A005060 -id:A005060 - cp's OEIS frontend

A059802 Numbers k such that 5^k - 4^k is prime.

3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937

Offset: 1

Mike Oakes, Feb 23 2001

Some of the larger terms may only correspond to probable primes.
5^1663 - 4^1663, a 1163-digit number, has been certified prime with Primo. - Rick L. Shepherd, Nov 13 2002
4 more terms found by Predrag Minovic in 2004: 35449, 36697, 45259, 91493. Corresponding numbers of decimal digits are 24778, 25651, 31635, 63951. - Alexander Adamchuk, Dec 02 2006

Cf. A005060.

Cf. A000043, A057468, A059801, A128335, etc.

Select[Range[1000], PrimeQ[5^# - 4^#] &] (* Alonso del Arte, Sep 09 2013 *)

forprime(p=2,1e5,if(ispseudoprime(5^p-4^p),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

New term 246497 found by Jean-Louis Charton in 2008 corresponding to a probable prime with 172295 digits - Jean-Louis Charton, Sep 02 2009
New term a(19) = 265007 found by Jean-Louis Charton, Feb 19 2013
a(20) = 289937 found by Jean-Louis Charton, Mar 15 2013

A005061 a(n) = 4^n - 3^n.

Original entry on oeis.org

0, 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807, 70274600998837, 281192547174175, 1125052618233181

Offset: 0

N. J. A. Sloane

Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row, see A359576. - R. H. Hardin, Mar 21 2002
Number of binary vectors (x_1, x_2, ..., x_{2n}) such that in at least one of the disjoint pairs (x_1, x_2), (x_3, x_4), ..., (x_{2n-1}, x_{2n}) both x_{2i-1} and x_{2i} are both 1. Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_3*x_4 + x_5*x_6 + ... +x_{2n-1}*x_{2n} = 1 in base-2 lunar arithmetic. - N. J. A. Sloane, Apr 23 2011
a(n)/4^n is the probability that two randomly selected (with replacement) subsets of [n] will have at least one element in common if the probability of selection is equal for all subsets. - Geoffrey Critzer, May 09 2009
This sequence is also the second column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). (See the e.g.f. given below.) - Wolfdieter Lang, Oct 08 2011
Also, the number of numbers with at most n digits whose largest digit equals 3. See A255463 for the first differences (i.e., ...with exactly n digits...). - M. F. Hasler, May 03 2015
If 2^k | n then a(2^k) | a(n). - Bernard Schott, Oct 08 2020
a(n) is the number of ordered n-tuples with elements from {0,1,2,3} in which any of these elements, say 0, appears at least once. For example, a(2)=7 since 01,10,02,20,03,30,00 are the ordered 2-tuples that contain 0. - Enrique Navarrete, Apr 05 2021
a(n) is the number of n-digit numbers whose smallest decimal digit is 6. - Stefano Spezia, Nov 15 2023

			G.f. = x + 7*x^2 + 37*x^3 + 175*x^4 + 781*x^5 + 3367*x^6 + 14197*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
David Applegate, Marc LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Dominique Désérable, Versatile Topology for Two-Dimensional Cellular Automata, Advances in Cellular Automata, Emergence, Complexity and Computation (ECC Vol 52) Springer, Cham (2025), Ch. 6, pp. 151-186.
John Elias, Illustration of initial terms: Unfolded Sierpinski triangle or 3-branches tree in square configuration
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
Vladeta Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (Russian, translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
Eric Weisstein's World of Mathematics, Power Fractional Parts
Index entries for linear recurrences with constant coefficients, signature (7,-12).
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A001047, A002250, A005060, A005062, A143495, A255463 (first differences), A359576.

Array column A047969(n-1, 3), or triangle's subdiagonal A047969(n+2, n-1), for n >= 1.

