A367955 -id:A367955 - cp's OEIS frontend

A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

1, 5, 23, 109, 544, 2876, 16113, 95495, 597155, 3929243, 27132324, 196122796, 1480531285, 11647194573, 95297546695, 809490850313, 7126717111964, 64930685865768, 611337506786061, 5940420217001199, 59502456129204083, 613689271227219015, 6510381400140132872

Offset: 0

Sergey Kirgizov, Nov 26 2016

nonn

Original name: Popularity of left children in treeshelves avoiding pattern T231 (with offset 2).
Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T231 illustrated below corresponds to a treeshelf constructed from permutation 231. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.
a(n) is also the sum of the last entries in all blocks of all set partitions of [n-1]. a(4) = 23 because the sum of the last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+5+5+4+6  = 23. - Alois P. Heinz, Apr 24 2017
a(n-2) is the number of lines that rhyme (with at least one earlier line) across all rhyme schemes counted by A000110. - Martin Fuller, Apr 20 2025

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T231:
     1
    /
   /
  2
   \
    3
Treeshelves of size 3 that avoid pattern T231:
      1  1      1       1        1
     /    \      \     / \      / \
    2      2      \   2   \    /   2
   /        \      2       3  3
  3          3    /
                 3
Popularity of left children here is 5.

Alois P. Heinz, Table of n, a(n) for n = 0..572
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), pp. 35-50.

Cf. A000110, A000111, A000142, A001286, A008292, A011971, A131178, A278678, A278679, A285595, A286897, A367955.

b:= proc(n, m) option remember; `if`(n=0, [1, 0],
     (p-> p+[0, p[1]*n])(b(n-1, m+1))+m*b(n-1, m))
    end:
a:= n-> b(n+1, 0)[2]:
seq(a(n), n=0..22);  # Alois P. Heinz, Dec 15 2023
# Using the generating function:
gf := ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2: ser := series(gf, z, 25):
seq((n+2)!*coeff(ser, z, n), n=0..22);  # Peter Luschny, Feb 01 2025

a[n_] := (n+3) BellB[n+2] - BellB[n+3];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Dec 01 2018 *)

from sympy import bell
HOW_MANY = 30
print([(n + 3) * bell(n+2) - bell(n + 3) for n in range(HOW_MANY)])

E.g.f.: ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2.
a(n) = (n + 3)*b(n + 2) - b(n + 3) where b(n) is the n-th Bell number (see A000110).
Asymptotics: a(n) ~ n*b(n).
a(n) = Sum_{k=1..n+1} A285595(n+1,k)/k. - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=0..n} Stirling2(n+2, k+1) * (n+1-k). - Ilya Gutkovskiy, Apr 06 2021
a(n) ~ n*Bell(n)*(1 - 1/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=n+1..(n+1)*(n+2)/2} k * A367955(n+1,k). - Alois P. Heinz, Dec 11 2023

New name and offset 0 by Peter Luschny, Feb 01 2025

A204856 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 9, 11, 22, 27, 49, 68, 115, 157, 279, 372, 628, 914, 1457, 2070, 3457, 4840, 7753, 11442, 17768, 25824, 41315, 59008, 92140, 137212, 208524, 305472, 477659, 691381, 1058019, 1575694, 2370618, 3491693, 5359888, 7796346, 11799263, 17583757

Offset: 0

Paul D. Hanna, Jan 20 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 9*x^7 + 11*x^8 +...
where A(x) = 1 + x/(1-x) + x^3/((1-x)*(1-2*x^2)) + x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..6279 (terms 0..250 from Vaclav Kotesovec)

Cf. A204855, A204858.

Column sums of A367955.

Table[SeriesCoefficient[Sum[x^Binomial[n + 1, 2]/Product[(1 - k*x^k), {k, 1, n}], {x, 0, 100}], {x, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 19 2017 *)

{a(n)=polcoeff(1+sum(m=1,n,x^(m*(m+1)/2)/prod(k=1,m,1-k*x^k+x*O(x^n))),n)}

G.f.: 1/(1 - x/(1 - x^2*(1-x)/(1-x^2 - x^3*(1-2*x^2)/(1-2*x^3 - x^4*(1-3*x^3)/(1-3*x^4 - x^5*(1-4*x^4)/(1-4*x^5 -...)))))), a continued fraction.
From Vaclav Kotesovec, Jun 18 2019: (Start)
a(n) ~ c * 3^(n/3), where
c = 23.5612420584121380174441491950859168338330954540437... if mod(n,3)=0
c = 23.5209031427848763179214171003561794127717213180726... if mod(n,3)=1
c = 23.5214569018665529984420312927586688667133017590049... if mod(n,3)=2
(End)

A124327 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 4, 2, 1, 3, 2, 1, 0, 1, 1, 0, 8, 4, 2, 10, 6, 7, 2, 5, 3, 2, 1, 0, 1, 1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1, 1, 0, 32, 16, 8, 85, 56, 64, 42, 101, 62, 75, 69, 47, 54, 38, 38, 24, 23, 10, 13, 7, 5, 3, 2, 1, 0, 1, 1, 0, 64, 32, 16

Offset: 1

Emeric Deutsch, Oct 31 2006

Row n has n(n+1)/2 terms. Row sums yield the Bell numbers (A000110). T(n,1)=1; T(n,2)=0; T(n,3)=2^(n-2). for n>=2; T(n,4)=2^(n-3) for n>=3; T(n,5)=2^(n-4) for n>=4.

