A060632 -id:A060632 - cp's OEIS frontend

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1

Offset: 0

N. J. A. Sloane

Restored the alternative spelling of Sierpinski to facilitate searching for this triangle using regular-expression matching commands in ASCII. - N. J. A. Sloane, Jan 18 2016
Also triangle giving successive states of cellular automaton generated by "Rule 60" and "Rule 102". - Hans Havermann, May 26 2002
Also triangle formed by reading triangle of Eulerian numbers (A008292) mod 2. - Philippe Deléham, Oct 02 2003
Self-inverse when regarded as an infinite lower triangular matrix over GF(2).
Start with [1], repeatedly apply the map 0 -> [00/00], 1 -> [10/11] [Allouche and Berthe]
Also triangle formed by reading triangles A011117, A028338, A039757, A059438, A085881, A086646, A086872, A087903, A104219 mod 2. - Philippe Deléham, Jun 18 2005
J. H. Conway writes (in Math Forum): at least the first 31 rows give odd-sided constructible polygons (sides 1, 3, 5, 15, 17, ... see A001317). The 1's form a Sierpiński sieve. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005
When regarded as an infinite lower triangular matrix, its inverse is a (-1,0,1)-matrix with zeros undisturbed and the nonzero entries in every column form the Prouhet-Thue-Morse sequence (1,-1,-1,1,-1,1,1,-1,...) A010060 (up to relabeling). - David Callan, Oct 27 2006
Triangle read by rows: antidiagonals of an array formed by successive iterates of running sums mod 2, beginning with (1, 1, 1, ...). - Gary W. Adamson, Jul 10 2008
T(n,k) = A057427(A143333(n,k)). - Reinhard Zumkeller, Oct 24 2010
The triangle sums, see A180662 for their definitions, link Sierpiński’s triangle A047999 with seven sequences, see the crossrefs. The Kn1y(n) and Kn2y(n), y >= 1, triangle sums lead to the Sierpiński-Stern triangle A191372. - Johannes W. Meijer, Jun 05 2011
Used to compute the total Steifel-Whitney cohomology class of the Real Projective space. This was an essential component of the proof that there are no product operations without zero divisors on R^n for n not equal to 1, 2, 4 or 8 (real numbers, complex numbers, quaternions, Cayley numbers), proved by Bott and Milnor. - Marcus Jaiclin, Feb 07 2012
T(n,k) = A134636(n,k) mod 2. - Reinhard Zumkeller, Nov 23 2012
T(n,k) = 1 - A219463(n,k), 0 <= k <= n. - Reinhard Zumkeller, Nov 30 2012
From Vladimir Shevelev, Dec 31 2013: (Start)
Also table of coefficients of polynomials s_n(x) of degree n which are defined by formula s_n(x) = Sum_{i=0..n} (binomial(n,i) mod 2)*x^k. These polynomials we naturally call Sierpiński's polynomials. They also are defined by the recursion: s_0(x)=1, s_(2*n+1)(x) = (x+1)*s_n(x^2), n>=0, and s_(2*n)(x) = s_n(x^2), n>=1.
Note that: s_n(1)  = A001316(n),
           s_n(2)  = A001317(n),
           s_n(3)  = A100307(n),
           s_n(4)  = A001317(2*n),
           s_n(5)  = A100308(n),