List([0..10^2], n->4*n - 3^n); # Muniru A Asiru, Feb 06 2018

[4^n - 3^n: n in [0..25]]; // Vincenzo Librandi, Jun 03 2011

seq(4^n - 3^n, n=0..10^2); # Muniru A Asiru, Feb 06 2018

Table[4^n - 3^n, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{7,-12},{0,1},30] (* Harvey P. Dale, May 04 2012 *)
Table[Numerator[1-(3/4)^n],{n,0,20}] (* see link Wolfram Mathworld - Fred Daniel Kline, Feb 05 2018 *)

a(n)=1<<(n+n)-3^n \\ Charles R Greathouse IV, Jun 16 2011

def a(n): return 4**n - 3**n
print([a(n) for n in range(23)]) # Michael S. Branicky, Sep 01 2021

a(n) = 4*a(n-1) + 3^(n-1) for n>=1. - Xavier Acloque, Oct 20 2003
Binomial transform of A001047. - Ross La Haye, Sep 17 2005
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-4*x)-1/(1-3*x).
E.g.f.: exp(4*x)-exp(3*x). (End)
a(n) = 2^n * Sum_{i=0...n} binomial(n,i)*(2^i-1)/2^i. - Geoffrey Critzer, May 09 2009
a(n) = 7*a(n-1) - 12*a(n-2) for n>=2. - Bruno Berselli, Jan 25 2011
From Joe Slater, Jan 15 2017: (Start)
a(n) = 3*a(n-1) + 4^(n-1) for n>=0.
a(n+1) = Sum_{k=0..n} 4^(n-k) * 3^k. (End)
a(n) = -a(-n) * 12^n for all n in Z. - Michael Somos, Jan 22 2017

A047969 Square array of nexus numbers a(n,k) = (n+1)^(k+1) - n^(k+1) (n >= 0, k >= 0) read by upwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 19, 15, 1, 1, 9, 37, 65, 31, 1, 1, 11, 61, 175, 211, 63, 1, 1, 13, 91, 369, 781, 665, 127, 1, 1, 15, 127, 671, 2101, 3367, 2059, 255, 1, 1, 17, 169, 1105, 4651, 11529, 14197, 6305, 511, 1, 1, 19, 217, 1695, 9031

Offset: 0

N. J. A. Sloane

If each row started with an initial 0 (i.e., a(n,k) = (n+1)^k - n^k) then each row would be the binomial transform of the preceding row. - Henry Bottomley, May 31 2001
a(n-1, k-1) is the number of ordered k-tuples of positive integers such that the largest of these integers is n. - Alford Arnold, Sep 07 2005
From Alford Arnold, Jul 21 2006: (Start)
The sequences in A047969 can also be calculated using the Eulerian Array (A008292) and Pascal's Triangle (A007318) as illustrated below: (cf. A101095).
  1       1       1       1       1       1
  1       1       1       1       1       1
  -----------------------------------------
  1       2       3       4       5       6
          1       2       3       4       5
  1       3       5       7       9      11
  -----------------------------------------
  1       3       6      10      15      21
          4      12      24      40      60
                  1       3       6      10
  1       7      19      37      61      91
  -----------------------------------------
  1       4      10      20      35      56
         11      44     110     220     385
                 11      44     110     220
                          1       4      10
  1      15      65     175     369     671
  -----------------------------------------  (End)
From Peter Bala, Oct 26 2008: (Start)
The above remarks of Alford Arnold may be summarized by saying that (the transpose of) this array is the Hilbert transform of the triangle of Eulerian numbers A008292 (see A145905 for the definition of the Hilbert transform). In this context, A008292 is best viewed as the array of h-vectors of permutohedra of type A. See A108553 for the Hilbert transform of the array of h-vectors of type D permutohedra. Compare this array with A009998.
The polynomials n^k - (n-1)^k, k = 1,2,3,..., which give the nonzero entries in the columns of this array, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re s = 1/2 in the complex plane. See A019538 for the connection between the polynomials n^k - (n-1)^k and the Stirling polynomials of the simplicial complexes dual to the type A permutohedra.
(End)
Empirical: (n+1)^(k+1) - n^(k+1) is the number of first differences of length k+1 arrays of numbers in 0..n, k > 0. - R. H. Hardin, Jun 30 2013
a(n-1, k-1) is the number of bargraphs of width k and height n. Examples: a(1,2) = 7 because we have [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], and [2,2,2]; a(2,1) = 5 because we have [1,3], [2,3], [3,1], [3,2], and [3,3] (bargraphs are given as compositions). This comment is equivalent to A. Arnold's Sep 2005 comment. - Emeric Deutsch, Jan 30 2017

