A000918 -id:A000918 - cp's OEIS frontend

A159741 a(n) = 8*(2^n - 1).

8, 24, 56, 120, 248, 504, 1016, 2040, 4088, 8184, 16376, 32760, 65528, 131064, 262136, 524280, 1048568, 2097144, 4194296, 8388600, 16777208, 33554424, 67108856, 134217720, 268435448, 536870904, 1073741816, 2147483640, 4294967288, 8589934584, 17179869176, 34359738360

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Fifth diagonal of the array which contains m-acci numbers in the m-th row.
The base array is constructed from m-acci numbers starting each with 1, 1, and 2 and filling one row of the table (see the examples).
The main and the upper diagonals of the table are the powers of 2, A000079.
The first subdiagonal is essentially A000225, followed by essentially A036563.
The next subdiagonal is this sequence here, followed by A159742, A159743, A159744, A159746, A159747, A159748.
a(n) written in base 2: 1000, 11000, 111000, 1111000, ..., i.e., n times 1 and 3 times 0 (A161770). - Jaroslav Krizek, Jun 18 2009
Also numbers for which n^8/(n+8) is an integer. - Vicente Izquierdo Gomez, Jan 03 2013

			From _R. J. Mathar_, Apr 22 2009: (Start)
The base table is
.1..1....1....1....1....1....1....1....1....1....1....1....1....1
.1..1....1....1....1....1....1....1....1....1....1....1....1....1
.2..2....2....2....2....2....2....2....2....2....2....2....2....2
.0..2....3....4....4....4....4....4....4....4....4....4....4....4
.0..2....5....7....8....8....8....8....8....8....8....8....8....8
.0..2....8...13...15...16...16...16...16...16...16...16...16...16
.0..2...13...24...29...31...32...32...32...32...32...32...32...32
.0..2...21...44...56...61...63...64...64...64...64...64...64...64
.0..2...34...81..108..120..125..127..128..128..128..128..128..128
.0..2...55..149..208..236..248..253..255..256..256..256..256..256
.0..2...89..274..401..464..492..504..509..511..512..512..512..512
.0..2..144..504..773..912..976.1004.1016.1021.1023.1024.1024.1024
.0..2..233..927.1490.1793.1936.2000.2028.2040.2045.2047.2048.2048
.0..2..377.1705.2872.3525.3840.3984.4048.4076.4088.4093.4095.4096
Columns: A000045, A000073, A000078, A001591, A001592 etc. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A000225, A036563, A159742, A159743, A159744, A159746, A159747, A159748.

Cf. A000918, A028399, A161770, A173787, A175161, A175166.

[8*(2^n -1): n in [1..50]]; // G. C. Greubel, May 22 2018

T := proc(n,m) option remember ; if n < 0 then 0; elif n <= 1 then 1; elif n = 2 then 2; else add(procname(n-i,m),i=1..m) ; fi: end: A159741 := proc(n) T(n+4,n+1) ; end: seq(A159741(n),n=1..40) ; # R. J. Mathar, Apr 22 2009

Table[8(2^n-1),{n,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{3,-2},{8,24},30] (* Harvey P. Dale, Jan 01 2019 *)

a(n)=8*(2^n-1) \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = 8*(2^n-1).
G.f.: 8*x/((2*x-1)*(x-1)). (End)
From Jaroslav Krizek, Jun 18 2009: (Start)
a(n) = Sum_{i=3..(n+2)} 2^i.
a(n) = Sum_{i=1..n} 2^(i+2).
a(n) = a(n-1) + 2^(n+2) for n >= 2. (End)
a(n) = A173787(n+3,3) = A175166(2*n)/A175161(n). - Reinhard Zumkeller, Feb 28 2010
From Elmo R. Oliveira, Jun 15 2025: (Start)
E.g.f.: 8*exp(x)*(exp(x) - 1).
a(n) = 8*A000225(n) = 4*A000918(n+1) = 2*A028399(n+2). (End)

More terms from R. J. Mathar, Apr 22 2009
Edited by Al Hakanson (hawkuu(AT)gmail.com), May 11 2009
Comments claiming negative entries deleted by R. J. Mathar, Aug 24 2009

A030130 Binary expansion contains a single 0.