           s_n(6)  = A100309(n),
           s_n(7)  = A100310(n),
           s_n(8)  = A100311(n),
           s_n(9)  = A100307(2*n),
           s_n(10) = A006943(n),
           s_n(16) = A001317(4*n),
           s_n(25) = A100308(2*n), etc.
The equality s_n(10) = A006943(n) means that sequence A047999 is obtained from A006943 by the separation by commas of the digits of its terms. (End)
Comment from N. J. A. Sloane, Jan 18 2016: (Start)
Take a diamond-shaped region with edge length n from the top of the triangle, and rotate it by 45 degrees to get a square S_n. Here is S_6:
  [1, 1, 1, 1, 1, 1]
  [1, 0, 1, 0, 1, 0]
  [1, 1, 0, 0, 1, 1]
  [1, 0, 0, 0, 1, 0]
  [1, 1, 1, 1, 0, 0]
  [1, 0, 1, 0, 0, 0].
Then (i) S_n contains no square (parallel to the axes) with all four corners equal to 1 (cf. A227133); (ii) S_n can be constructed by using the greedy algorithm with the constraint that there is no square with that property; and (iii) S_n contains A064194(n) 1's. Thus A064194(n) is a lower bound on A227133(n). (End)
See A123098 for a multiplicative encoding of the rows, i.e., product of the primes selected by nonzero terms; e.g., 1 0 1 => 2^1 * 3^0 * 5^1. - M. F. Hasler, Sep 18 2016
From Valentin Bakoev, Jul 11 2020: (Start)
The Sierpinski's triangle with 2^n rows is a part of a lower triangular matrix M_n of dimension 2^n X 2^n. M_n is a block matrix defined recursively: M_1= [1, 0], [1, 1], and for n>1, M_n = [M_(n-1), O_(n-1)], [M_(n-1), M_(n-1)], where M_(n-1) is a block matrix of the same type, but of dimension 2^(n-1) X 2^(n-1), and O_(n-1) is the zero matrix of dimension 2^(n-1) X 2^(n-1). Here is how M_1, M_2 and M_3 look like:
1 0     1 0 0 0      1 0 0 0 0 0 0 0
1 1     1 1 0 0      1 1 0 0 0 0 0 0   - It is seen the self-similarity of the
        1 0 1 0      1 0 1 0 0 0 0 0     matrices M_1, M_2, ..., M_n, ...,
        1 1 1 1      1 1 1 1 0 0 0 0     analogously to the Sierpinski's fractal.
                     1 0 0 0 1 0 0 0
                     1 1 0 0 1 1 0 0
                     1 0 1 0 1 0 1 0
                     1 1 1 1 1 1 1 1
M_n can also be defined as M_n = M_1 X M_(n-1) where X denotes the Kronecker product. M_n is an important matrix in coding theory, cryptography, Boolean algebra, monotone Boolean functions, etc. It is a transformation matrix used in computing the algebraic normal form of Boolean functions. Some properties and links concerning M_n can be seen in LINKS. (End)
Sierpinski's gasket has fractal (Hausdorff) dimension log(A000217(2))/log(2) = log(3)/log(2) = 1.58496... (and cf. A020857). This gasket is the first of a family of gaskets formed by taking the Pascal triangle (A007318) mod j, j >= 2 (see CROSSREFS). For prime j, the dimension of the gasket is log(A000217(j))/log(j) = log(j(j + 1)/2)/log(j) (see Reiter and Bondarenko references). - Richard L. Ollerton, Dec 14 2021