			Array a begins:
  [n\k][0  1   2    3    4   5  6  ...
  [0]   1  1   1    1    1   1  1  ...
  [1]   1  3   7   15   31  63  ...
  [2]   1  5  19   65  211  ...
  [3]   1  7  37  175  ...
  ...
Triangle T begins:
  n\m   0   1    2     3     4      5      6      7      8     9  10 ...
  0:    1
  1:    1   1
  2:    1   3    1
  3:    1   5    7     1
  4:    1   7   19    15     1
  5:    1   9   37    65    31      1
  6:    1  11   61   175   211     63      1
  7:    1  13   91   369   781    665    127      1
  8:    1  15  127   671  2101   3367   2059    255      1
  9:    1  17  169  1105  4651  11529  14197   6305    511     1
  10:   1  19  217  1695  9031  31031  61741  58975  19171  1023   1
  ...  - _Wolfdieter Lang_, May 07 2021

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

T. D. Noe, Rows n = 0..100 of triangle, flattened
A. Blecher, C. Brennan, A. Knopfmacher and Helmut Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
Eric Weisstein's World of Mathematics, Nexus Number

Cf. A047970.
Cf. A009998, A108553 (Hilbert transform of array of h-vectors of type D permutohedra), A145904, A145905.
Row n sequences of array a: A000012, A000225(k+1), A001047(k+1), A005061(k+1), A005060(k+1), A005062(k+1), A016169(k+1), A016177(k+1), A016185(k+1), A016189(k+1), A016195(k+1), A016197(k+1).
Column k sequences of array a: (nexus numbers): A000012, A005408, A003215, A005917(n+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Cf. A343237 (row reversed triangle).

Flatten[Table[n = d - e; k = e; (n + 1)^(k + 1) - n^(k + 1), {d, 0, 100}, {e, 0, d}]] (* T. D. Noe, Feb 22 2012 *)

T(n,m):=if m=0 then 1 else sum(k!*(-1)^(m+k)*stirling2(m,k)*binomial(n+k-1,n),k,0,m); /* Vladimir Kruchinin, Jan 28 2018 */

From Vladimir Kruchinin: (Start)
O.g.f. of e.g.f of rows of array: ((1-x)*exp(y))/(1-x*exp(y))^2.
T(n,m) = Sum_{k=0..m} k!*(-1)^(m+k)*Stirling2(m,k)*C(n+k-1,n), T(n,0)=1.(End)
From Wolfdieter Lang, May 07 2021: (Start)
T(n,m) = a(n-m,m) = (n-m+1)^(m+1) - (n-m)^(m+1), n >= 0, m = 0, 1,..., n.
O.g.f. column k of the array: polylog(-(k+1), x)*(1-x)/x. See the Peter Bala comment above, and the Eulerian triangle A008292 formula by Vladeta Jovovic, Sep 02 2002.
E.g.f. of e.g.f. of row of the array: exp(y)*(1 + x*(exp(y) - 1))*exp(x*exp(y)).
O.g.f. of triangle's exponential row polynomials R(n, y) = Sum_{m=0} T(n, m)*(y^m)/m!: G(x, y) = exp(x*y)*(1 - x)/(1 - x*exp(x*y))^2. (End)

A016189 a(n) = 10^n - 9^n.

