutation has exactly one descent. - Ran Pan, Apr 17 2015
a(0)-a(30) appear, with a(26)-a(30) in error, in tablet M 08613 (see CDLI link) from the Old Babylonian period (c. 1900-1600 BC). - Charles R Greathouse IV, Sep 03 2015
Subsequence of A028982 (the squares or twice squares sequence). - Timothy L. Tiffin, Jul 18 2016
A000120(a(n)) = 1. A000265(a(n)) = 1. A000593(a(n)) = 1. - Juri-Stepan Gerasimov, Aug 16 2016
Number of monotone maps f : [0..n] -> [0..n] which are order-increasing (i <= f(i)) and idempotent (f(f(i)) = f(i)). In other words, monads on the n-th ordinal (seen as a posetal category). Any monad f determines a subset of [0..n] that contains n, by considering its set of monad algebras = fixed points { i | f(i) = i }. Conversely, any subset S of [0..n] containing n determines a monad on [0..n], by the function i |-> min { j | i <= j, j in S }. - Noam Zeilberger, Dec 11 2016
Consider n points lying on a circle. Then for n>=2 a(n-2) gives the number of ways to connect two adjacent points with nonintersecting chords. - Anton Zakharov, Dec 31 2016
Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
Also the number of independent vertex sets and vertex covers in the n-empty graph. - Eric W. Weisstein, Sep 21 2017
Also the number of maximum cliques in the n-halved cube graph for n > 4. - Eric W. Weisstein, Dec 04 2017
Number of pairs of compositions of n corresponding to a seaweed algebra of index n-1. - Nick Mayers, Jun 25 2018
The multiplicative group of integers modulo a(n) is cyclic if and only if n = 0, 1, 2. For n >= 3, it is a product of two cyclic groups. - Jianing Song, Jun 27 2018
k^n is the determinant of n X n matrix M_(i, j) = binomial(k + i + j - 2, j) - binomial(i+j-2, j), in this case k=2. - Tony Foster III, May 12 2019
Solutions to the equation Phi(2n + 2*Phi(2n)) = 2n. - M. Farrokhi D. G., Jan 03 2020
a(n-1) is the number of subsets of {1,2,...,n} which have an element that is the size of the set. For example, for n = 4, a(3) = 8 and the subsets are {1}, {1,2}, {2,3}, {2,4}, {1,2,3}, {1,3,4}, {2,3,4}, {1,2,3,4}. - Enrique Navarrete, Nov 21 2020
a(n) is the number of self-inverse (n+1)-order permutations with 231-avoiding. E.g., a(3) = 8: [1234, 1243, 1324, 1432, 2134, 2143, 3214, 4321]. - Yuchun Ji, Feb 26 2021
For any fixed k > 0, a(n) is the number of ways to tile a strip of length n+1 with tiles of length 1, 2, ... k, where the tile of length k can be black or white, with the restriction that the first tile cannot be black. - Greg Dresden and Bora Bursalı, Aug 31 2023

			There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }.

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N. J. A. Sloane, Table of n, 2^n for n = 0..1000
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Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Eric Weisstein's World of Mathematics, Empty Graph
Eric Weisstein's World of Mathematics, Erf
Eric Weisstein's World of Mathematics, Fractional Part
Eric Weisstein's World of Mathematics, Halved Cube Graph
Eric Weisstein's World of Mathematics, Hypercube
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Least Deficient Number
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, PowerFractional Parts
Eric Weisstein's World of Mathematics, Subset
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Index entries for sequences related to binary expansion of n
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for "core" sequences
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to Benford's law

Cf. A000225, A038754, A133464, A140730, A037124, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A000041, A152537, A001405, A007318, A000120, A000265, A000593, A001227, A077020, A077021.
This is the Hankel transform (see A001906 for the definition) of A000984, A002426, A026375, A026387, A026569, A026585, A026671 and A032351. - John W. Layman, Jul 31 2000
Euler transform of A001037, A209406 (multisets), inverse binomial transform of A000244, binomial transform of A000012.
Complement of A057716.
Boustrophedon transforms: A000734, A000752.
Range of values of A006519, A007875, A011782, A030001, A034444, A037445, A053644, and A054243.
Cf. A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (sum of 2, ..., 9 distinct powers of 2).
Cf. A090129.
The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).

a000079 = (2 ^)
a000079_list = iterate (* 2) 1
-- Reinhard Zumkeller, Jan 22 2014, Mar 05 2012, Dec 29 2011

[2^n: n in [0..40]]; // Vincenzo Librandi, Feb 17 2014

[n le 2 select n else 5*Self(n-1)-6*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 17 2014

A000079 := n->2^n; [ seq(2^n,n=0..50) ];
isA000079 := proc(n)
    local fs;
    fs := numtheory[factorset](n) ;
    if n = 1 then
        true ;
    elif nops(fs) <> 1 then
        false;
    elif op(1,fs) = 2 then
        true;
    else
        false ;
    end if;
end proc: # R. J. Mathar, Jan 09 2017