			Triangle begins:
              1,
             1,1,
            1,0,1,
           1,1,1,1,
          1,0,0,0,1,
         1,1,0,0,1,1,
        1,0,1,0,1,0,1,
       1,1,1,1,1,1,1,1,
      1,0,0,0,0,0,0,0,1,
     1,1,0,0,0,0,0,0,1,1,
    1,0,1,0,0,0,0,0,1,0,1,
   1,1,1,1,0,0,0,0,1,1,1,1,
  1,0,0,0,1,0,0,0,1,0,0,0,1,
  ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Brand, Neal; Das, Sajal; Jacob, Tom. The number of nonzero entries in recursively defined tables modulo primes. Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990). Congr. Numer. 78 (1990), 47--59. MR1140469 (92h:05004).
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, pp. 43-49 (sequence appears on p. 46).
H.-O. Peitgen, H. Juergens and D. Saupe: Chaos and Fractals (Springer-Verlag 1992), p. 408.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

N. J. A. Sloane, Table of n, a(n) for n = 0..10584 [First 144 rows, flattened; first 50 rows from T. D. Noe].
J.-P. Allouche and V. Berthe, Triangle de Pascal, complexité et automates, Bulletin of the Belgian Mathematical Society Simon Stevin 4.1 (1997): 1-24.
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.],
J. Baer, Explore patterns in Pascal's Triangle
Valentin Bakoev, Fast Bitwise Implementation of the Algebraic Normal Form Transform, Serdica J. of Computing 11 (2017), No 1, 45-57.
Valentin Bakoev, Properties and links concerning M_n
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
Thomas Baruchel, A non-symmetric divide-and-conquer recursive formula for the convolution of polynomials and power series, arXiv:1912.00452 [math.NT], 2019.
A. Bogomolny, Dot Patterns and Sierpinski Gasket
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 9.
S. Butkevich, Pascal Triangle Applet
David Callan, Sierpinski's triangle and the Prouhet-Thue-Morse word, arXiv:math/0610932 [math.CO], 2006.
B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part I
B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part II
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014; Journal of Difference Equations and Applications, Vol. 20, #11, 2014.
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Brady Haran, Chaos Game, Numberphile video, YouTube (April 27, 2017).
I. Kobayashi et al., Pascal's Triangle
Dr. Math, Regular polygon formulas [Broken link?]
Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.
National Curve Bank, Sierpinski Triangles
Hieu D. Nguyen, A Digital Binomial Theorem, arXiv:1412.3181 [math.NT], 2014.
S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
F. Richman, Javascript for computing Pascal's triangle modulo n. Go to this page, then under "Modern Algebra and Other Things", click "Pascal's triangle modulo n".
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization, J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29. Also arXiv:1011.6083, 2010.
N. J. A. Sloane, Illustration of rows 0 to 32 (encoignure style)
N. J. A. Sloane, Illustration of rows 0 to 64 (encoignure style)
N. J. A. Sloane, Illustration of rows 0 to 128 (encoignure style)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Sierpiński Sieve, Rule 60, Rule 102
Index entries for sequences related to cellular automata
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences generated by sieves

Sequences based on the triangles formed by reading Pascal's triangle mod m: (this sequence) (m = 2), A083093 (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).
Other versions: A090971, A038183.
Cf. A007318, A054431, A001317, A008292, A083093, A034931, A034930, A008975, A034932, A166360, A249133, A064194, A227133.
From Johannes W. Meijer, Jun 05 2011: (Start)
A106344 is a skew version of this triangle.
Triangle sums (see the comments): A001316 (Row1; Related to Row2), A002487 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23), A007306 (Kn3, Kn4), A060632 (Fi1, Fi2), A120562 (Ca1, Ca2), A112970 (Gi1, Gi2), A127830 (Ze3, Ze4). (End)

import Data.Bits (xor)
a047999 :: Int -> Int -> Int
a047999 n k = a047999_tabl !! n !! k
a047999_row n = a047999_tabl !! n
a047999_tabl = iterate (\row -> zipWith xor ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Dec 11 2011, Oct 24 2010

A047999:= func< n,k | BitwiseAnd(n-k, k) eq 0 select 1 else 0 >;
[A047999(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 03 2024

# Maple code for first M rows (here M=10) - N. J. A. Sloane, Feb 03 2016
ST:=[1,1,1]; a:=1; b:=2; M:=10;
for n from 2 to M do ST:=[op(ST),1];
for i from a to b-1 do ST:=[op(ST), (ST[i+1]+ST[i+2]) mod 2 ]; od:
ST:=[op(ST),1];
a:=a+n; b:=a+n; od:
ST; # N. J. A. Sloane
# alternative
A047999 := proc(n,k)
    modp(binomial(n,k),2) ;
end proc:
seq(seq(A047999(n,k),k=0..n),n=0..12) ; # R. J. Mathar, May 06 2016

Mod[ Flatten[ NestList[ Prepend[ #, 0] + Append[ #, 0] &, {1}, 13]], 2] (* Robert G. Wilson v, May 26 2004 *)
rows = 14; ca = CellularAutomaton[60, {{1}, 0}, rows-1]; Flatten[ Table[ca[[k, 1 ;; k]], {k, 1, rows}]] (* Jean-François Alcover, May 24 2012 *)
Mod[#,2]&/@Flatten[Table[Binomial[n,k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 26 2019 *)
A047999[n_,k_]:= Boole[BitAnd[n-k,k]==0];
Table[A047999[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 03 2025 *)

\\ Recurrence for Pascal's triangle mod p, here p = 2.
p = 2; s=13; T=matrix(s,s); T[1,1]=1;
for(n=2,s, T[n,1]=1; for(k=2,n, T[n,k] = (T[n-1,k-1] + T[n-1,k])%p ));
for(n=1,s,for(k=1,n,print1(T[n,k],", "))) \\ Gerald McGarvey, Oct 10 2009

A011371(n)=my(s);while(n>>=1,s+=n);s
T(n,k)=A011371(n)==A011371(k)+A011371(n-k) \\ Charles R Greathouse IV, Aug 09 2013