Original entry on oeis.org

0, 1, 19, 271, 3439, 40951, 468559, 5217031, 56953279, 612579511, 6513215599, 68618940391, 717570463519, 7458134171671, 77123207545039, 794108867905351, 8146979811148159, 83322818300333431, 849905364703000879, 8649148282327007911, 87842334540943071199, 890581010868487640791

Offset: 0

N. J. A. Sloane

Almost all numbers contain any given sequence of digits (in any base) [Theorem 143 of Hardy and Wright]. a(7) = 5217031, more than 52% of the numbers < 10^7 contain any given nonzero decimal digit. - Frank Ellermann, May 30 2001
a(n) gives the number of integers from 0 to 10^n-1 which contain (at least) any one given decimal digit except 0. - Michael Taktikos, Aug 24 2004
These are the numerators of a(n)=(integral{x=0 to 0.2} (1-0.5*x)^n dx). E.g., a(3)=3439/20000. The denominators are b(n)=5*(n+1)*10^n. E.g., b(3)=20000. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004
Binomial transforms of sequences defined by a(n)=(C+1)^n-C^n are the sequences (C+2)^n-(C+1)^n. The binomial transform of this here is in A016195, for example. - R. J. Mathar, Nov 27 2008
First differences are given in A088924. - M. F. Hasler, May 04 2015

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 143

Vincenzo Librandi, Table of n, a(n) for n = 0..130
Alexander Bogomolny, Almost every integer has a digit 3 in it
John Elias, Illustration of Initial Terms
James Grime, 3 is everywhere, Numberphile video
Index entries for linear recurrences with constant coefficients, signature (19, -90).

Base 2: A000225, 3: A001047, 4: A005061, 5: A005060, 6: A005062, base 7: A016169, 8: A016177, 9: A016185 11: A016195 12: A016197.
Equals A155671 - 1.
Cf. A011557, A011533.

a016189 n = 10 ^ n - 9 ^ n
a016189_list = 0 : zipWith (+) (map (* 9) a016189_list) a011557_list
-- Reinhard Zumkeller, Apr 03 2015

[10^n - 9^n: n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=10^n-9^n;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

a(n)=10^n-9^n \\ M. F. Hasler, May 04 2015

G.f.: x/((1-9x)(1-10x)).
a(0) = 0, a(1) = 1, then a(n+1) = 9*a(n) + 10^n.
a(n) = 19*a(n-1) - 90*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: e^(10*x) - e^(9*x). - Mohammad K. Azarian, Jan 14 2009

A005062 a(n) = 6^n - 5^n.

Original entry on oeis.org

0, 1, 11, 91, 671, 4651, 31031, 201811, 1288991, 8124571, 50700551, 313968931, 1932641711, 11839990891, 72260648471, 439667406451, 2668522016831, 16163719991611, 97745259402791, 590286253682371

Offset: 0

N. J. A. Sloane

These are the numerators of a(n) = (Integral_{x=0..1/3} (1-x/2)^n dx). E.g., a(3)=671/2592. The denominators are b(n) = 3*(n+1)*6^n. E.g., b(3)=2592. the subscripts in both cases are 0. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004
Number of numbers with at most n digits whose largest digit is 5. For the first 5 terms, the first differences (i.e., ...with exactly n digits...) are given in A125373. - M. F. Hasler, May 03 2015
a(n) is the number of n-digit numbers whose smallest decimal digit is 4. - Stefano Spezia, Nov 15 2023

			G.f. = x + 11*x^2 + 91*x^3 + 671*x^4 + 4651*x^5 + 31031*x^6 + 201811*x^7 + ... - _Michael Somos_, Jul 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-30).

Cf. A005060 (5^n - 4^n), A125373.

[6^n - 5^n: n in [0..25]]; // Vincenzo Librandi, Jun 03 2011

restart:a:=n->sum(5^(n-j)*binomial(n,j),j=1..n): seq(a(n), n=0..19); # Zerinvary Lajos, Apr 18 2009

f[n_]:=6^n-5^n;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
LinearRecurrence[{11,-30},{0,1},20] (* Harvey P. Dale, May 28 2015 *)

a(n)=6^n-5^n \\ M. F. Hasler, May 03 2015

for(d=0,9,print1(sum(n=1,10^d-1,vecmax(digits(n))==5)",")) \\ Only to illustrate the comment about "largest digit equals 5".