Table[2^n, {n, 0, 50}]
2^Range[0, 50] (* Wesley Ivan Hurt, Jun 14 2014 *)
LinearRecurrence[{2}, {2}, {0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)
CoefficientList[Series[1/(1 - 2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
NestList[2# &, 1, 40] (* Harvey P. Dale, Oct 07 2019 *)

A000079(n):=2^n$ makelist(A000079(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A000079(n)=2^n \\ Edited by M. F. Hasler, Aug 27 2014

unimodal(n)=local(x,d,um,umc); umc=0; for (c=0,n!-1, x=numtoperm(n,c); d=0; um=1; for (j=2,n,if (x[j]x[j-1] && d==1,um=0); if (um==0,break)); if (um==1,print(x)); umc+=um); umc

def a(n): return 1<Michael S. Branicky, Jul 28 2022

def is_powerof2(n) -> bool: return n and (n & (n - 1)) == 0  # Peter Luschny, Apr 10 2025

(List.fill(20)(2: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, Jan 16 2020

(define (A000079 n) (expt 2 n)) ;; Antti Karttunen, Mar 21 2017

a(n) = 2^n.
a(0) = 1; a(n) = 2*a(n-1).
G.f.: 1/(1 - 2*x).
E.g.f.: exp(2*x).
a(n)= Sum_{k = 0..n} binomial(n, k).
a(n) is the number of occurrences of n in A000523. a(n) = A001045(n) + A001045(n+1). a(n) = 1 + Sum_{k = 0..(n - 1)} a(k). The Hankel transform of this sequence gives A000007 = [1, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Feb 25 2004
n such that phi(n) = n/2, for n > 1, where phi is Euler's totient (A000010). - Lekraj Beedassy, Sep 07 2004
a(n + 1) = a(n) XOR 3*a(n) where XOR is the binary exclusive OR operator. - Philippe Deléham, Jun 19 2005
a(n) = StirlingS2(n + 1, 2) + 1. - Ross La Haye, Jan 09 2008
a(n+2) = 6a(n+1) - 8a(n), n = 1, 2, 3, ... with a(1) = 1, a(2) = 2. - Yosu Yurramendi, Aug 06 2008
a(n) = ka(n-1) + (4 - 2k)a(n-2) for any integer k and n > 1, with a(0) = 1, a(1) = 2. - Jaume Oliver Lafont, Dec 05 2008
a(n) = Sum_{l_1 = 0..n + 1} Sum_{l_2 = 0..n}...Sum_{l_i = 0..n - i}...Sum_{l_n = 0..1} delta(l_1, l_2, ..., l_i, ..., l_n) where delta(l_1, l_2, ..., l_i, ..., l_n) = 0 if any l_i <= l_(i+1) and l_(i+1) != 0 and delta(l_1, l_2, ..., l_i, ..., l_n) = 1 otherwise. - Thomas Wieder, Feb 25 2009
a(0) = 1, a(1) = 2; a(n) = a(n-1)^2/a(n-2), n >= 2. - Jaume Oliver Lafont, Sep 22 2009
a(n) = A173786(n, n)/2 = A173787(n + 1, n). - Reinhard Zumkeller, Feb 28 2010
If p[i] = i - 1 and if A is the Hessenberg matrix of order n defined by: A[i, j] = p[j - i + 1], (i <= j), A[i, j] = -1, (i = j + 1), and A[i, j] = 0 otherwise. Then, for n >= 1, a(n-1) = det A. - Milan Janjic, May 02 2010
If p[i] = Fibonacci(i-2) and if A is the Hessenberg matrix of order n defined by: A[i, j] = p[j - i + 1], (i <= j), A[i, j] = -1, (i = j + 1), and A[i, j] = 0 otherwise. Then, for n >= 2, a(n-2) = det A. - Milan Janjic, May 08 2010
The sum of reciprocals, 1/1 + 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 2. - Mohammad K. Azarian, Dec 29 2010
a(n) = 2*A001045(n) + A078008(n) = 3*A001045(n) + (-1)^n. - Paul Barry, Feb 20 2003
a(n) = A118654(n, 2).
a(n) = A140740(n+1, 1).
a(n) = A131577(n) + A011782(n) = A024495(n) + A131708(n) + A024493(n) = A000749(n) + A038503(n) + A038504(n) + A038505(n) = A139761(n) + A139748(n) + A139714(n) + A133476(n) + A139398(n). - Paul Curtz, Jul 25 2011
a(n) = row sums of A007318. - Susanne Wienand, Oct 21 2011
a(n) = Hypergeometric([-n], [], -1). - Peter Luschny, Nov 01 2011
G.f.: A(x) = B(x)/x, B(x) satisfies B(B(x)) = x/(1 - x)^2. - Vladimir Kruchinin, Nov 10 2011
a(n) = Sum_{k = 0..n} A201730(n, k)*(-1)^k. - Philippe Deléham, Dec 06 2011
2^n = Sum_{k = 1..floor((n+1)/2)} C(n+1, 2k-1). - Dennis P. Walsh, Dec 15 2011
A209229(a(n)) = 1. - Reinhard Zumkeller, Mar 07 2012
A001227(a(n)) = 1. - Reinhard Zumkeller, May 01 2012
Sum_{n >= 1} mobius(n)/a(n) = 0.1020113348178103647430363939318... - R. J. Mathar, Aug 12 2012
E.g.f.: 1 + 2*x/(U(0) - x) where U(k) = 6*k + 1 + x^2/(6*k+3 + x^2/(6*k + 5 + x^2/U(k+1) )); (continued fraction, 3-step). - Sergei N. Gladkovskii, Dec 04 2012
a(n) = det(|s(i+2,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013
a(n) = det(|ps(i+1,j)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013
G.f.: W(0), where W(k) = 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/( 2*x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
a(n-1) = Sum_{t_1 + 2*t_2 + ... + n*t_n = n} multinomial(t_1 + t_2 + ... + t_n; t_1, t_2, ..., t_n). - Mircea Merca, Dec 06 2013
Construct the power matrix T(n,j) = [A^*j]*[S^*(j-1)] where A(n)=(1,1,1,...) and S(n)=(0,1,0,0,...) (where * is convolution operation). Then a(n-1) = Sum_{j=1..n} T(n,j). - David Neil McGrath, Jan 01 2015
a(n) = A000005(A002110(n)). - Ivan N. Ianakiev, May 23 2016
From Ilya Gutkovskiy, Jul 18 2016: (Start)
Exponential convolution of A000012 with themselves.
a(n) = Sum_{k=0..n} A011782(k).
Sum_{n>=0} a(n)/n! = exp(2) = A072334.
Sum_{n>=0} (-1)^n*a(n)/n! = exp(-2) = A092553. (End)
G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = A090129(x) = (1 + 2x + 2x^2 + 4x^3 + 8x^4 + ...). - Gary W. Adamson, Sep 13 2016
a(n) = A000045(n + 1) + A000045(n) + Sum_{k = 0..n - 2} A000045(k + 1)*2^(n - 2 - k). - Melvin Peralta, Dec 22 2017
a(n) = 7*A077020(n)^2 + A077021(n)^2, n>=3. - Ralf Steiner, Aug 08 2021
a(n)= n + 1 + Sum_{k=3..n+1} (2*k-5)*J(n+2-k), where Jacobsthal number J(n) = A001045(n). - Michael A. Allen, Jan 12 2022
Integral_{x=0..Pi} cos(x)^n*cos(n*x) dx = Pi/a(n) (see Nahin, pp. 69-70). - Stefano Spezia, May 17 2023