T(n,k)=bitand(n-k,k)==0 \\ Charles R Greathouse IV, Aug 11 2016

def A047999_T(n,k):
    return int(not ~n & k) # Chai Wah Wu, Feb 09 2016

Lucas's Theorem is that T(n,k) = 1 if and only if the 1's in the binary expansion of k are a subset of the 1's in the binary expansion of n; or equivalently, k AND NOT n is zero, where AND and NOT are bitwise operators. - Chai Wah Wu, Feb 09 2016 and N. J. A. Sloane, Feb 10 2016
Sum_{k>=0} T(n, k) = A001316(n) = 2^A000120(n).
T(n,k) = T(n-1,k-1) XOR T(n-1,k), 0 < k < n; T(n,0) = T(n,n) = 1. - Reinhard Zumkeller, Dec 13 2009
T(n,k) = (T(n-1,k-1) + T(n-1,k)) mod 2 = |T(n-1,k-1) - T(n-1,k)|, 0 < k < n; T(n,0) = T(n,n) = 1. - Rick L. Shepherd, Feb 23 2018
From Vladimir Shevelev, Dec 31 2013: (Start)
For polynomial {s_n(x)} we have
s_0(x)=1; for n>=1, s_n(x) = Product_{i=1..A000120(n)} (x^(2^k_i) + 1),
if the binary expansion of n is n = Sum_{i=1..A000120(n)} 2^k_i;
G.f. Sum_{n>=0} s_n(x)*z^n = Product_{k>=0} (1 + (x^(2^k)+1)*z^(2^k)) (0Let x>1, t>0 be real numbers. Then
Sum_{n>=0} 1/s_n(x)^t = Product_{k>=0} (1 + 1/(x^(2^k)+1)^t);
Sum_{n>=0} (-1)^A000120(n)/s_n(x)^t = Product_{k>=0} (1 - 1/(x^(2^k)+1)^t).
In particular, for t=1, x>1, we have
Sum_{n>=0} (-1)^A000120(n)/s_n(x) = 1 - 1/x. (End)
From Valentin Bakoev, Jul 11 2020: (Start)
(See my comment about the matrix M_n.) Denote by T(i,j) the number in the i-th row and j-th column of M_n (0 <= i, j < 2^n). When i>=j, T(i,j) is the j-th number in the i-th row of the Sierpinski's triangle. For given i and j, we denote by k the largest integer of the type k=2^m and kT(i,0) = T(i,i) = 1, or
T(i,j) = 0 if i < j, or
T(i,j) = T(i-k,j), if j < k, or
T(i,j) = T(i-k,j-k), if j >= k.
Thus, for given i and j, T(i,j) can be computed in O(log_2(i)) steps. (End)

Additional links from Lekraj Beedassy, Jan 22 2004

A001790 Numerators in expansion of 1/sqrt(1-x).

Original entry on oeis.org

1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719

Offset: 0

N. J. A. Sloane

Also numerator of e(n-1,n-1) (see Maple line).
Leading coefficient of normalized Legendre polynomial.
Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).
Also the numerator of binomial(2*n,n)/2^n. - T. D. Noe, Nov 29 2005
This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - Stephen Crowley, Apr 03 2007
Truncations of rational expressions like those given by the numerator operator are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - Peter Luschny, Aug 01 2009
It appears that a(n) = A060818(n)*A001147(n)/A000142(n). - James R. Buddenhagen, Jan 20 2010
The convolution of sequence binomial(2*n,n)/4^n with itself is the constant sequence with all terms = 1.
a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011
a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012
a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013
Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016
From Ralf Steiner, Apr 07 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
a(n)/A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} a(k)/A060818(k) = -i.
Sum_{k>=0} (-1)^k*a(k)/A060818(k) = 1/sqrt(3).
Sum_{k>=0} (-1)^(k+1)*a(k)/A060818(k) = -1/sqrt(3).
a(n)/A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/sqrt(2).
Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/sqrt(2). (End)
a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017
Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/A344402. - Peter Luschny, May 20 2021
Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022
a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - Mohammed Yaseen, Jul 29 2023
a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - Mohammed Yaseen, Sep 19 2023

			1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...
binomial(2*n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...

P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:6 at page 51.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102.