[lucas_number1(n,11,30) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009

G.f.: x/((1-5*x)(1-6*x)).
a(n) = 11*a(n-1) - 30*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: exp(6*x) - exp(5*x). - Mohammad K. Azarian, Jan 14 2009
a(n) = -(30)^n * a(-n) for all n in Z. - Michael Somos, Jul 14 2018

A143496 4-Stirling numbers of the second kind.

Original entry on oeis.org

1, 4, 1, 16, 9, 1, 64, 61, 15, 1, 256, 369, 151, 22, 1, 1024, 2101, 1275, 305, 30, 1, 4096, 11529, 9751, 3410, 545, 39, 1, 16384, 61741, 70035, 33621, 7770, 896, 49, 1, 65536, 325089, 481951, 305382, 95781, 15834, 1386, 60, 1, 262144, 1690981, 3216795

Offset: 4

Peter Bala, Aug 20 2008

This is the case r = 4 of the r-Stirling numbers of the second kind. The 4-Stirling numbers of the second kind count the ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1, 2, 3 and 4 belong to distinct subsets. For remarks on the general case see A143494 (r = 2). The corresponding array of 4-Stirling numbers of the first kind is A143493. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For 4-Lah numbers refer to A143499.
From Wolfdieter Lang, Sep 29 2011: (Start)
T(n,k) = S(n,k,4), n >= k >= 4, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column k from (A20) with k->4, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(4*x),exp(x)-1) with e.g.f. of column no. m >= 0: exp(4*x)*((exp(x)-1)^m)/m!. See one of the formulas given below. For Sheffer matrices see the W. Lang link under A006232 with the S. Roman reference, also found in A132393.
(End)

			Triangle begins
n\k|.....4.....5.....6.....7.....8.....9
========================================
4..|.....1
5..|.....4.....1
6..|....16.....9.....1
7..|....64....61....15.....1
8..|...256...369...151....22.....1
9..|..1024..2101..1275...305....30.....1
...
T(6,5) = 9. The set {1,2,3,4,5,6} can be partitioned into five subsets such that 1, 2, 3 and 4 belong to different subsets in 9 ways: {{1,5}{2}{3}{4}{6}}, {{1,6}{2}{3}{4}{5}}, {{2,5}{1}{3}{4}{6}}, {{2,6}{1}{3}{4}{5}}, {{3,5}{1}{2}{4}{6}}, {{3,6}{1}{2}{4}{5}}, {{4,5}{1}{2}{3}{6}}, {{4,6}{1}{2}{3}{5}} and {{5,6}{1}{2}{3}{4}}.

Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, Stanford University, Department of Computer Science; see also, Discrete Math. 49, 241-259 (1984).
Askar Dzhumadil'daev and Damir Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001)
Lily L. Liu and Yi Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.

Cf. A003468 (column 7), A005060 (column 5), A008277, A016103 (column 6), A045379 (row sums), A049459 (matrix inverse), A143493, A143494, A143495, A143499.

with combinat: T := (n, k) -> 1/(k-4)!*add ((-1)^(k-i)*binomial(k-4,i)*(i+4)^(n-4),i = 0..k-4): for n from 4 to 13 do seq(T(n, k), k = 4..n) end do;

t[n_, k_] := StirlingS2[n, k] - 6*StirlingS2[n-1, k] + 11*StirlingS2[n-2, k] - 6*StirlingS2[n-3, k]; Flatten[ Table[ t[n, k], {n, 4, 13}, {k, 4, n}]] (* Jean-François Alcover, Dec 02 2011 *)