Clarified a comment T. D. Noe, Aug 30 2009
Edited by Daniel Forgues, May 12 2010
Incorrect comment deleted by Matthew Vandermast, May 17 2014
Comment corrected to match offset by Geoffrey Critzer, Nov 28 2014

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

A091894 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having k ddu's [here u = (1,1) and d = (1,-1)].

Original entry on oeis.org

1, 1, 2, 4, 1, 8, 6, 16, 24, 2, 32, 80, 20, 64, 240, 120, 5, 128, 672, 560, 70, 256, 1792, 2240, 560, 14, 512, 4608, 8064, 3360, 252, 1024, 11520, 26880, 16800, 2520, 42, 2048, 28160, 84480, 73920, 18480, 924, 4096, 67584, 253440, 295680, 110880, 11088, 132

Offset: 0

Emeric Deutsch, Mar 10 2004

Number of Dyck paths of semilength n, having k uu's with midpoint at even height. Example: T(4,1) = 6 because we have u(uu)duddd, u(uu)udddd, udu(uu)ddd, u(uu)dddud, u(uu)ddudd and uud(uu)ddd [here u = (1,1), d = (1,-1) and the uu's with midpoint at even height are shown between parentheses]. Row sums are the Catalan numbers (A000108). T(2n+1,n) = A000108(n) (the Catalan numbers). Sum_{k>=0} k*T(n,k) = binomial(2n-2,n-3) = A002694(n-1).
Sometimes called the Touchard distribution (after Touchard's Catalan number identity). T(n,k) = number of full binary trees on 2n edges with k deep interior vertices (deep interior means you have to traverse at least 2 edges to reach a leaf) = number of binary trees on n-1 edges with k vertices having a full complement of 2 children. - David Callan, Jul 19 2004
From David Callan, Oct 25 2004: (Start)
T(n,k) = number of ordered trees on n edges with k prolific edges. A prolific edge is one whose child vertex has at least two children. For example with n=3, drawing ordered trees down from the root, /|\ has no prolific edge and the only tree with one prolific edge has the shape of an inverted Y, so T(3,1)=1.
Proof: Consider the following bijection, recorded by Emeric Deutsch, from ordered trees on n edges to Dyck n-paths. For a given ordered tree, traverse the tree in preorder (walk-around from root order). To each node of outdegree r there correspond r upsteps followed by 1 downstep; nothing corresponds to the last leaf. This bijection sends prolific edges to noninitial ascents of length >=2, that is, to DUU's. Then reverse the resulting Dyck n-path so that prolific edges correspond to DDU's. (End)
T(n,k) is the number of Łukasiewicz paths of length n having k fall steps (1,-1) that start at an even level. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(3,1) = 1 because we have U(2)(D)D, where U(2) = (1,2), D = (1,-1) and the fall steps that start at an even level are shown between parentheses. Row n contains ceiling(n/2) terms (n >= 1). - Emeric Deutsch, Jan 06 2005
Number of binary trees with n-1 edges and k+1 leaves (a binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child). - Emeric Deutsch, Jul 31 2006
Number of full binary trees with 2n edges and k+1 vertices both children of which are leaves (n >= 1; a full binary tree is a rooted tree in which each vertex has either 0 or two children). - Emeric Deutsch, Dec 26 2006
Number of ordered trees with n edges and k jumps. In the preorder traversal of an ordered tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch, Jan 18 2007
It is remarkable that we can generate the coefficients of the right hand columns of triangle A175136 with the aid of the coefficients in the rows of the triangle given above. See A175136 for more information. - Johannes W. Meijer, May 06 2011
The antidiagonal sums equal A152225. - Johannes W. Meijer, Sep 13 2012
This array also counts 231-avoiding permutations according to the number of peaks, i.e., positions w[i-1] < w[i] > w[i+1]. For example, 123, 213, 312, and 321 have no peaks, while 132 has one peak. Note also T(n,k) = 2^(n - 1 - 2*k)*A055151(n-1,k). - Kyle Petersen, Aug 02 2013