Original entry on oeis.org

0, 2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039

Offset: 1

Toby Donaldson (tjdonald(AT)uwaterloo.ca)

From Reinhard Zumkeller, Aug 29 2009: (Start)
A023416(a(n)) = 1;
apart from the initial term the sequence can be seen as a triangle read by rows, see A164874;
A055010 and A086224 are subsequences, see also A000918 and A036563. (End)
Zero and numbers of form 2^m-2^k-1, 2 <= m, 0 <= k <= m-2. - Zak Seidov, Aug 06 2010

			23 is OK because it is '10111' in base 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000918, A023416, A036563, A055010, A086224, A140977, A160502, A164874, A190620.

long int element (long int i) { return (pow(2,g(i))-1-pow(2,(pow(2*g(i)-1,2)-1-8*i)/8));} long int g(long int m) {if (m==0) return(1); return ((sqrt(8*m-7)+3)/2);}

a030130 n = a030130_list !! (n-1)
a030130_list = filter ((== 1) . a023416) [0..]
-- Reinhard Zumkeller, Mar 31 2015, Dec 07 2012

[0] cat [k:k in [0..2050]| Multiplicity(Intseq(k,2),0) eq 1]; // Marius A. Burtea, Feb 06 2020

Sort[Flatten[{{0}, Table[2^n - 2^m - 1, {n, 2, 50}, {m, 0, n - 2}]}]] (* Zak Seidov, Aug 06 2010 *)
Select[Range[0,2100],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Dec 19 2021 *)

print1("0, ");for(k=1,2039,my(v=digits(k,2));if(vecsum(v)==#v-1,print1(k,", "))) \\ Hugo Pfoertner, Feb 06 2020

from math import isqrt, comb
def A030130(n): return (1<<(a:=(isqrt(n-1<<3)+1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

a(n) = 2^(g(n))-1-2^(((2*g(n)-1)^2-1-8*n)/8) with g(n)=int((sqrt(8*n-7)+3)/2) for all n>0 and g(0)=1. - Ulrich Schimke (ulrschimke(AT)aol.com)
a(n+1) = A140977(a(n)) for any n > 1. - Rémy Sigrist, Feb 06 2020
Sum_{n>=2} 1/a(n) = A160502. - Amiram Eldar, Oct 06 2020
a(n) = (A190620(n-1)-1)/2. - Chai Wah Wu, Dec 19 2024

More terms from Erich Friedman

Offset fixed by Reinhard Zumkeller, Aug 24 2009

A056453 Number of palindromes of length n using exactly two different symbols.

Original entry on oeis.org

0, 0, 2, 2, 6, 6, 14, 14, 30, 30, 62, 62, 126, 126, 254, 254, 510, 510, 1022, 1022, 2046, 2046, 4094, 4094, 8190, 8190, 16382, 16382, 32766, 32766, 65534, 65534, 131070, 131070, 262142, 262142, 524286, 524286, 1048574, 1048574, 2097150, 2097150, 4194302

Offset: 1

Marks R. Nester

Also the number of bitstrings of length n+2 where the last two runs have the same length. (A run is a maximal subsequence of (possibly just one) identical bits.) - David Nacin, Mar 03 2012

Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Apr 22 2017

			The palindromes in two symbols of length three take the forms 000, 111, 010, 101. Of these only two have exactly two symbols.  Thus a(3) = 2. - _David Nacin_, Mar 03 2012

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A000918, A016116, A056391, A208900, A208901, A208902, A208903.

[2^Floor((n+1)/2)-2: n in [1..40]]; // Vincenzo Librandi, Aug 16 2011

Table[2^(Floor[n/2] + 1) - 2, {n, 0, 40}] (* David Nacin, Mar 03 2012 *)
LinearRecurrence[{1, 2, -2}, {0, 0, 2}, 40] (* David Nacin, Mar 03 2012 *)
k=2; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 2^((n+1)\2)-2; \\ Altug Alkan, Sep 25 2018

def A056453(n): return (1<<(n+1>>1))-2 # Chai Wah Wu, Feb 18 2024

a(n) = 2^floor((n+1)/2) - 2.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3). - David Nacin, Mar 03 2012
G.f.: 2*x^3/((1-x)*(1-2*x^2)). - David Nacin, Mar 03 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=2 is the number of symbols. - Robert A. Russell, Sep 25 2018
a(n) = k! S2(ceiling(n/2),k), where k=2 is the number of symbols and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018
E.g.f.: 1 - 2*cosh(x) + cosh(sqrt(2)*x) - 2*sinh(x) + sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Jun 06 2023

