A047970 -id:A047970 - cp's OEIS frontend

A112532 First differences of [0, A047970].

1, 1, 3, 9, 29, 101, 379, 1525, 6549, 29889, 144419, 736241, 3947725, 22201549, 130624587, 802180701, 5131183301, 34121977865, 235486915507, 1683925343929, 12458499203901, 95237603403381, 751291094637083, 6108883628141189, 51144808472958709, 440444879385258001

Offset: 0

Alford Arnold, Sep 10 2005

nonn

Number of sequences of length n in [n] (endofunctions) whose first run has length equal to the maximum of the sequence.

			The 9 sequences for n=4 (sorted by maximum)
1121,1122,2211,2212, 1113,2223,3331,3332, 4444
The 29 sequences for n=5 (sorted by maximum)
11211,11212,11221,11222, 22111,22112,22121,22122, 11123,11131,11132,11133, 22213,22231,22232,22233, 33311,33312,33313,33321,33322,33323, 11114, 22224, 33334, 44441,44442,44443, 55555

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A047970, A112531, A229046.

First differences of column 0 of triangle A089246 (beginning at row 1). With offset 1, first differences of column 0 of triangle A242431. Second differences of column 0 of triangle A101494.

a[n_]:= If[n==0, 1, n + Sum[(i-1)^2*i^(n-i), {i,0,n}]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 12 2022 *)

a(n) = n + sum(i = 0, n, (n-i-1)^2 * (n-i)^i); \\ Michel Marcus, Mar 01 2021

[n +sum((j-1)^2*j^(n-j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jan 12 2022

G.f.: (1-x)^2*( Sum_{n >= 0} x^n/(1 - (n+2)*x) ). - Peter Bala, Jul 09 2014
From Mathew Englander, Feb 28 2021: (Start)
a(n) = A089246(n+2,0) - A089246(n+1,0).
a(n) = n + Sum_{i = 0..n} (n-i-1)^2 * (n-i)^i. (End)

Corrected by D. S. McNeil, Aug 20 2010

Combinatorial interpretation and examples by Olivier Gérard, Jan 29 2023

A112508 Counts the objects described in A047969 and A089246 when grouped by minimum part. (Row sums give A047970).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 9, 1, 1, 3, 9, 29, 1, 1, 3, 9, 29, 101

Offset: 1

Alford Arnold, Sep 08 2005

			Row 4 of the triangular array is (1 1 3 9) because there are nine tuples with a minimum value of 1, three tuples with a minimum value of 2, one tuple with a minimum value of 3 and one tuple with a minimum value of 4; the relevant unordered partitions are illustrated in the below Gaussian polynomial template:
4
31 32 33
211 221 222
1111

Cf. A047969, A089246.

A112852 Table with row lengths 1 1 2 3 5 9 17 33 65 ... which counts the objects described in A047970 and A112508.

Original entry on oeis.org

1, 2, 4, 1, 7, 6, 1, 11, 20, 8, 3, 1, 16, 50, 33, 21, 10, 3, 6, 4, 1, 22, 105, 98, 81, 49, 21, 48, 36, 12, 3, 6, 12, 10, 4, 10, 5, 1, 29, 196, 238, 231, 168, 81, 210, 168, 68

Offset: 0

Alford Arnold, Sep 24 2005

The row sums are 1 2 5 14 43 144 523 2048 8597 ... A047970. The columns are essentially A000124, A002415, A051836, A112851, A051947, ...

			The table begins
1
2
4 1
7 6 1
11 20 8 3 1
16 50 33 21 10 3 6 4 1
22 105 98 81 49 21

Cf. A000124, A002415, A051836, A112851, A051947, A047970, A112508.

A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 2, 1, 10, 14, 12, 6, 1, 15, 30, 39, 39, 20, 1, 21, 55, 95, 138, 142, 71, 1, 28, 91, 195, 364, 548, 551, 270, 1, 36, 140, 357, 804, 1564, 2317, 2278, 1100

Offset: 1

Alford Arnold, Apr 16 2009

The first five diagonals are essentially A000012, A000217, A000330, A086602 and A159571.
From Alford Arnold, Apr 20 2009: (Start)
After the first two diagonals, each additional diagonal is computed using blocks of source partitions (defined in A053445).
The size of each block increases by powers of two; e.g. 22, 33 222, 44 332 333 2222; etc.
Each source partition can be associated with a specific sequence as illustrated in the below example using partition 332: grow the leftmost value to form 432 then append "1" to form 3321. in like manner, generate 532 4321 and 33211 from the previously formed cases. Note that the number of arrangements are 3, 6+12, and 6+24+30 respectively and that we now have three terms of A006011: 3 18 and 60.
Next we note that 6 39 138 364 804 ... A159571 resulted from summing term by term, the sequences associated with partitions 44 332 333 and 2222:
1...5..14...30...55
3..18..60..150..315
1...7..25...65..140
1...9..39..119..294
(End)

Cf. A047970 (row sums), A000012, A000217, A000330, A086602, A159571.