			T(4,1) = 6 because we have uduu(ddu)d, uu(ddu)dud, uuu(ddu)dd, uu(ddu)udd, uudu(ddu)d and uuud(ddu)d [here u = (1,1), d = (1,-1) and the ddu's are shown between parentheses].
Triangle begins:
    1;
    1;
    2;
    4,    1;
    8,    6;
   16,   24,    2;
   32,   80,   20;
   64,  240,  120,   5;
  128,  672,  560,  70;
  256, 1792, 2240, 560, 14;
  ...

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 4.3.

Alois P. Heinz, Rows n = 0..200, flattened
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 7.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vanjovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, Disc. Math. 341 (2018) 2608-2615, Table 1.
Andrew M. Baxter, Refining enumeration schemes to count according to permutation statistics, arXiv:1401.0337 [math.CO], 2014.
Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018.
David Callan, A variant of Touchard's Catalan number identity, arXiv:1204.5704 [math.CO], 2012. - _N. J. A. Sloane_, Oct 10 2012
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Colin Defant, Postorder Preimages, arXiv:1604.01723 [math.CO], 2016.
Colin Defant, Stack-Sorting Preimages of Permutation Classes, arXiv:1809.03123 [math.CO], 2018.
Colin Defant, Counting 3-Stack-Sortable Permutations, arXiv:1903.09138 [math.CO], 2019.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 22.
Emeric Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202 (see pp. 192-193)..
Maciej Dziemiańczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See p. 6.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Nonleft peaks in Dyck paths: a combinatorial approach, Discrete Math., 337 (2014), 97-105.
Lara Pudwell, On the distribution of peaks (and other statistics), 16th International Conference on Permutation Patterns, Dartmouth College, 2018. [Broken link]
John Riordan, A Note on Catalan Parentheses, Amer. Math. Monthly, Vol. 80, No. 8 (1973), pp. 904-906.
Tom Roberts and Thomas Prellberg, Improving Convergence of Generalised Rosenbluth Sampling for Branched Polymer Models by Uniform Sampling, arXiv:2401.12201 [cond-mat.stat-mech], 2024. See p. 12.
A. Sapounakis, I. Tasoulas, and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
L. W. Shapiro, A short proof of an identity of Touchard's concerning Catalan numbers, J. Combin. Theory Ser. A 20 (1976) 375-376.
Guoce Xin and Jing-Feng Xu, A short approach to Catalan numbers modulo 2^r, Electronic Journal of Combinatorics Vol. 18 Issue 1 (2011), #P177 (see page 2).
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
Index entries for sequences related to Łukasiewicz.

The first few columns equal A011782, A001788, 2*A003472, 5*A002409, 14*A140325 and 42*A172242. - Johannes W. Meijer, Sep 13 2012

Cf. A000108, A002694, A127529, A236406, A097610, A125181, A133156, A134264, A207538.