Robert G. Wilson v, Table of n, a(n) for n = 0..1666 (first 201 terms from T. D. Noe)
Horst Alzer and Bent Fuglede, Normalized binomial mid-coefficients and power means, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 284-294.
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993 (see V_n with N=1).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table [Annotated scanned copy].
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
S. Łukaszyk and W. Bieniawski, Assembly Theory of Binary Messages, Mathematics, 12(10) (2024), 1600.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages).
Eric Weisstein's World of Mathematics, Binomial Series.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Wikipedia, Lorentz Factor.

Cf. A000142, A000984, A001147, A001316, A001800, A001801, A001803.
Cf. A005187, A007814, A008316, A046161, A056040, A086117, A094638.
Cf. A101926, A123854, A273194, A344402.
Cf. A060818 (denominator of binomial(2*n,n)/2^n), A061549 (denominators).
Cf. A123854 (denominators).
Cf. A161198 (triangle of coefficients for (1-x)^((-1-2*n)/2)).
Cf. A163590 (odd part of the swinging factorial).
Cf. A001405.
First column and diagonal 1 of triangle A100258.
Bisection of A036069.
Bisections give A061548 and A063079.
Inverse Moebius transform of A180403/A046161.
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), this sequence (p=-1), A001803 (p=-3), A161199 (p=-5), A161201 (p=-7).

A001790:= func< n | Numerator((n+1)*Catalan(n)/4^n) >;
[A001790(n): n in [0..40]]; // G. C. Greubel, Sep 23 2024

e := proc(l,m) local k; add(2^(k-2*m)*binomial(2*m-2*k,m-k)*binomial(m+k,m)*binomial(k,l),k=l..m); end;
# From Peter Luschny, Aug 01 2009: (Start)
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i=convert(iquo(n,2),base,2))):
a := n -> swing(2*n)/sigma(2*n); # (End)
A001790 := proc(n) binomial(2*n, n)/4^n ; numer(%) ; end proc : # R. J. Mathar, Jan 18 2013

Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}]   (* John M. Campbell, Jul 04 2011 *)
Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]),{n,0,20}]] (* Ralf Steiner, Apr 07 2017 *)
Numerator[Table[Binomial[2n,n]/2^n, {n, 0, 25}]] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* Andres Cicuttin, Jan 22 2018 *)
A = {1}; Do[A = Append[A, 2^IntegerExponent[n, 2]*(2*n - 1)*A[[n]]/n], {n, 1, 25}]; Print[A] (* John Lawrence, Jul 17 2020 *)

{a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))};

a(n)=binomial(2*n,n)>>hammingweight(n); \\ Gleb Koloskov, Sep 26 2021

# uses[A000120]
@CachedFunction
def swing(n):
    if n == 0: return 1
    return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
A001790 = lambda n: swing(2*n)/2^A000120(2*n)
[A001790(n) for n in (0..25)]  # Peter Luschny, Nov 19 2012

a(n) = numerator( binomial(2*n,n)/4^n ) (cf. A046161).
a(n) = A000984(n)/A001316(n) where A001316(n) is the highest power of 2 dividing C(2*n, n) = A000984(n). - Benoit Cloitre, Jan 27 2002
a(n) = denominator of (2^n/binomial(2*n,n)). - Artur Jasinski, Nov 26 2011
a(n) = numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).
L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0.
Numerator in (1-2t)^(-1/2) = 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - Tom Copeland, Dec 04 2013
From Ralf Steiner, Apr 08 2017: (Start)
a(n)/A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
Sum_{k>=0} a(k)/A061549(k) = 2/sqrt(3).
Sum_{k>=0} (-1)^k*a(k)/A061549(k) = 2/sqrt(5).
Sum_{k>=0} (-1)^(k+1)*a(k)/A061549(k) = -2/sqrt(5).
a(n)/A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)).
Sum_{k>=0} a(k)/A123854(k) = sqrt(2).
Sum_{k>=0} (-1)^k*a(k)/A123854(k) = sqrt(2/3).
Sum_{k>=0} (-1)^(k+1)*a(k)/A123854(k) = -sqrt(2/3). (End)
a(n) = 2^A007814(n)*(2*n-1)*a(n-1)/n. - John Lawrence, Jul 17 2020
Sum_{k>=0} A086117(k+3)/a(k+2) = Pi. - Antonio Graciá Llorente, Aug 31 2024
a(n) = A001803(n)/(2*n+1). - G. C. Greubel, Sep 23 2024

A001803 Numerators in expansion of (1 - x)^(-3/2).