			T(4,7) = 2 because we have 13|2|4 and 1|23|4.
Triangle starts:
  1;
  1, 0,  1;
  1, 0,  2, 1, 0,  1;
  1, 0,  4, 2, 1,  3,  2,  1,  0,  1;
  1, 0,  8, 4, 2, 10,  6,  7,  2,  5,  3,  2,  1, 0, 1;
  1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..40, flattened

Cf. A000110, A000217, A124325, A367955.
Antidiagonal sums give A365821.
Row maxima give A365903.
T(n,n) gives A368204.

Q[1]:=t*s: for n from 2 to 8 do Q[n]:=expand(s*diff(Q[n-1],s)+t^n*s*Q[n-1]) od: for n from 1 to 8 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 8 do seq(coeff(P[n],t,k),k=1..n*(n+1)/2) od; # yields sequence in triangular form

Q[1, t_, s_] := t s;
Q[n_, t_, s_] := Q[n, t, s] = s D[Q[n-1, t, s], s] + s t^n Q[n-1, t, s] // Expand;
P[n_, t_] := Q[n, t, s] /. s -> 1;
T[n_] := Rest@CoefficientList[P[n, t], t];
Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Jun 10 2024 *)

The generating polynomial of row n is P(n,t)=Q(n,t,1), where Q(n,t,s)=s*dQ(n-1,t,s)/ds + st^n*Q(n-1,t,s); Q(1,t,s)=ts.

Sum_{k=1..n*(n+1)/2} k * T(n,k) = A124325(n+1). - Alois P. Heinz, Dec 05 2023

A365441 Number of partitions of [2n] whose block maxima sum to 3n.

Original entry on oeis.org

1, 1, 2, 5, 10, 23, 47, 103, 209, 449, 908, 1909, 3864, 8011, 16165, 33244, 66933, 136628, 274876, 558107, 1121160, 2268526, 4552291, 9183569, 18417449, 37073504, 74300048, 149334422, 299134695, 600481001, 1202436958, 2411536369, 4827532935, 9675363921, 19364235775

Offset: 0

Alois P. Heinz, Dec 06 2023

nonn

Rows of A367955 and the reversed columns of A367955 converge to this sequence.

			a(0) = 1: the empty partition.
a(1) = 1: 1|2.
a(2) = 2: 12|34, 134|2.
a(3) = 5: 123|456, 12456|3, 13|2456, 1456|23, 1|2|3456.
a(4) = 10: 1234|5678, 1235678|4, 124|35678, 125678|34, 134|25678, 135678|24, 14|235678, 15678|234, 1|23|45678, 1|245678|3.
a(5) = 23: 12345|6789(10), 12346789(10)|5, 1235|46789(10), 1236789(10)|45, 1245|36789(10), 1246789(10)|35, 125|346789(10), 126789(10)|345, 12|3|456789(10), 1345|26789(10), 1346789(10)|25, 135|246789(10), 136789(10)|245, 13|2|456789(10), 145|236789(10), 146789(10)|235, 15|2346789(10), 16789(10)|2345, 1|234|56789(10), 1|2356789(10)|4, 1456789(10)|2|3, 1|24|356789(10), 1|256789(10)|34.

Alois P. Heinz, Table of n, a(n) for n = 0..3321
Wikipedia, Partition of a set

Cf. A000110, A367955, A367969, A368204, A368675.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> coeff(b(2*n, 0), x, 3*n):
seq(a(n), n=0..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(3*n, 2*n, 0):
seq(a(n), n=0..42);

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[3n, 2n, 0]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

a(n) = A367955(2n,3n).

A200660 Sum of the number of arcs describing the set partitions of {1,2,...,n}.