T:=Concatenation([1],Flat(List([1..13],n->List([0..Int((n-1)/2)],k->2^(n-2*k-1)*Binomial(n-1,2*k)*Binomial(2*k,k)/(k+1))))); # Muniru A Asiru, Nov 29 2018

a := proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)*binomial(n-1,2*k)*binomial(2*k,k)/(k+1) fi end: 1,seq(seq(a(n,k),k=0..(n-1)/2),n=1..15);

A091894[n_] := Prepend[Table[ CoefficientList[ 2^i (1 - z)^((2 i + 3)/2) Hypergeometric2F1[(i + 3)/2, (i + 4)/2, 2, z], z], {i, 0, n}], {1}] (* computes a table of the first n rows. Stumbled accidentally on it. Perhaps someone can find a relationship here? Thies Heidecke (theidecke(AT)astrophysik.uni-kiel.de), Sep 23 2008 *)
Join[{1},Select[Flatten[Table[2^(n-2k-1) Binomial[n-1,2k] Binomial[2k,k]/ (k+1), {n,20},{k,0,n}]],#!=0&]] (* Harvey P. Dale, Mar 05 2012 *)
p[n_] := 2^n Hypergeometric2F1[(1 - n)/2, -n/2, 2, x]; Flatten[Join[{{1}}, Table[CoefficientList[p[n], x], {n, 0, 12}]]] (* Peter Luschny, Jan 23 2018 *)

{T(n, k) = if( n<1, n==0 && k==0, polcoeff( polcoeff( serreverse( x / (1 + 2*x*y + x^2) + x * O(x^n)), n), n-1 - 2*k))} /* Michael Somos, Sep 25 2006 */

[1] + [[2^(n-2*k-1)*binomial(n-1,2*k)*binomial(2*k,k)/(k+1) for k in (0..floor((n-1)/2))] for n in (1..12)] # G. C. Greubel, Nov 30 2018

T(n,k) = 2^(n - 2*k - 1)*binomial(n-1,2*k)*binomial(2*k,k)/(k + 1), T(0,0) = 1, for 0 <= k <= floor((n-1)/2).
G.f.: G = G(t,z) satisfies: t*z*G^2 - (1 - 2*z + 2*t*z)*G + 1 - z + t*z = 0.
With first row zero, the o.g.f. is g(x,t) = (1 - 2*x - sqrt((1 - 2*x)^2 - 4*t*x^2)) / (2*t*x) with the inverse ginv(x,t) = x / (1 + 2*x + t*x^2), an o.g.f. for shifted A207538 and A133156 mod signs, so A134264 and A125181 can be used to interpret the polynomials of this entry. Cf. A097610. - Tom Copeland, Feb 08 2016
If we delete the first 1 from the data these are the coefficients of the polynomials p(n) = 2^n*hypergeom([(1 - n)/2, - n/2], [2], x). - Peter Luschny, Jan 23 2018

A054849 a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.

Original entry on oeis.org

1, 12, 84, 448, 2016, 8064, 29568, 101376, 329472, 1025024, 3075072, 8945664, 25346048, 70189056, 190513152, 508035072, 1333592064, 3451650048, 8820883456, 22284337152, 55710842880, 137950658560, 338606161920

Offset: 5

Henry Bottomley, Apr 14 2000

With 5 leading zeros, binomial transform of binomial(n,5). - Paul Barry, Apr 10 2003

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>4, a(n) is equal to the number of (n+5)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

G. C. Greubel, Table of n, a(n) for n = 5..1000
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (12, -60, 160, -240, 192, -64).

Cf. A000079, A001787, A001788, A001789, A003472, A002409, A054851.
a(n) = A038207(n,5).
Equals 2 * A082139. First differences are in A006975.

List([5..30], n-> 2^(n-5)*Binomial(n,5)); # G. C. Greubel, Aug 27 2019

[2^(n-5)*Binomial(n,5): n in [5..30]]; // G. C. Greubel, Aug 27 2019

seq(binomial(n+5,5)*2^n,n=0..22); # Zerinvary Lajos, Jun 13 2008

Table[2^(n-5)*Binomial[n,5], {n,5,30}] (* G. C. Greubel, Aug 27 2019 *)

vector(25, n, 2^(n-1)*binomial(n+4,5)) \\ G. C. Greubel, Aug 27 2019

[lucas_number2(n, 2, 0)*binomial(n,5)/32 for n in range(5, 28)] # Zerinvary Lajos, Mar 10 2009

a(n) = 2*a(n-1) + A003472(n-1).
From Paul Barry, Apr 10 2003: (Start)
O.g.f.: x^5/(1-2*x)^6.
E.g.f.: exp(2*x)*(x^5/5!) (with 5 leading zeros). (End)
a(n) = Sum_{i=5..n} binomial(i,5)*binomial(n,i). Example: for n=8, a(8) = 1*56 + 6*28 + 21*8 + 56*1 = 448. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=5} 1/a(n) = 10*log(2) - 35/6.
Sum_{n>=5} (-1)^(n+1)/a(n) = 810*log(3/2) - 655/2. (End)

More terms from James Sellers, Apr 15 2000

A130809 If X_1, ..., X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 3-subsets of X containing none of X_i, (i=1,...,n).