Original entry on oeis.org

1, 3, 15, 35, 315, 693, 3003, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 300540195, 9917826435, 20419054425, 83945001525, 172308161025, 1412926920405, 2893136075115, 11835556670925

Offset: 0

N. J. A. Sloane

a(n) is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). - James R. Buddenhagen, Aug 17 2008
a(n) is the denominator of (2n)!!/(2*n + 1)!! = 2^(2*n)*n!*n!/(2*n + 1)! (see Andersson). - N. J. A. Sloane, Jun 27 2011
a(n) = (2*n + 1)*A001790(n). A046161(n)/a(n) = 1, 2/3, 8/15, 16/35, 128/315, 256/693, ... is binomial transform of Madhava-Gregory-Leibniz series for Pi/4 (i.e., 1 - 1/3 + 1/5 - 1/7 + ... ). See A173384 and A173396. - Paul Curtz, Feb 21 2010
a(n) is the denominator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding numerator is A101926(n). - Mohammed Yaseen, Jul 25 2023

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:9 at page 51.

T. D. Noe, Table of n, a(n) for n = 0..200
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Mats Erik Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Eric Weisstein's World of Mathematics, Heads-Minus-Tails Distribution, Random Walk--1-Dimensional, Circle Line Picking.

The denominator is given in A046161.
Largest odd divisors of A001800, A002011, A002457, A005430, A033876, A086228.
Bisection of A004731, A004735, A086116.
Second column of triangle A100258.
Cf. A161199, A161201, A173384, A173396.
Cf. A002596 (numerators in expansion of (1-x)^(1/2)).
Cf. A161198 (triangle related to the series expansions of (1-x)^((-1-2*n)/2)).
A163590 is the odd part of the swinging factorial, A001790 at even indices. - Peter Luschny, Aug 01 2009

A001803(n) = sum(<<(A001790(k), A005187(n) - A005187(k)) for k in 0:n) # Peter Luschny, Oct 03 2019

A001803:= func< n | Numerator(Binomial(n+2,2)*Catalan(n+1)/4^n) >;
[A001803(n): n in [0..30]]; // G. C. Greubel, Apr 27 2025

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
a := n -> swing(2*n+1)/sigma(2*n+1); # Peter Luschny, Aug 01 2009
A001803 := proc(n) (2*n+1)*binomial(2*n,n)/4^n ; numer(%) ; end proc: # R. J. Mathar, Jul 06 2011
a := n -> denom(Pi*binomial(n, -1/2)): seq(a(n), n = 0..22); # Peter Luschny, Dec 06 2024

Numerator/@CoefficientList[Series[(1-x)^(-3/2),{x,0,25}],x]  (* Harvey P. Dale, Feb 19 2011 *)
Table[Denominator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *)

a(n) = numerator((2*n+1)*binomial(2*n,n)/(4^n)); \\ Altug Alkan, Sep 06 2018

def A001803(n): return numerator((n+1)*binomial(2*n+2,n+1)/2^(2*n+1))
print([A001803(n) for n in range(31)]) # G. C. Greubel, Apr 27 2025

a(n) = (2*n + 1)! /(n!^2*2^A000120(n)) = (n + 1)*binomial(2*n+2,n+1)/2^(A000120(n)+1). - Ralf Stephan, Mar 10 2004
From Johannes W. Meijer, Jun 08 2009: (Start)
a(n) = numerator( (2*n+1)*binomial(2*n,n)/(4^n) ).
(1 - x)^(-3/2) = Sum_{n>=0} ((2*n+1)*binomial(2*n,n)/4^n)*x^n. (End)
Truncations of rational expressions like those given by the numerator or denominator operators are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n+1)$ / sigma(2*n+1) = A056040(2*n+1) / A060632(2*n+2). Simply said: This sequence gives the odd part of the swinging factorial at odd indices. - Peter Luschny, Aug 01 2009
a(n) = denominator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 06 2024

A038534 Numerators of coefficients of EllipticK/Pi.