More terms from Vincenzo Librandi, Aug 16 2011

Name clarified by Michel Marcus and Felix Fröhlich, Jul 09 2018

A094262 Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 14, 61, 124, 131, 70, 15, 1, 30, 240, 890, 1830, 2226, 1600, 630, 105, 1, 62, 841, 5060, 16990, 35216, 47062, 40796, 22225, 6930, 945, 1, 126, 2772, 25410, 127953, 401436, 836976, 1196532, 1182195, 795718, 349020, 90090, 10395

Offset: 1

André F. Labossière, Jun 01 2004

The original name for this sequence was "Triangle read by rows giving the coefficients of formulas generating each variety of S2(n,k) (Stirling numbers of 2nd kind). The p-th row (p>=1) contains T(i,p) for i=1 to 2*p-1, where T(i,p) satisfies Sum_{i=1..2*p-1} T(i,p) * C(n-p,i-1)".
The terms of the n-th diagonal sequence of the triangle of Stirling numbers of the second kind A008277, i.e., (Stirling2(N + n - 1,N)), N>=1, are given by a polynomial in N of degree 2*n - 2. This polynomial may be expressed as a linear combination of the falling factorial polynomials binomial(N - n,0), binomial(N - n,1), ... , binomial(N - n,2*n - 2). This table gives the coefficients in these expansions.
The formulas obtained are those for Stirling2(N+1,N) (A000217), Stirling2(N+2,N) (A001296), Stirling2(N+3,N) (A001297), Stirling2(N+4,N) (A001298), Stirling2(N+5,N) (A112494), Stirling2(N+6,N) (A144969) and so on.

			Row 5 contains 1,30,240,890,1830,2226,1600,630,105, so the formula generating Stirling2(n+4,n) numbers (A001298) will be the following: 1 + 30*(n-5) + 240*C(n-5,2) + 890*C(n-5,3) + 1830*C(n-5,4) + 2226*C(n-5,5) + 1600*C(n-5,6) + 630*C(n-5,7) + 105*C(n-5,8). For example, taking n = 9 gives Stirling2(13,9) = 359502.
Triangle starts:
  1;
  1,  2,   1;
  1,  6,  12,  10,    3;
  1, 14,  61, 124,  131,   70,   15;
  1, 30, 240, 890, 1830, 2226, 1600, 630, 105;
  ...
From _Peter Bala_, Jun 14 2016: (Start)
Connection with row polynomials of A134991:
  R(2,z) = (1 + z)^2*z
  R(3,z) = (1 + z)^2*(z + 3*z^2)
  R(4,z) = (1 + z)^4*(z + 10*z^2 + 15*z^3)
  R(5,z) = (1 + z)^5*(z + 25*z^2 + 105*z^3 + 105*z^4). (End)
From _Andrew Howroyd_, Mar 28 2025: (Start)
The T(3,3) = 12 trees up to relabeling have one of the following 3 forms:
     {}         {1}        {1}
    /  \       /   \        |
  {1} {2,3}   {2}  {3}     {2}
                            |
                           {3}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..2500 (rows 1..50)
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].
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions (with Formulas, Graphs and Mathematical Tables), U.S. Dept. of Commerce, National Bureau of Standards, Applied Math. Series 55, 1964, 1046 pages (9th Printing: November 1970) - Combinatorial Analysis, Table 24.4, Stirling Numbers of the Second Kind (author: Francis L. Miksa), p. 835.
J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.
M. Kazarian, KP hierarchy for Hodge integrals, p. 2, arxiv:0809.3263 [math.AG], 18 Sep 2008. [From _Tom Copeland_, Jun 12 2015]
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.

Row sums are A005172.
Columns 1..4 are A000012, A000918, A005173, A005174, A005175.
Cf. A008277, A000217, A001296, A001297, A001298, A008517, A134991, A112494, A144969.

row_poly := n -> (1+z)^(n+1)*add(z^k*add((-1)^(m+k)*binomial(n+k,n+m)*Stirling2(n+m,m), m=0..k), k=0..n): T_row := n -> seq(coeff(row_poly(n),z,j),j=1..2*n+1):
seq(T_row(n),n=0..6); # Peter Luschny, Jun 15 2016

Clear[T, q, u]; T[0] = q[1];T[n_] := Sum[m*(u^2*q[m] + 2*u*q[m+1] + q[m+2])*D[T[n-1], q[m]], {m, 1, 2*n+1}]; row[n_] := List @@ Expand[T[n-1]] /. {u -> 1, q[] -> 1}; Table[row[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover, Jun 12 2015 *)