Original entry on oeis.org

8, 32, 80, 160, 280, 448, 672, 960, 1320, 1760, 2288, 2912, 3640, 4480, 5440, 6528, 7752, 9120, 10640, 12320, 14168, 16192, 18400, 20800, 23400, 26208, 29232, 32480, 35960, 39680, 43648, 47872, 52360, 57120, 62160, 67488, 73112, 79040, 85280

Offset: 3

Milan Janjic, Jul 16 2007

Uncentered octahedral numbers: take a simple cubical grid of size n X n X n where n = 2k is an even number, n >= 6. Retain all points that are at Manhattan distance n or greater from all 8 corners of the cube, and discard all other points. The number of points that remain is a(k). If n were to be an odd number, the same operation would yield the centered octahedral numbers A001845. - Arun Giridhar, Mar 06 2014
For an (n+2)-dimensional Rubik's cube, the number of cubes that have exactly 3 exposed facets. - Phil Scovis, Aug 03 2009
a(n) is the number of 2-simplices in an n-cross polytope. - Arkadiusz Wesolowski, Oct 16 2012
a(n) is also the number of unit tetrahedra in an (n+1)-scaled octahedron composed of the tetrahedral-octahedral honeycomb. - Jason Pruski, Aug 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Harlan J. Brothers, Pascal's Prism: Supplementary Material, 2012.
Milan Janjic, Two Enumerative Functions
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000079, A001787, A001788, A001789, A001845, A002409, A003472, A038207, A046092, A054849, A054851, A056220, A140325, A140354.

[(4/3)*n*(n-1)*(n-2): n in [3..60]]; // Vincenzo Librandi, Oct 03 2017

Maple
```
a:=n->4/3*n*(n-1)*(n-2);
```

Table[(4/3) n (n - 1) (n - 2), {n, 3, 41}] (* or *)
Table[Binomial[n, n - 3] 2^3, {n, 3, 41}] (* or *)
DeleteCases[#, 0] &@ CoefficientList[Series[8 x^3/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 31 2017 *)

a(n) = 4*n*(n-1)*(n-2)/3; \\ Andrew Howroyd, Nov 06 2018

a(n) = (4/3)*n*(n-1)*(n-2).
a(n) = C(n,n-3)*8, n >= 3. - Zerinvary Lajos, Dec 07 2007
G.f.: 8*x^3/(1-x)^4. - Colin Barker, Apr 14 2012
For n>1, a(n) = a(n-1) + A056220(n-1) + A056220(n-2). - Bruce J. Nicholson, Feb 14 2018
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/16.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3*log(2)/2 - 15/16. (End)
E.g.f.: 4*x^3*exp(x)/3. - Stefano Spezia, Apr 02 2024

A133156 Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order.

Original entry on oeis.org

1, 2, 4, -1, 8, -4, 16, -12, 1, 32, -32, 6, 64, -80, 24, -1, 128, -192, 80, -8, 256, -448, 240, -40, 1, 512, -1024, 672, -160, 10, 1024, -2304, 1792, -560, 60, -1, 2048, -5120, 4608, -1792, 280, -12, 4096, -11264, 11520, -5376, 1120, -84, 1

Offset: 0

Gary W. Adamson, Dec 16 2007

The Chebyshev polynomials of the second kind are defined by the recurrence relation: U(0,x) = 1; U(1,x) = 2x; U(n+1,x) = 2x*U(n,x) - U(n-1,x).
From Gary W. Adamson, Nov 28 2008: (Start)
Triangle read by rows, unsigned = A000012 * A028297.
Row sums of absolute values give the Pell series, A000129.
(End)
The row sums are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}.
Triangle, with zeros omitted, given by (2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 27 2011
Coefficients in the expansion of sin((n+1)*x)/sin(x) in descending powers of cos(x). The length of the n-th row is A008619(n). - Jianing Song, Nov 02 2018

			The first few Chebyshev polynomials of the second kind are
    1;
    2x;
    4x^2 -    1;
    8x^3 -    4x;
   16x^4 -   12x^2 +   1;
   32x^5 -   32x^3 +   6x;
   64x^6 -   80x^4 +  24x^2 -   1;
  128x^7 -  192x^5 +  80x^3 -   8x;
  256x^8 -  448x^6 + 240x^4 -  40x^2 +  1;
  512x^9 - 1024x^7 + 672x^5 - 160x^3 + 10x;
  ...
From _Roger L. Bagula_ and _Gary W. Adamson_: (Start)
     1;
     2;
     4,    -1;
     8,    -4;
    16,   -12,    1;
    32,   -32,    6;
    64,   -80,   24,   -1;
   128,  -192,   80,   -8;
   256,  -448,  240,  -40,  1;
   512, -1024,  672, -160, 10;
  1024, -2304, 1792, -560, 60, -1; (End)
From  _Philippe Deléham_, Dec 27 2011: (Start)
Triangle (2, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, ...) begins:
   1;
   2,   0;
   4,  -1,  0;
   8,  -4,  0,  0;
  16, -12,  1,  0,  0;
  32, -32,  6,  0,  0,  0;
  64, -80, 24, -1,  0,  0,  0; (End)

Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, The Pfaffian Transformation, Journal of Integer Sequences, Vol. 12 (2009), page 25
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From _Tom Copeland_, Oct 11 2014
Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120-135. MR3149030
Caglar Koca and Ozgur B. Akan, Modelling 1D Partially Absorbing Boundaries for Brownian Molecular Communication Channels, arXiv:2402.15888 [q-bio.MN], 2024. See p. 9.
Wikipedia, Chebyshev polynomials