Cf. A053445. - Alford Arnold, Apr 20 2009

Submitted without a definition. - N. J. A. Sloane, Apr 18 2009

More terms from Alford Arnold, Oct 06 2009

A089355 A quarter-square shaped (A002620) staircase array distributing the frequency of objects counted by A047970.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 4, 2, 2, 1, 4, 3, 8, 7, 9, 5, 5, 1, 1, 5, 4, 13, 14, 23, 20, 25, 16, 14, 7, 2, 1, 1, 6, 5, 19, 23, 44, 49, 71, 65, 75, 60, 48, 32, 18, 6, 1, 1, 7, 6, 26, 34, 73, 94, 154, 172, 232, 234, 249, 228, 195, 145, 101, 59, 27, 9, 2

Offset: 1

Alford Arnold, Dec 26 2003

			The array begins:
1..1..1..1..1..1..1....
...1..2..3..4..5..6....
......1..2..3..4..5....
......1..4..8.13.19....
where the fourth row counts the ordered partitions of type 221^k and type 41^k since (1 3 6 10 15 ...) + (0 1 2 3 4 ...) yields the required result.

A112996 A table distributing the values 1 2 5 14 43 144 523 2048 38486 182905 ... A047970 based on row sums of subtables counting permutations of partitions.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 4, 2, 1, 15, 12, 10, 5, 1, 31, 32, 36, 30, 14, 1, 63, 80, 112, 124, 100, 43, 1, 127, 192, 320, 432, 472, 360, 144, 1, 255, 448, 864, 1360, 1832, 1924, 1390, 523, 1, 511, 1024, 2240, 3958, 6320, 8280

Offset: 1

Alford Arnold, Nov 03 2005

The next diagonal, 2 10 36 112 320 864 2240 ... is 1 4 12 32 80 192 448 ... plus 1 6 24 80 240 672 1792 ...

			The table begins
1
1 1
1 3 1
1 7 4 2
1 15 12 10 5
1 31 32 36 30 14
the third diagonal is 1 4 12 32 80 192 448 ... A001787 which, in turn,
is the row sum of the below table
1
2 2
3 6 3
4 12 12 4
5 20 30 20 5
which counts the permutations of
3; 32, 31; 322, 321, 311; 3222, 3221, 3211, 3111; etc.

Cf. A047969, A047970.

A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.

Original entry on oeis.org

1, 2, 4, 9, 23, 66, 210, 733, 2781, 11378, 49864, 232769, 1151915, 6018786, 33087206, 190780213, 1150653921, 7241710930, 47454745804, 323154696185, 2282779990495, 16700904488706, 126356632390298, 987303454928973, 7957133905608837, 66071772829247410

Offset: 0

N. J. A. Sloane

nonn

Row sums of A004248, A009998, A009999.
First differences are in A047970.
First differences of A103439.
Antidiagonal sums of array A003992.
a(n-1), for n>=1, is the number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, that are simultaneously projections as maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x); see example and the two comments (Arndt, Apr 30 2011 Jan 04 2013) in A000110. - Joerg Arndt, Mar 07 2015
Number of finite sequences s of length n+1 whose discriminator sequence is s itself. Here the discriminator sequence of s is the one where the n-th term (n>=1) is the least positive integer k such that the first n terms are pairwise incongruent, modulo k. - Jeffrey Shallit, May 17 2016
From Gus Wiseman, Jan 08 2019: (Start)
Also the number of set partitions of {1,...,n+1} whose minima form an initial interval of positive integers. For example, the a(3) = 9 set partitions are:
  {{1},{2},{3},{4}}
  {{1},{2},{3,4}}
  {{1},{2,4},{3}}
  {{1,4},{2},{3}}
  {{1},{2,3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,3,4},{2}}
  {{1,2,3,4}}
Missing from this list are:
  {{1},{2,3},{4}}
  {{1,2},{3},{4}}
  {{1,3},{2},{4}}
  {{1,2},{3,4}}
  {{1,2,3},{4}}
  {{1,2,4},{3}}
(End)
a(n) is the number of m-tuples of nonnegative integers less than or equal to n-m (including the "0-tuple"). - Mathew Englander, Apr 11 2021