T(n)={my(g=serreverse(log(((1+1/y)*x+1)/exp(x + O(x*x^n))))); [Vecrev(p/y) | p<-Vec(serlaplace(g))]}
{ my(A=T(5)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 28 2025

Apparently, a raising operator for bivariate polynomials P(n,u,z) having these coefficients is R = (u+z)^2 * z * d/dz with P(0,u,z) = z. E.g., R P(1,u,z) = R^2 P(0,u,z) = R^2 z = u^4 z + 6 u^3 z^2 + 12 u^2 z^3 + 10 u z^4 + 3 z^5 = P(2,u,z). See the Kazarian link. - Tom Copeland, Jun 12 2015
Reverse polynomials seem to be generated by 1 + exp[t*(x+1+z)^2*(1+z)d/dz]z evaluated at z = 0. - Tom Copeland, Jun 13 2015
From Peter Bala, Jun 14 2016: (Start)
T(n,k) = k*T(n,k) + 2*(k - 1)*T(n,k-1) + (k - 2)*T(n,k-2).
n-th diagonal of A008277: Stirling2(N + n - 1,N) = Sum_{k = 1..2*n - 1} T(n,k)*binomial(N - n,k - 1) for N = 1,2,3,....
Row polynomials R(n,z) = Sum_{k >= 1} k^(n+k-1)*( z/(1 + z)*exp(-z/(1 + z)) )^k/k!, n = 1,2,..., follows from the formula given in A008277 for the o.g.f.'s of the diagonals of the Stirling numbers of the second kind.
Consequently, R(n+1,z) = (1 + z)^2*z*d/dz(R(n,z)) for n >= 1 as conjectured above by Copeland.
R(n,z) = (1 + z)^n*P(n,z) where P(n,z) are the row polynomials of A134991.
R(n,z) = (1 + z)^(2*n+1)*B(n,z/(1 + z)), where B(n,z) are the row polynomials of the triangle of second-order Eulerian numbers A008517 (see Barbero et al., Section 6, equation 27). (End)
Based on the comment of Bala the row polynomials have the explicit form R(n, z) = (1+z)^(n+1)*Sum_{k=0..n}(z^k*Sum_{m=0..k}((-1)^(m+k)*binomial(n+k, n+m)* Stirling2(n+m,m))). - Peter Luschny, Jun 15 2016
E.g.f. G(x,y) satisfies G(x,y) = y*(exp(x)*exp(G(x,y)) - G(x,y) - 1). - Andrew Howroyd, Mar 28 2025

Edited and Name changed by Peter Bala, Jun 16 2016

Name changed by Andrew Howroyd, Mar 28 2025

A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20

Offset: 1

Antti Karttunen, Dec 18 2015

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).
Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

			The top left corner of the array:
   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...
   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...
   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...
   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...
   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...
  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...
  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...
  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...
  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...
  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...
  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...
  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...
  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...
  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...
  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...
  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A267102.
Transpose: A265903.
Cf. A265900 (main diagonal).
Cf. A162598 (row index of n in array), A265332 (column index of n in array).
Cf. A004001, A051135, A088359, A087686.
Column 1: A188163.
Column 2: A266109.
Row 1: A000079 (2^n).
Row 2: A000225 (2^n - 1, from 3 onward).
Row 3: A000325 (2^n - n, from 5 onward).
Row 4: A000918 (2^n - 2, from 6 onward).
Row 5: A084634 (?, from 9 onward).
Row 6: A132732 (2^n - 2n + 2, from 10 onward).
Row 7: A000295 (2^n - n - 1, from 11 onward).
Row 8: A036563 (2^n - 3).
Row 9: A084635 (?, from 17 onward).
Row 12: A048492 (?, from 20 onward).
Row 13: A249453 (?, from 22 onward).
Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).
Row 15: A000247 (2^n - n - 2, from 25 onward).
Row 16: A028399 (2^n - 4).
Cf. also permutations A267111, A267112.

(define (A265901 n) (A265901bi (A002260 n) (A004736 n)))
(define (A265901bi row col) (if (= 1 col) (A188163 row) (A087686 (+ 1 (A265901bi row (- col 1))))))

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

A000920 Differences of 0: 6!*Stirling2(n,6).