Original entry on oeis.org

0, 1, 8, 49, 284, 1658, 9974, 62375, 406832, 2769493, 19668054, 145559632, 1121153604, 8974604065, 74553168520, 641808575961, 5718014325296, 52653303354906, 500515404889978, 4905937052293759, 49530189989912312, 514541524981377909, 5494885265473192914

Offset: 1

Nantel Bergeron, Nov 20 2011

nonn

Supercharacter theory of unipotent upper triangular matrices over a finite field F(2) is indexed by set partitions S(n) of {1,2,...,n} where a set partition P of {1,2,...,n} is a subset { (i,j) : 1 <= i < j <= n} such that (i,j) in P implies (i,k),(k,j) are not in P for all i < l < j.
One of the statistics used to compute the supercharacter table is the number of arcs in P (that is, the cardinality |P| of P).
The sequence we have is arcs(n) = Sum_{P in S(n)} |P|.

Seiichi Manyama, Table of n, a(n) for n = 1..500
M. Aguiar, C. Andre, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I. M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, T. Le, S. Lewis, H. Li, K. Magaard, E. Marberg, J-C. Novelli, A. Pang, F. Saliola, L. Tevlin, J-Y. Thibon, N. Thiem, V. Venkateswaran, C. R. Vinroot, N. Yan, and M. Zabrocki, Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras, arXiv:1009.4134 [math.CO], 2010-2011.
C. André, Basic characters of the unitriangular group, Journal of Algebra, 175 (1995), 287-319.

Cf. A011971 (sequence is computed from Aitken's array b(n,k) arcs(n) = Sum_{k=1..n-1} k*b(n,k)).
Cf. A200580, A200673 (other statistics related to supercharacter table).
Cf. A367955.

b:=proc(n,k) option remember;
  if n=1 and k=1 then RETURN(1) fi;
  if k=1 then RETURN(b(n-1,n-1)) fi;
  b(n,k-1)+b(n-1,k-1)
end:
arcs:=proc(n) local res,k;
  res:=0;
  for k to n-1 do res:=res+ k*b(n,k) od;
  res
end:
seq(arcs(n),n=1..34);

b[n_, k_] := b[n, k] = Which[n == 1 && k == 1, 1, k == 1, b[n - 1, n - 1], True, b[n, k - 1] + b[n - 1, k - 1]];
arcs[n_] := Module[{res = 0, k}, For[k = 1, k <= n-1, k++, res = res + k * b[n, k]]; res];
Array[arcs, 34] (* Jean-François Alcover, Nov 25 2017, translated from Maple *)

a(n) = Sum_{k=1..n} Stirling2(n,k) * k * (n-k). - Ilya Gutkovskiy, Apr 06 2021

a(n) = Sum_{k=n..n*(n+1)/2} (k-n) * A367955(n,k). - Alois P. Heinz, Dec 11 2023

A367969 Number of partitions of [n] whose block maxima sum to k, where k is chosen so as to maximize this number.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 37, 129, 431, 1921, 9544, 43844, 223512, 1407320, 8519457, 52422985, 373424140, 2685768084, 20354852852, 160370778238, 1318493838635, 11239312718146, 98700416575613, 916309760098349, 8735277842452542, 84921152781222758, 860903677319960583

Offset: 0

Alois P. Heinz, Dec 06 2023

nonn

			a(5) = 11 = A367955(5,12) is the largest value in row 5 of A367955 and counts the partitions of [5] having block maxima sum 12: 123|4|5, 124|3|5, 125|3|4, 13|24|5, 13|25|4, 14|23|5, 15|23|4, 14|25|3, 15|24|3, 1|2|34|5, 1|2|35|4.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition of a set

Row maxima of A367955.

Cf. A365441, A365903.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> max(coeffs(b(n, 0))):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 max(seq(b(k, n, 0), k=n..n*(n+1)/2)):
seq(a(n), n=0..30);

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m]*m + Expand[x^n*b[n-1, m+1]]];
a[n_] := Max[CoefficientList[b[n, 0], x]];
Table[a[n], {n, 0, 30}]
(* second program: *)
b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, Max[Table[b[k, n, 0], { k, n, n*(n + 1)/2}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)

A368675 Number of partitions of [n] whose block maxima sum to 2n.

Original entry on oeis.org

1, 0, 0, 1, 2, 7, 15, 39, 81, 193, 396, 885, 1816, 3915, 7973, 16860, 34165, 71092, 143804, 295963, 596872, 1219950, 2455139, 4989265, 10028841, 20296288, 40745616, 82225558, 164916967, 332045545, 665566046, 1337794545, 2680049287, 5380396625, 10774301183

Offset: 0

Alois P. Heinz, Jan 02 2024

nonn

			a(0) = 1: the empty partition.
a(3) = 1: 1|2|3.
a(4) = 2: 1|23|4, 1|24|3.
a(5) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 145|2|3, 1|24|35, 1|25|34.
a(6) = 15: 12|34|56, 12|356|4, 134|2|56, 1356|2|4, 1|2345|6, 1|2346|5, 1|235|46, 1|236|45, 14|2|356, 1|245|36, 1|246|35, 156|2|34, 1|25|346, 1|26|345, 1|2|3|456.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Partition of a set