Cf. A038207, A053117.
Cf. A018297, A000129. - Gary W. Adamson, Nov 28 2008
Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851. - Philippe Deléham, Sep 12 2009
Cf. A091894, A097610, A099089, A207538.

t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);
Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Dec 19 2008 *)

A generating function for U(n) is 1/(1 - 2tx + t^2). Given A038207, shift down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row, then insert alternate signs.
T(n,m) = (-1)^m*binomial(n - m, m)*2^(n - 2*m). - Roger L. Bagula and Gary W. Adamson, Dec 19 2008
From Tom Copeland, Feb 11 2016: (Start)
Shifted o.g.f.: G(x,t) = x/(1 - 2x + tx^2).
A053117 is a reflected, aerated version of this entry; A207538, an unsigned version; and A099089, a reflected, shifted version.
The compositional inverse of G(x,t) is Ginv(x,t) = ((1 + 2x) - sqrt((1 + 2x)^2 - 4tx^2))/(2tx) = x - 2x^2 + (4 + t)x^3 - (8 + 6t)x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0.). Cf. A097610 with h_1 = -2 and h_2 = t. (End)

More terms from Philippe Deléham, Sep 12 2009

A054851 a(n) = 2^(n-7)*binomial(n,7). Number of 7D hypercubes in an n-dimensional hypercube.

Original entry on oeis.org

1, 16, 144, 960, 5280, 25344, 109824, 439296, 1647360, 5857280, 19914752, 65175552, 206389248, 635043840, 1905131520, 5588385792, 16066609152, 45364543488, 126012620800, 344876646400, 931166945280, 2483111854080

Offset: 7

Henry Bottomley, Apr 14 2000

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>6, a(n) is equal to the number of (n+7)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

G. C. Greubel, Table of n, a(n) for n = 7..1000
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792, 1024,-256).

Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409.

a(n) = A038207(n,7).

List([7..30], n-> 2^(n-7)*Binomial(n,7)); # G. C. Greubel, Aug 27 2019

[2^(n-7)*Binomial(n,7): n in [7..30]]; // G. C. Greubel, Aug 27 2019

seq(binomial(n+7,7)*2^n,n=0..21); # Zerinvary Lajos, Jun 23 2008

Table[2^(n-7)*Binomial[n,7], {n,7,30}] (* G. C. Greubel, Aug 27 2019 *)

vector(23, n, 2^(n-1)*binomial(n+6, 7)) \\ G. C. Greubel, Aug 27 2019

a(n) = 2*a(n-1) + A002409(n-1).
a(n+8) = A082141(n+1)/2.
G.f.: x^7/(1-2*x)^8. - Colin Barker, Sep 04 2012
a(n) = Sum_{i=7..n} binomial(i,7)*binomial(n,i). Example: for n=11, a(11) = 1*330 + 8*165 + 36*55 + 120*11 + 330*1 = 5280. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=7} 1/a(n) = 14*log(2) - 259/30.
Sum_{n>=7} (-1)^(n+1)/a(n) = 10206*log(3/2) - 124117/30. (End)

More terms from James Sellers, Apr 15 2000

A006976 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+12, n), n >= 0.

Original entry on oeis.org

1, 13, 98, 560, 2688, 11424, 44352, 160512, 549120, 1793792, 5637632, 17145856, 50692096, 146227200, 412778496, 1143078912, 3111714816, 8341487616, 22052208640, 57567870976, 148562247680, 379364311040, 959384125440

Offset: 0

Simon Plouffe

Binomial transform of A069039. - Paul Barry, Feb 19 2003

If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n >= 5, a(n-5) is the number of (n+6)-subsets of X intersecting each X_i, (i = 1, 2, ..., n). - Milan Janjic, Nov 18 2007

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
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].
Milan Janjic, Two Enumerative Functions
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

a(n) = A039991(n+12, 12), A053120.

Partial sums are in A002409.

List([0..25], n-> 2^(n-1)*Binomial(n+5,5)*(n+12)/6); # G. C. Greubel, Aug 27 2019

[2^(n-1)/6*Binomial(n+5,5)*(n+12) : n in [0..25]]; // Brad Clardy, Mar 10 2012

seq(2^(n-1)*binomial(n+5,5)*(n+12)/6, n=0..25); # G. C. Greubel, Aug 27 2019

Table[2^(n-1)*Binomial[n+5,5]*(n+12)/6, {n,0,25}] (* G. C. Greubel, Aug 27 2019 *)
LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,13,98,560,2688,11424,44352},30] (* Harvey P. Dale, Sep 26 2024 *)

vector(26, n, 2^(n-2)*binomial(n+4,5)*(n+11)/6) \\ G. C. Greubel, Aug 27 2019

[2^(n-1)*binomial(n+5,5)*(n+12)/6 for n in (0..25)] # G. C. Greubel, Aug 27 2019

G.f.: (1-x)/(1-2*x)^7.
a(n) = 2^n*binomial(n+5, 5)*(n+12)/12. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]
a(n) = Sum_{k = 0..floor((n+12)/2)} C(n+12,2*k)*C(k,6). - Paul Barry, May 15 2003
E.g.f.: (1/45)*exp(2*x)*(45 + 495*x + 1125*x^2 + 900*x^3 + 300*x^4 + 42*x^5 + 2*x^6). - Stefano Spezia, Jan 03 2020

More terms from James Sellers, Aug 21 2000

Name clarified by Wolfdieter Lang, Nov 26 2019

A082140 A transform of binomial(n,6).