Original entry on oeis.org

0, 0, 0, 0, 0, 720, 15120, 191520, 1905120, 16435440, 129230640, 953029440, 6711344640, 45674188560, 302899156560, 1969147121760, 12604139926560, 79694820748080, 499018753280880, 3100376804676480, 19141689213218880, 117579844328562000

Offset: 1

N. J. A. Sloane

Number of surjections from an n-element set onto a six-element set, with n >= 6. - Mohamed Bouhamida, Dec 15 2007

Number of rows of n colors using exactly six colors. For n=6, the 720 rows are the 720 permutations of ABCDEF. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A001117, A001118, A000918, A000919, A000770.

Column 6 of A019538.

[6^n-Binomial(6,5)*5^n+Binomial(6,4)*4^n-Binomial(6,3)*3^n+Binomial(6,2)*2^n-Binomial(6,1): n in [1..30]]; // Vincenzo Librandi, May 18 2015

720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);

CoefficientList[Series[(720*x^5)/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 6!*stirling(n, 6, 2); \\ Altug Alkan, Sep 25 2018

a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).
a(n) = 6^n-C(6,5)*5^n+C(6,4)*4^n-C(6,3)*3^n+C(6,2)*2^n-C(6,1) with n>=6. - Mohamed Bouhamida, Dec 15 2007
G.f.: 720*x^6/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]
a(n) = 720*A000770(n). - R. J. Mathar, Apr 30 2015
E.g.f.: (exp(x) - 1)^6. - Geoffrey Critzer, May 17 2015

A052386 Number of integers from 1 to 10^n-1 that lack 0 as a digit.

Original entry on oeis.org

0, 9, 90, 819, 7380, 66429, 597870, 5380839, 48427560, 435848049, 3922632450, 35303692059, 317733228540, 2859599056869, 25736391511830, 231627523606479, 2084647712458320, 18761829412124889, 168856464709124010, 1519708182382116099, 13677373641439044900

Offset: 0

Odimar Fabeny, Mar 10 2000

			For n=2, the numbers from 1 to 99 which *have* 0 as a digit are the 9 numbers 10, 20, 30, ..., 90. So a(1) = 99 - 9 = 90.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A024101, A052379.

Row n=9 of A228275.

[9*(9^n-1)/8: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

Table[9(9^n - 1)/8, {n, 0, 20}]
LinearRecurrence[{10,-9},{0,9},30] (* Harvey P. Dale, Mar 22 2019 *)

a(n)=9^(n+1)\8 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = 9*a(n-1) + 9.
a(n) = 9*(9^n-1)/8 = sum_{j=1..n} 9^j = a(n-1)+9^n = 9*A002452(n) = A002452(n+1)-1; write A000918(n+1) in base 2 and read as if written in base 9. - Henry Bottomley, Aug 30 2001
a(n) = 10*a(n-1)-9*a(n-2). G.f.: 9*x / ((x-1)*(9*x-1)). - Colin Barker, Sep 26 2013

More terms and revised description from James Sellers, Mar 13 2000
More terms and revised description from Robert G. Wilson v, Apr 14 2003
More terms from Colin Barker, Sep 26 2013

A168604 a(n) = 2^(n-2) - 1.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591

Offset: 3

Martin Griffiths, Dec 01 2009

Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly two nonempty parts.

An elephant sequence, see A175655. For the central square six A[5] vectors, with decimal values between 26 and 176, lead to this sequence. For the corner squares these vectors lead to the companion sequence A000325 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010

			The partitions of {1,1,1,2,3} into exactly two nonempty parts are {{1},{1,1,2,3}}, {{2},{1,1,1,3}}, {{3},{1,1,1,2}}, {{1,1},{1,2,3}}, {{1,2},{1,1,3}}, {{1,3},{1,1,2}} and {{2,3},{1,1,1}}.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5
Index entries for linear recurrences with constant coefficients, signature (3,-2).

The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly three and four nonempty parts are given in A168605 and A168606, respectively.

[2^(n-2)-1 : n in [3..35]]; // Vincenzo Librandi, May 13 2011

f4[n_] := 2^(n - 2) - 1; Table[f4[n], {n, 3, 30}]
LinearRecurrence[{3,-2},{1,3},40] (* Harvey P. Dale, Oct 20 2013 *)

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

E.g.f.: 2*exp(2*x)-exp(x).
a(n) = A000225(n-2).
G.f.: x^3/((1-x)*(1-2*x))
a(n) = A126646(n-3). - R. J. Mathar, Dec 11 2009
a(n) = 3*a(n-1) - 2*a(n-2). - Arkadiusz Wesolowski, Jun 14 2013
a(n) = A000918(n-2) + 1. - Miquel Cerda, Aug 09 2016

A007179 Dual pairs of integrals arising from reflection coefficients.