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 23*x^4 + 66*x^5 + 210*x^6 + ...
where we have the identity:
A(x) = 1/(1-x) + x/(1-2*x) + x^2/(1-3*x) + x^3/(1-4*x) + x^4/(1-5*x) + ...
is equal to
A(x) = 1/(1-x) + x/((1-x)^2*(1+x)) + 2!*x^2/((1-x)^3*(1+x)*(1+2*x)) + 3!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^4/((1-x)^5*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
From _Joerg Arndt_, Mar 07 2015: (Start)
The a(5-1) = 23 RGS described in the comment are (dots denote zeros):
01:  [ . . . . . ]
02:  [ . 1 . . . ]
03:  [ . 1 . . 1 ]
04:  [ . 1 . 1 . ]
05:  [ . 1 . 1 1 ]
06:  [ . 1 1 . . ]
07:  [ . 1 1 . 1 ]
08:  [ . 1 1 1 . ]
09:  [ . 1 1 1 1 ]
10:  [ . 1 2 . . ]
11:  [ . 1 2 . 1 ]
12:  [ . 1 2 . 2 ]
13:  [ . 1 2 1 . ]
14:  [ . 1 2 1 1 ]
15:  [ . 1 2 1 2 ]
16:  [ . 1 2 2 . ]
17:  [ . 1 2 2 1 ]
18:  [ . 1 2 2 2 ]
19:  [ . 1 2 3 . ]
20:  [ . 1 2 3 1 ]
21:  [ . 1 2 3 2 ]
22:  [ . 1 2 3 3 ]
23:  [ . 1 2 3 4 ]
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 12.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.
Sajed Haque, Discriminators of Integer Sequences, 2017, See p. 33 Corollary 29.
Mathematics Stack Exchange, Asymptotics of ..., 2011.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019. See p. 3.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See p. 4.

Cf. A000110, A000258, A000670, A003101, A008277, A038125, A062810.

Cf. A105795, A179928, A287215, A287216.

a026898 n = sum $ zipWith (^) [n + 1, n .. 1] [0 ..]
-- Reinhard Zumkeller, Sep 14 2014

[(&+[(n-k+1)^k: k in [0..n]]): n in [0..50]]; // Stefano Spezia, Jan 09 2019

a:= n-> add((n+1-j)^j, j=0..n): seq(a(n), n=0..23); # Zerinvary Lajos, Apr 18 2009

Table[Sum[(n-k+1)^k, {k,0,n}], {n, 0, 25}] (* Michael De Vlieger, Apr 01 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^m/(1-(m+1)*x+x*O(x^n))),n)} /* Paul D. Hanna, Sep 13 2011 */

{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{a(n)=local(A=1+x);A=sum(k=0,n,INTEGRATE(k,exp((k+1)*x+x*O(x^n))));n!*polcoeff(A,n)} \\ Paul D. Hanna, Dec 28 2013
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff( sum(m=0, n, m!*x^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x +x*O(x^n))), n)}  /* From o.g.f. (Paul D. Hanna, Jul 20 2014) */
for(n=0, 25, print1(a(n), ", "))

[sum((n-j+1)^j for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n) = A003101(n) + 1.
G.f.: Sum_{n>=0} x^n/(1 - (n+1)*x). - Paul D. Hanna, Sep 13 2011
G.f.: G(0) where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
E.g.f.: Sum_{n>=0} Integral^n exp((n+1)*x) dx^n, where Integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013
O.g.f.: Sum_{n>=0} n! * x^n/(1-x)^(n+1) / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Jul 20 2014
a(n) = A101494(n+1,0). - Vladimir Kruchinin, Apr 01 2015
a(n-1) = Sum_{k = 1..n} k^(n-k). - Gus Wiseman, Jan 08 2019
log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021
a(n) ~ sqrt(2*Pi/(n+1 + (n+1)/w(n))) * ((n+1)/w(n))^(n+2 - (n+1)/w(n)), where w(n) = LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

a(23)-a(25) from Paul D. Hanna, Dec 28 2013

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)

A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.

Original entry on oeis.org

0, 1, 3, 8, 22, 65, 209, 732, 2780, 11377, 49863, 232768, 1151914, 6018785, 33087205, 190780212, 1150653920, 7241710929, 47454745803, 323154696184, 2282779990494, 16700904488705, 126356632390297, 987303454928972, 7957133905608836, 66071772829247409