T(n+4,k+4) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*(i+4)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - 6*Stirling2(n-1,k) + 11*Stirling2(n-2,k) - 6*Stirling2(n-3,k) for n,k >= 4.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 4 with boundary conditions: T(n,3) = T(3,n) = 0 for all n; T(4,4) = 1; T(4,k) = 0 for k > 4. Special cases: T(n,4) = 4^(n-4); T(n,5) = 5^(n-4) - 4^(n-4).
E.g.f. (k+4)-th column (with offset 4): (1/k!)*exp(4*x)*(exp(x)-1)^k.
O.g.f. k-th column: Sum_{n>=k} T(n,k)*x^n = x^k/((1-4*x)*(1-5*x)*...*(1-k*x)).
E.g.f.: exp(4*t + x*(exp(t)-1)) = Sum_{n = 0..infinity} Sum_(k = 0..n) T(n+4,k+4)*x^k*t^n/n! = Sum_{n = 0..infinity} B_n(4;x)*t^n/n! = 1 + (4+x)*t/1! + (16+9*x+x^2)*t^2/2! + ..., where the row polynomials, B_n(4;x) := Sum_{k = 0..n} T(n+4,k+4)*x^k, may be called the 4-Bell polynomials.
Dobinski-type identities: Row polynomial B_n(4;x) = exp(-x)*Sum_{i = 0..infinity} (i+4)^n*x^i/i!; Sum_{k = 0..n} k!*T(n+4,k+4)*x^k = Sum_{i = 0..infinity} (i+4)^n*x^i/(1+x)^(i+1).
The T(n,k) are the connection coefficients between the falling factorials and the shifted monomials (x+4)^(n-4). For example, 16 + 9*x + x*(x-1) = (x+4)^2; 64 + 61*x + 15*x*(x-1) + x*(x-1)*(x-2) = (x+4)^3.
This array is the matrix product P^3 * S, where P denotes Pascal's triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).
The inverse array is A049459, the signed 4-Stirling numbers of the first kind.
From Peter Bala, Sep 19 2008: (Start)
Let D be the derivative operator d/dx and E the Euler operator x*d/dx. Then x^(-4)*E^n*x^4 = Sum_{k = 0..n} T(n+4,k+4)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k=4..n} T(n,k)*x^k satisfy the recurrence R_(n+1)(x) = x*R_n(x) + x*d/dx(R_n(x)) with R_4(x) = x^4. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 4-Eulerian numbers E_4(n,j) := A144698(n,j): T(n,k) = 4!/k!*Sum_{j = n-k..n-4} E_4(n,j)*binomial(j,n-k) for n >= k >= 4.
(End)

A003468 Number of minimal 3-covers of a labeled n-set.

Original entry on oeis.org

1, 22, 305, 3410, 33621, 305382, 2619625, 21554170, 171870941, 1337764142, 10216988145, 76862115330, 571247591461, 4203844925302, 30687029023865, 222518183370890, 1604626924403181, 11518132293452862

Offset: 3

N. J. A. Sloane

nonn

This is also the fourth column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See the e.g.f. given below. See also the Sheffer comments in A193685. - Wolfdieter Lang, Oct 08 2011

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 = 3..1000
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Minimal cover.
Index entries for linear recurrences with constant coefficients, signature (22, -179, 638, -840).

Cf. A000392, A003468, A016111, A046166-A046169, A057668, A005783-A005786, A055066.

Cf. A143496, A000302, A005060, A016103.

[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6: n in [3..30]]; // Vincenzo Librandi, May 03 2013

A003468:=1/(6*z-1)/(4*z-1)/(7*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Table[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6, {n, 3, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)
LinearRecurrence[{22,-179,638,-840},{1,22,305,3410},20] (* Harvey P. Dale, Jan 09 2024 *)

G.f.: x^3/((1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)). - N. J. A. Sloane, May 12 1994, corrected by Vaclav Kotesovec, Nov 19 2012
E.g.f.: (exp(4*x)*(exp(x) - 1)^3)/6. More generally, e.g.f. for number of minimal m-covers of a labeled n-set is (exp((2^m - m - 1)*x)*(exp(x) - 1)^m)/m!. - Vladeta Jovovic, May 09 2004
If we define f(m, j, x) = sum(binomial(m, k)*stirling2(k, j)*x^(m - k),k = j .. m) then a(n) = f(n, 3, 4), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 7^n/6 - 6^n/2 + 5^n/2 - 4^n/6. - Vaclav Kotesovec, Nov 19 2012