Original entry on oeis.org

0, 1, 1, 4, 6, 16, 28, 64, 120, 256, 496, 1024, 2016, 4096, 8128, 16384, 32640, 65536, 130816, 262144, 523776, 1048576, 2096128, 4194304, 8386560, 16777216, 33550336, 67108864, 134209536, 268435456, 536854528, 1073741824, 2147450880, 4294967296, 8589869056

Offset: 0

N. J. A. Sloane, Simon Plouffe

			From _Gus Wiseman_, Feb 26 2022: (Start)
Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
  (1)  (1,1)  (3)      (1,3)      (5)
              (1,2)    (3,1)      (1,4)
              (2,1)    (1,1,2)    (2,3)
              (1,1,1)  (1,2,1)    (3,2)
                       (2,1,1)    (4,1)
                       (1,1,1,1)  (1,1,3)
                                  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
(End)

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Column k=2 of A309748.
Odd bisection is A000302.
Even bisection is A006516 = 2^(n-1)*(2^n - 1).
The complement is counted by A077957, internal version A027383.
The internal case is A274230, even bisection A134057.
A000045(n-1) counts compositions without odd parts, non-singleton A077896.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A034871, A097805, and A345197 count compositions by alternating sum.
A052952 (or A074331) counts non-singleton compositions without even parts.
Cf. A000918, A033484, A052955, A060867, A116406, A138364.

[Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011

f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;

LinearRecurrence[{2,2,-4},{0,1,1},30] (* Harvey P. Dale, Nov 30 2015 *)
Table[2^(n-1)-If[EvenQ[n],2^(n/2-1),0],{n,0,15}] (* Gus Wiseman, Feb 26 2022 *)

Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016

From Paul Barry, Apr 28 2004: (Start)
  Binomial transform is (A000244(n)+A001333(n))/2.
  G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
  a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
  a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
a(n) = A011782(n+2) - A077957(n) - Gus Wiseman, Feb 26 2022

A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 4, 4, 2, 1, 0, 4, 6, 4, 2, 1, 0, 8, 12, 8, 4, 2, 1, 0, 8, 18, 14, 8, 4, 2, 1, 0, 16, 36, 28, 16, 8, 4, 2, 1, 0, 16, 54, 48, 30, 16, 8, 4, 2, 1, 0, 32, 108, 96, 60, 32, 16, 8, 4, 2, 1, 0, 32, 162, 164, 110, 62, 32, 16, 8, 4, 2, 1, 0, 64, 324, 328, 220, 124, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Henry Bottomley, Mar 06 2002

			Rows start:
  1,  0,  0,  0,  0, ...
  1,  2,  2,  4,  4, ...
  1,  2,  4,  6, 12, ...
  1,  2,  4,  8, 14, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Cf. early rows: A000007, A016116 (without initial term), A068911, A068912, A216212, A216241, A235701.

Central and lower diagonals are A000079, higher diagonals include A000918, A028399.

T[n_,0]=1; T[n_,k_]:=2^k/(n+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (n+1))]^k Cot[(Pi (1-2r))/(4 (n+1))],{r,1,n+1}]; Table[T[r,n-r],{n,0,20},{r,0,n}]//Round//Flatten (* Herbert Kociemba, Sep 23 2020 *)

Starting with T(n, 0) = 1, if (k-n) is negative or even then T(n, k) = 2*T(n, k-1), otherwise T(n, k) = 2*T(n, k-1) - A061897(n+1, (k-n-1)/2). So for n>=k, T(n, k) = 2^k. [Corrected by Sean A. Irvine, Mar 23 2024]

T(n,0) = 1, T(n,k) = (2^k/(n+1))*Sum_{r=1..n+1} (-1)^r*cos((Pi*(2*r-1))/(2*(n+1)))^k*cot((Pi*(1-2*r))/(4*(n+1))). - Herbert Kociemba, Sep 23 2020

cp's OEIS Frontend

A159741 a(n) = 8*(2^n - 1).

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A030130 Binary expansion contains a single 0.

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A056453 Number of palindromes of length n using exactly two different symbols.

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A094262 Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

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A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

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A000920 Differences of 0: 6!*Stirling2(n,6).

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