Original entry on oeis.org

1, 7, 56, 336, 1680, 7392, 29568, 109824, 384384, 1281280, 4100096, 12673024, 38019072, 111132672, 317521920, 889061376, 2444918784, 6615662592, 17641766912, 46425702400, 120706826240, 310388981760, 790081044480

Offset: 0

Paul Barry, Apr 06 2003

Seventh row of number array A082137. C(n,6) has e.g.f. (x^6/6!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*binomial(6,0) = 2*(1/2) = 1 (use 0^0 = 1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A082137, A082139, A080951.

For n>0, a(n) = 1/2 * A002409(n).

[(2^(n-1) + 0^n/2)*Binomial(n+6,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

[seq (ceil(binomial(n+6,6)*2^(n-1)),n=0..22)]; # Zerinvary Lajos, Nov 01 2006

Drop[With[{nmax = 56}, CoefficientList[Series[x^6*Exp[x]*Cosh[x]/6!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+6,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,7,56,336,1680,7392,29568,109824},30] (* Harvey P. Dale, Jul 18 2023 *)

my(x='x+O('x^30)); Vec(serlaplace(x^6*exp(x)*cosh(x)/6!)) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(n+6,n).
a(n) = Sum_{j=0..n} C(n+6, j+6)*C(j+6, 6)*(1+(-1)^j)/2.
G.f.: (1 - 7*x + 42*x^2 - 140*x^3 + 280*x^4 - 336*x^5 + 224*x^6 - 64*x^7)/ (1-2*x)^7.
E.g.f.: (x^6/6!)*exp(x)*cosh(x) (with 6 leading zeros).
a(n) = ceiling(binomial(n+6,6)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 89/5 - 24*log(2).
Sum_{n>=0} (-1)^n/a(n) = 5832*log(3/2) - 11819/5. (End)

A130810 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 4-subsets of X containing none of X_i, (i=1,...,n).

Original entry on oeis.org

16, 80, 240, 560, 1120, 2016, 3360, 5280, 7920, 11440, 16016, 21840, 29120, 38080, 48960, 62016, 77520, 95760, 117040, 141680, 170016, 202400, 239200, 280800, 327600, 380016, 438480, 503440, 575360, 654720, 742016, 837760, 942480, 1056720

Offset: 4

Milan Janjic, Jul 16 2007

Number of n permutations (n>=4) of 3 objects u,v,z, with repetition allowed, containing n-4 u's. Example: if n=4 then n-4 =(0) zero u, a(1)=16 because we have vvvv zzzz vvvz zzzv vvzv zzvz vzvv zvzz zvvv vzzz vvzz zzvv vzvz zvzv zvvz vzzv. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 3-dimensional elements in an n-cross polytope where n>=4. - Patrick J. McNab, Jul 06 2015

Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Milan Janjić, Two Enumerative Functions. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Cross Polytope.

Cf. A000332, A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809. Equals twice A162668.

a:= n-> binomial(2*n,4) +binomial(n,2) -n*binomial(2*n-2,2);
seq(binomial(n, n-4)*2^4, n=4..37); # Zerinvary Lajos, Dec 07 2007

a[n_] := 16 * Binomial[n, 4]; Array[a, 34, 4] (* Amiram Eldar, Jul 25 2025 *)

a(n) = 16 * binomial(n, 4); \\ Amiram Eldar, Jul 25 2025

a(n) = binomial(2*n,4) + binomial(n,2) - n*binomial(2*n-2,2).
a(n) = binomial(n,4)*16. - Zerinvary Lajos, Dec 07 2007
G.f.: 16*x^4/(1-x)^5. - Colin Barker, Apr 14 2012
a(n) = 2*n*(n-1)*(n-2)*(n-3)/3 = 2*A162668(n-3). - Robert Israel, Jul 06 2015
a(n) = 16 * A000332(n). - Alois P. Heinz, Oct 26 2020
E.g.f.: 2*exp(x)*x^4/3. - Stefano Spezia, Jul 17 2025
From Amiram Eldar, Jul 25 2025: (Start)
Sum_{n>=4} 1/a(n) = 1/12.
Sum_{n>=4} (-1)^n/a(n) = 2*log(2) - 4/3. (End)

cp's OEIS Frontend

A000079 Powers of 2: a(n) = 2^n.

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

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A091894 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having k ddu's [here u = (1,1) and d = (1,-1)].

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A054849 a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.

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A130809 If X_1, ..., X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 3-subsets of X containing none of X_i, (i=1,...,n).

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