A287216 -id:A287216 - cp's OEIS frontend

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

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

A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 21, 7, 1, 1, 65, 86, 39, 11, 1, 1, 209, 361, 209, 77, 19, 1, 1, 732, 1584, 1123, 493, 171, 35, 1, 1, 2780, 7315, 6153, 3124, 1293, 413, 67, 1, 1, 11377, 35635, 34723, 20019, 9320, 3709, 1059, 131, 1, 1, 49863, 183080, 202852, 130916, 66992, 30396, 11373, 2837, 259, 1

Offset: 0

Alois P. Heinz, May 21 2017

The maximal absolute difference is assumed to be zero if there are fewer than two blocks.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.

			T(4,0) = 1: 1234.
T(4,1) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4.
T(4,3) = 1: 123|4.
Triangle T(n,k) begins:
  1;
  1;
  1,   1;
  1,   3,    1;
  1,   8,    5,    1;
  1,  22,   21,    7,   1;
  1,  65,   86,   39,  11,   1;
  1, 209,  361,  209,  77,  19,  1;
  1, 732, 1584, 1123, 493, 171, 35, 1;

Alois P. Heinz, Rows n = 0..141, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A003101(n-1), A322875, A322876, A322877, A322878, A322879, A322880, A322881, A322882, A322883.
Row sums give A000110.
T(2n,n) gives A322884.
Cf. A287213, A287216, A287416, A287640.

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);

b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m*b[n - 1, k, m, l]];
A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[n - 1, 0]}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,k) = A287216(n,k) - A287216(n,k-1) for k>0, T(n,0) = 1.

A287641 Number A(n,k) of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 1, 1, 1, 2, 5, 14, 1, 1, 1, 2, 5, 15, 42, 1, 1, 1, 2, 5, 15, 51, 132, 1, 1, 1, 2, 5, 15, 52, 191, 429, 1, 1, 1, 2, 5, 15, 52, 202, 773, 1430, 1, 1, 1, 2, 5, 15, 52, 203, 861, 3336, 4862, 1, 1, 1, 2, 5, 15, 52, 203, 876, 3970, 15207, 16796, 1

Offset: 0

Alois P. Heinz, May 28 2017

			A(5,0) = 1: 12345.
A(5,1) = 42 = 52 - 10 = A000110(5) - 10 counts all set partitions of [5] except: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5, 134|2|5.
A(5,2) = 51 = 52 - 1 = A000110(5) - 1 counts all set partitions of [5] except: 134|2|5.
Square array A(n,k) begins:
  1,   1,   1,   1,   1,   1,   1,   1, ...
  1,   1,   1,   1,   1,   1,   1,   1, ...
  1,   2,   2,   2,   2,   2,   2,   2, ...
  1,   5,   5,   5,   5,   5,   5,   5, ...
  1,  14,  15,  15,  15,  15,  15,  15, ...
  1,  42,  51,  52,  52,  52,  52,  52, ...
  1, 132, 191, 202, 203, 203, 203, 203, ...
  1, 429, 773, 861, 876, 877, 877, 877, ...

Alois P. Heinz, Antidiagonals n = 0..34, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000108, A275605, A287666, A287667, A287668, A287669, A287670, A287671, A287672, A287673.
Main diagonal gives A000110.
Cf. A064645, A287214, A287216, A287417, A287640.

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[0, ] = 1; b[n, l_List] := b[n, l] = Sum[b[n - 1, Append[ Table[ Max[ l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}];
A[n_, k_] := If[k == 0, 1, b[n, Table[0, k]]];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k} A287640(n,j).

A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 8, 1, 1, 1, 2, 5, 13, 16, 1, 1, 1, 2, 5, 15, 34, 32, 1, 1, 1, 2, 5, 15, 47, 89, 64, 1, 1, 1, 2, 5, 15, 52, 150, 233, 128, 1, 1, 1, 2, 5, 15, 52, 188, 481, 610, 256, 1, 1, 1, 2, 5, 15, 52, 203, 696, 1545, 1597, 512, 1

Offset: 0

Alois P. Heinz, May 21 2017

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			A(4,0) = 1: 1|2|3|4.
A(4,1) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
A(4,2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,   1,   1, ...
  1,  1,   1,   1,   1,   1,   1,   1, ...
  1,  2,   2,   2,   2,   2,   2,   2, ...
  1,  4,   5,   5,   5,   5,   5,   5, ...
  1,  8,  13,  15,  15,  15,  15,  15, ...
  1, 16,  34,  47,  52,  52,  52,  52, ...
  1, 32,  89, 150, 188, 203, 203, 203, ...
  1, 64, 233, 481, 696, 825, 877, 877, ...