Offset: 0

N. J. A. Sloane, Henry W. Gould

For n > 0: a(n) = sum of row n of triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014
a(n-1) is the number of set partitions of [n] into two or more blocks such that all absolute differences between least elements of consecutive blocks are 1. a(3) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, May 22 2017
Min_{n >= 1} a(n+1)/a(n) = 8/3. This is the answer to the 4th problem proposed during the first day of the final round of the 16th Austrian Mathematical Olympiad in 1985 (see link IMO Compendium). - Bernard Schott, Jan 07 2019
If n rings of different internal diameter can fit close together on a tapering column, a(n) is the number of different arrangements of at least one ring. For example, if the rings increasing in size are 1, 2 and 3, then a(3) = 8 corresponding to the possible arrangements from the point on the column of smallest diameter (1XX), (X2X), (XX3), (12X), (32X), (1X3), (X23) and (123), where X denotes a space on the column. - Ian Duff, Jun 23 2025

			For n = 3 we get a(3) = 3^1 + 2^2 + 1^3 = 8. For n = 4 we get a(4) = 4^1 + 3^2 + 2^3 + 1^4 = 22.

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

Charles R Greathouse IV, Table of n, a(n) for n = 0..598
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
Mathematics Stack Exchange, Asymptotics of 1^n+2^(n-1)+3^(n-2)+...+(n-1)^2+n^1, 2011.
Daniel Ropp, Problem 2 - 16th Austrian Mathematical Olympiad (Final round), Crux Mathematicorum, page 7, Vol. 14, Jun. 88.
The IMO compendium, Problem 2, 16th Austrian Mathematical Olympiad, 1985.
Index to sequences related to Olympiads.

Cf. A026898, A051129, A062810, A247358, A287215.

First differences are in A047970.

a003101 n = sum $ zipWith (^) [0 ..] [n + 1, n .. 1]
-- Reinhard Zumkeller, Sep 14 2014

[n eq 0 select 0 else (&+[(n-j+1)^j: j in [1..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

A003101 := n->add((n-k+1)^k, k=1..n);
a:= n-> add((n-j+1)^j, j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Jun 07 2008

Table[Sum[(n-k+1)^k,{k,n}],{n,0,25}] (* Harvey P. Dale, Aug 14 2011 *)

a(n)=sum(k=1,n,(n-k+1)^k) \\ Charles R Greathouse IV, Oct 31 2011

def A003101(n): return sum( (n-k+1)^k for k in range(1,n+1))
[A003101(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

a(n) = A026898(n) - 1.
G.f.: G(0)/x-1/(1-x)/x where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
G.f.: Sum_{k>=1} x^k/(1 - (k + 1)*x). - Ilya Gutkovskiy, Oct 09 2018
a(n) = n^1 + (n-1)^2 + (n-2)^3 + ... + 3^(n-2) + 2^(n-1) + 1^n. - Bernard Schott, Jan 07 2019
log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021
a(n) ~ sqrt(2*Pi/(n+1 + w(n))) * w(n)^(n+2 - w(n)), where w(n) = (n+1)/LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

Original entry on oeis.org

5, 32, 113, 299, 664, 1309, 2366, 4002, 6423, 9878, 14663, 21125, 29666, 40747, 54892, 72692, 94809, 121980, 155021, 194831, 242396, 298793, 365194, 442870, 533195, 637650, 757827, 895433, 1052294, 1230359, 1431704, 1658536, 1913197

Offset: 1

Alford Arnold, Dec 29 2003; extended May 04 2005

The diagonals are finite and sum to A047970.
Values appear to be a transformation of A006468 (rooted planar maps). Also known as well-labeled trees (cf. A000168).
First differences of the conjectured polynomial formula for A006468. [From R. J. Mathar, Jun 26 2010]

			The array begins
1
2
4
7 1
11 5
16 14 2
22 30 12
29 55 39 5
37 91 95 32 1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A105552, A006468.
Cf. A006011, A034261.
Cf. A000124 (column 1), A000330 (column 2), A086602 (column 3), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

Row sums are powers of 2.
a(n) = A000330(n) + A006011(n+1) + A034263(n-1).
a(n)= +6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). G.f.: x*(5+2*x-4*x^2+x^3)/(x-1)^6. a(n) = n*(n+1)*(4*n^3+51*n^2+159*n+86)/120. [From R. J. Mathar, Jun 26 2010]

Extended beyond a(8) by R. J. Mathar, Jun 26 2010

cp's OEIS Frontend

A112532 First differences of [0, A047970].

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A112508 Counts the objects described in A047969 and A089246 when grouped by minimum part. (Row sums give A047970).

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A112852 Table with row lengths 1 1 2 3 5 9 17 33 65 ... which counts the objects described in A047970 and A112508.

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A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

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A089355 A quarter-square shaped (A002620) staircase array distributing the frequency of objects counted by A047970.

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A112996 A table distributing the values 1 2 5 14 43 144 523 2048 38486 182905 ... A047970 based on row sums of subtables counting permutations of partitions.

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A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.

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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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A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.