A087685 Primes p such that 5^p - 4^p is composite.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Cino Hilliard, Oct 26 2003

nonn

Cf. A005060.

apmb(a,b,n) = { forprime(x=2,n, y=a^x-b^x; if(!ispseudoprime(y), print1(x","); ) ) }

Checked by Ray Chandler, Apr 24 2007

A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 15, 151, 1275, 9751, 70035, 481951, 3216795, 20991751, 134667555, 852639151, 5343198315, 33212784151, 205111785075, 1260114546751, 7708980203835, 46999640806951, 285743822630595, 1733261544204751

Offset: 0

Robert G. Wilson v

2*a(n-2) = 6^n - 2*5^n + 4^n is the number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) first and third row have a common 1, (c) second and third row have no common 1. - Andi Fugard and Vladeta Jovovic, Jul 26 2008

This is the third column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See A193685 for general comments. - Wolfdieter Lang, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Andi Fugard, Counting first-order models (with n individuals) of syllogisms.
Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

Cf. A051588, A000302, A005060, A003468.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-6*x)))); // Vincenzo Librandi, Jun 24 2013

I:=[1, 15, 151]; [n le 3 select I[n] else 15*Self(n-1)-74*Self(n-2)+120*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 24 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 6 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

Vec(1/((1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 2^(3 + 2*n) + 2^(1 + n) * 3^(2 + n) - 5^(2 + n). - Andi Fugard, Jul 22 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,4), n >= 2. - Milan Janjic, Apr 26 2009
O.g.f.: 1/((1-4*x)*(1-5*x)*(1-6*x)).
E.g.f.: (d^2/dx^2)(exp(4*x)*((exp(x)-1)^2)/2!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 08 2011
a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3). - Vincenzo Librandi, Jun 24 2013

A016197 a(n) = 12^n - 11^n.

Original entry on oeis.org

0, 1, 23, 397, 6095, 87781, 1214423, 16344637, 215622815, 2801832661, 35979939623, 457696700077, 5777672071535, 72470493235141, 904168630965623, 11229773405170717, 138934529031464255, 1713164078241143221

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (23, -132).

Cf. k^n-(k-1)^n: A000225 (k=2), A001047 (k=3), A005061 (k=4), A005060 (k=5), A005062 (k=6), A016169 (k=7), A016177 (k=8), A016185 (k=9), A016189 (k=10), A016195 (k=11), this sequence (k=12).

[12^n-11^n: n in [0..40]]; // Vincenzo Librandi, Aug 03 2014

f[n_]:= 12^n - 11^n; f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
CoefficientList[Series[x/((1 - 11 x) (1 - 12 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 03 2014 *)

a(n)=12^n-11^n \\ Charles R Greathouse IV, Aug 03 2014

G.f.: x/((1-11x)(1-12x)).
E.g.f.: e^(12*x)-e^(11*x). - Mohammad K. Azarian, Jan 14 2009
a(0)=0, a(n)=12*a(n-1)+11^(n-1). - _Vincenzo Librandi-, Feb 09 2011
a(0)=0, a(1)=1, a(n)=23*a(n-1)-132*a(n-2). - Vincenzo Librandi, Feb 09 2011

cp's OEIS Frontend

A059802 Numbers k such that 5^k - 4^k is prime.

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A005061 a(n) = 4^n - 3^n.

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A047969 Square array of nexus numbers a(n,k) = (n+1)^(k+1) - n^(k+1) (n >= 0, k >= 0) read by upwards antidiagonals.

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A016189 a(n) = 10^n - 9^n.

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A005062 a(n) = 6^n - 5^n.

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A143496 4-Stirling numbers of the second kind.

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