Alois P. Heinz, Antidiagonals n = 0..45, flattened
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A011782, A001519, A287275, A287276, A287277, A287278, A287279, A287280, A287281, A287282.
Main diagonal gives A000110.
Cf. A287213, A287216, A287417, A287641.

b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->
      `if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->
      `if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))
    end:
A:= (n, k)-> b(n, min(k, n-1), []):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[0, , ] = 1; b[n_, k_, l_List] := b[n, k, l] = b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}];
A[n_, k_] := b[n, Min[k, n - 1], {}];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k} A287213(n,j).

A287417 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks are not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 2, 5, 4, 0, 1, 1, 2, 5, 12, 5, 0, 1, 1, 2, 5, 15, 27, 6, 0, 1, 1, 2, 5, 15, 46, 58, 7, 0, 1, 1, 2, 5, 15, 52, 139, 121, 8, 0, 1, 1, 2, 5, 15, 52, 187, 410, 248, 9, 0, 1, 1, 2, 5, 15, 52, 203, 677, 1189, 503, 10, 0

Offset: 0

Alois P. Heinz, May 24 2017

			A(5,3) = 46 = 52 - 6 = A000110(5) - 6 counts all set partitions of [5] except: 1234|5, 15|234, 15|23|4, 15|24|3, 15|2|34, 15|2|3|4.
Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1, ...
  1, 1,   1,   1,   1,   1,   1,   1, ...
  0, 2,   2,   2,   2,   2,   2,   2, ...
  0, 3,   5,   5,   5,   5,   5,   5, ...
  0, 4,  12,  15,  15,  15,  15,  15, ...
  0, 5,  27,  46,  52,  52,  52,  52, ...
  0, 6,  58, 139, 187, 203, 203, 203, ...
  0, 7, 121, 410, 677, 824, 877, 877, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened
Wikipedia, Partition of a set

Columns k=1-10 give: A028310, A000325, A287582, A287583, A287584, A287585, A287586, A287587, A287588, A287589.
Main diagonal gives A000110.
Cf. A287214, A287216, A287416, A287641.

b:= proc(n, k, l, t) option remember; `if`(n<1, 1, `if`(t-n>k, 0,
       b(n-1, k, map(x-> `if`(x-n>=k, [][], x), [l[], n]), n)) +add(
       b(n-1, k, sort(map(x-> `if`(x-n>=k, [][], x), subsop(j=n, l))),
       `if`(t-n>k, infinity, t)), j=1..nops(l)))
    end:
A:= (n, k)-> b(n, min(k, n-1), [], n):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_, l_, t_] := b[n, k, l, t] = If[n < 1, 1, If[t - n > k, 0, b[n - 1, k, If[# - n >= k, Nothing, #]& /@  Append[l, n], n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]], If[t - n > k, Infinity, t]], {j, 1, Length[l]}]];
A[n_, k_] := b[n, Min[k, n - 1], {}, n];
Table[A[n, d - n], {d, 0, 14}, { n, 0, d}] // Flatten (* Jean-François Alcover, May 24 2018, translated from Maple *)

A(n,k) = Sum_{j=0..k} A287416(n,j).

A287252 Number of set partitions of [n] such that all absolute differences between least elements of consecutive blocks are <= two.

Original entry on oeis.org

1, 1, 2, 5, 14, 44, 152, 571, 2317, 10096, 47013, 232944, 1223428, 6786936, 39640947, 243060305, 1560340480, 10461611439, 73094563140, 531127372268, 4006242743228, 31316162403165, 253292622192153, 2116823651781702, 18255325000268015, 162261535224570326

Offset: 0

Alois P. Heinz, May 22 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..590
Wikipedia, Partition of a set

Column k=2 of A287216.

Cf. A000110.

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
a:= n-> b(n-1, min(2, n-1), 1, n):
seq(a(n), n=0..30);

b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m*b[n - 1, k, m, l]];
a[n_] := b[n - 1, Min[2, n - 1], 1, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

a(n) = A287216(n,2).

a(n) = A000110(n) for n <= 3.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

Extensions

A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287641 Number A(n,k) of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287417 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks are not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287252 Number of set partitions of [n] such that all absolute differences between least elements of consecutive blocks are <= two.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287253 Number of set partitions of [n] such that all absolute differences between least elements of consecutive blocks are <= three.

Views

Author

Keywords