Original entry on oeis.org

1, 1, 9, 25, 1225, 3969, 53361, 184041, 41409225, 147744025, 2133423721, 7775536041, 457028729521, 1690195005625, 25145962430625, 93990019574025, 90324408810638025, 340357374376418025, 5147380044581630625, 19520119892056100625, 1187604094232693162025

Offset: 0

Wouter Meeussen, revised Jan 03 2001

The denominators are given in A038533.
Also numerators in expansion of the hypergeometric series 2F1(1/2,1/2; 1; x).
This means numerators of the expansion coefficients of 2*K(k)/Pi = 2F1(1/2,1/2; 1; k^2) in powers of k^2, with K(k) the complete elliptic integral of the first kind. The denominators are given in A056982. The period T of the plane pendulum (mass m, length L, Earth's gravity g, energy E) is 4*sqrt(L/g)*K(sin(phi_0/2)) with cos(phi_0) = -E/(m*g*L) (maximal phi value). See the Landau - Lifschitz reference, p. 30. - Wolfdieter Lang, May 29 2016
It is easy and inexpensive to make a satisfactory precision measurement of a(1)/4, a(2)/64, and a(3)/256 using a pendulum rigged from a computer mouse. In "Digital Pendulum Data Analysis" (see links) amplitude vs. time data is transformed to period vs. sin(phi_0/2)^2 data, thus allowing extraction of expansion coefficients as fit parameters. - Bradley Klee, Dec 25 2016

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 91, Eq. 2.1.
L. D. Landau und E. M. Lifschitz, Mechanik, Akademie Verlag, Berlin, 1967, p. 30 (Exercise 1 in chapter III, paragraph 11.)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
B. Klee, Digital Pendulum Data Analysis: Output, Github, 2016.
David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, arXiv:2109.00027 [math.AG], 2021. See (1.2) p. 1.
G. N. Watson, A Note on Gamma Functions.Edinburgh Mathematical Notes, 42, 1959, pp 7-9.

Cf. A001790, A038533, A038535, A056040, A056982, A060632.

swing := proc(n) option remember; if n = 0 then 1 elif n mod 2 = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i, i = convert(iquo(n, 2), base, 2))):
a := n -> (swing(2*n)/sigma(2*n))^2; seq(a(n),n=0..20); # Peter Luschny, Aug 06 2014

Numerator@ CoefficientList[ Series[ EllipticK@x, {x, 0, 19}]/Pi, x] (* Robert G. Wilson v, Jul 19 2007 *)

a(n) = 2^(-2*w(n))*binomial(2*n,n)^2 with w(n) = A000120(n), the number of 1's in binary expansion of n.
a(n) = A001790(n)^2.
a(n) = (A056040(2*n)/A060632(2*n))^2. - Peter Luschny, Aug 06 2014
a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n-1/2,n). - Peter Luschny, Apr 08 2016
a(n) = numerator(((2*n)!/(2^(2*n)*(n!)^2))^2). - Stefano Spezia, May 01 2025

A151566 Leftist toothpicks (see Comments for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 14, 18, 20, 22, 26, 30, 34, 38, 46, 54, 56, 58, 62, 66, 70, 74, 82, 90, 94, 98, 106, 114, 122, 130, 146, 162, 164, 166, 170, 174, 178, 182, 190, 198, 202, 206, 214, 222, 230, 238, 254, 270, 274, 278, 286, 294, 302, 310, 326, 342, 350, 358, 374, 390, 406

Offset: 0

N. J. A. Sloane, May 23 2009

Similar to A139250, except that when we add toothpicks to horizontal toothpicks, we only add them at the left-hand end.
Sequence gives total number of toothpicks in the n-th generation. First differences are in A060632.
This is equivalent to the Sierpinski triangle A047999. Each inverted T formed by two toothpicks is equivalent to a triangle in the Sierpinski sieve. See Gould's sequence A001316. [From Omar E. Pol, May 23 2009]

Seiichi Manyama, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A001316, A006046.

a(2n) = 2*A006046(n), a(2n+1) = a(2n) + A001316(n) = 2*A006046(n) + A001316(n).

G.f.: (x*(1+x)/(1-x)) * Product_{k>0} (1 + 2 * x^(2^k)). - Seiichi Manyama, Oct 12 2019

A163590 Odd part of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395, 12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075, 35102025, 5014575, 145422675, 9694845, 300540195, 300540195, 9917826435, 583401555, 20419054425, 2268783825

Offset: 0

Peter Luschny, Aug 01 2009

nonn

Let n$ denote the swinging factorial. a(n) = n$ / 2^sigma(n) where sigma(n) is the exponent of 2 in the prime-factorization of n$. sigma(n) can be computed as the number of '1's in the base 2 representation of floor(n/2).

If n is even then a(n) is the numerator of the reduced ratio (n-1)!!/n!! = A001147(n-1)/A000165(n), and if n is odd then a(n) is the numerator of the reduced ratio n!!/(n-1)!! = A001147(n)/A000165(n-1). The denominators for each ratio should be compared to A060818. Here all ratios are reduced. - Anthony Hernandez, Feb 05 2020 [See the Mathematica program for a more compact form of the formula. Peter Luschny, Mar 01 2020 ]

			11$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 3^2*7*11 = 2772/4 = 693.
From _Anthony Hernandez_, Feb 04 2019: (Start)
a(7) = numerator((1*3*5*7)/(2*4*6)) = 35;
a(8) = numerator((1*3*5*7)/(2*4*6*8)) = 35;
a(9) = numerator((1*3*5*7*9)/(2*4*6*8)) = 315;
a(10) = numerator((1*3*5*7*9)/(2*4*6*8*10)) = 63. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A060632, A001790, A001803.

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
a := n -> swing(n)/sigma(n);

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/ f!]; a[n_] := With[{s = sf[n]}, s/2^IntegerExponent[s, 2]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *)
r[n_] := (n - Mod[n - 1, 2])!! /(n - 1 + Mod[n - 1, 2])!! ;
Table[r[n], {n, 0, 36}] // Numerator (* Peter Luschny, Mar 01 2020 *)

A163590(n) = {
    my(a = vector(n+1)); a[1] = 1;
    for(n = 1, n,
        a[n+1] = a[n]*n^((-1)^(n+1))*2^valuation(n, 2));
a } \\ Peter Luschny, Sep 29 2019

# uses[A000120]
@CachedFunction
def swing(n):
    if n == 0: return 1
    return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
A163590 = lambda n: swing(n)/2^A000120(n//2)
[A163590(n) for n in (0..31)]  # Peter Luschny, Nov 19 2012
# Alternatively:

@cached_function
def A163590(n):
    if n == 0: return 1
    return A163590(n - 1) * n^((-1)^(n + 1)) * 2^valuation(n, 2)
print([A163590(n) for n in (0..31)]) # Peter Luschny, Sep 29 2019

a(2*n) = A001790(n).
a(2*n+1) = A001803(n).
a(n) = a(n-1)*n^((-1)^(n+1))*2^valuation(n, 2) for n > 0. - Peter Luschny, Sep 29 2019

A088042 Number of permutations in the symmetric group S_n such that the size of their conjugacy class is odd.

Original entry on oeis.org

1, 2, 4, 4, 16, 76, 232, 106, 946, 5716, 27776, 63856, 272416, 2390480, 10349536, 2027026, 34459426, 344594404, 2618916472, 10475679736, 54997260256, 568305978472, 3132225435824, 1807129471456, 12047128545376, 175289251587776, 1326384554695552

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 02 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A088994, A060632.

Cf. A000085, A001189.

a:= n-> n!*add((binomial(n-(n mod 2), 2*k) mod 2)/((n-2*k)!*k!*2^k),
        k=0..floor(n/2)):
seq(a(n), n=1..30);  # Alois P. Heinz, May 01 2013

a[n_] := n!*Sum[Mod[Binomial[n-Mod[n, 2], 2*k], 2]/((n-2*k)!*k!*2^k), {k, 0, Floor[n/2]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

a(n) = Sum_{k=0..floor(n/2)} n!/(n-2*k)!/k!/2^k*(C(n-(n mod 2), 2*k) mod 2). - Vladeta Jovovic, Nov 06 2003

More terms from Vladeta Jovovic, Nov 03 2003

Showing 1-10 of 16 results. Next

cp's OEIS Frontend

A173531 a(0)=0: For n>0, a(n) = A060632(n)*A060632(n-1).

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A127973 a(2n)=A060632(n); a(2n+1)=A048896(n)/2.

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A151565 Duplicate of A060632.

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A047999 Sierpiński's [Sierpinski's] triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 2.

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A001790 Numerators in expansion of 1/sqrt(1-x).

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A001803 Numerators in expansion of (1 - x)^(-3/2).

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A038534 Numerators of coefficients of EllipticK/Pi.

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