Cf. A367955, A365441, A368678.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> coeff(b(n, 0), x, 2*n):
seq(a(n), n=0..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(2*n, n, 0):
seq(a(n), n=0..42);

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[2n, n, 0]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

a(n) = A367955(n,2n).

a(n) ~ c * 2^n, where c = 0.636808431228827742738441592748953932083264824206324529619378074873607293... - Vaclav Kotesovec, Jan 13 2024

A369596 Number T(n,k) of permutations of [n] whose fixed points sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1, 44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1, 265, 44, 44, 53, 53, 62, 64, 29, 22, 24, 16, 16, 8, 6, 5, 4, 2, 1, 1, 0, 0, 1, 1854, 265, 265, 309, 309, 353, 362, 406, 150, 159, 126, 126, 93, 86, 44, 36, 29, 19, 19, 9, 7, 5, 4, 2, 1, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 02 2024

			T(3,0) = 2: 231, 312.
T(3,1) = 1: 132.
T(3,2) = 1: 321.
T(3,3) = 1: 213.
T(3,6) = 1: 123.
T(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Triangle T(n,k) begins:
   1;
   0, 1;
   1, 0, 0,  1;
   2, 1, 1,  1,  0,  0, 1;
   9, 2, 2,  3,  3,  2, 1, 1, 0, 0, 1;
  44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Permutation

Column k=0 gives A000166.
Column k=3 gives A000255(n-2) for n>=2.
Row sums give A000142.
Row lengths give A000124.
Reversed rows converge to A331518.
T(n,n) gives A369796.
Cf. A000217, A001710, A008289, A008290, A038720, A053632, A062869, A124327, A143946, A143947, A161680, A263753, A263756, A317527, A367955, A368338.

b:= proc(s) option remember; (n-> `if`(n=0, 1, add(expand(
      `if`(j=n, x^j, 1)*b(s minus {j})), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..7);
# second Maple program:
g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
    end:
T:= (n, k)-> b(k, min(n, k), n):
seq(seq(T(n, k), k=0..n*(n+1)/2), n=0..7);

g[n_] := g[n] = If[n == 0, 1, n*g[n - 1] + (-1)^n];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0,
   If[n == 0, g[m], b[n, i-1, m] + b[n-i, Min[n-i, i-1], m-1]]];
T[n_, k_] := b[k, Min[n, k], n];
Table[Table[T[n, k], {k, 0, n*(n + 1)/2}], {n, 0, 7}] // Flatten (* Jean-François Alcover, May 24 2024, after Alois P. Heinz *)

Sum_{k=0..A000217(n)} k * T(n,k) = A001710(n+1) for n >= 1.
Sum_{k=0..A000217(n)} (1+k) * T(n,k) = A038720(n) for n >= 1.
Sum_{k=0..A000217(n)} (n*(n+1)/2-k) * T(n,k) = A317527(n+1).
T(n,A161680(n)) = A331518(n).
T(n,A000217(n)) = 1.

A368102 Total number of partitions of [n-s] whose block maxima sum to s, summed over all s.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 3, 2, 3, 6, 5, 9, 14, 13, 20, 40, 31, 56, 86, 105, 127, 227, 244, 394, 520, 665, 911, 1536, 1565, 2449, 3507, 4719, 5931, 9061, 11151, 16911, 21774, 29798, 39804, 60411, 71865, 104399, 144999, 197907, 253430, 370044, 478764, 694807

Offset: 0

Alois P. Heinz, Dec 11 2023

nonn

			a(11) = 6: 123|4, 124|3, 13|24, 14|23, 1|2|34, 1|2345.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Partition of a set

Antidiagonal sums of A367955.

Cf. A365821.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> add(coeff(b(n-k, 0), x, k), k=ceil(n/2)..n):
seq(a(n), n=0..80);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 add(b(k, n-k, 0), k=ceil(n/2)..n):
seq(a(n), n=0..80);

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, Sum[b[k, n - k, 0], {k, Ceiling[n/2], n}]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

A368401 Number T(n,k) of permutations of [n] whose sum of cycle maxima minus cycle minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 2, 1, 4, 12, 28, 53, 12, 10, 1, 5, 18, 52, 135, 289, 84, 72, 58, 6, 1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42, 1, 7, 33, 125, 429, 1407, 4545, 12983, 3520, 3976, 4292, 3950, 3422, 790, 486, 330, 24

Offset: 0

Alois P. Heinz, Dec 22 2023

			T(3,0) = 1: (1)(2)(3).
T(3,1) = 2: (12)(3), (1)(23).
T(3,2) = 3: (123), (132), (13)(2).
Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 11,   2;
  1, 4, 12, 28,  53,  12,   10;
  1, 5, 18, 52, 135, 289,   84,  72,  58,   6;
  1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42;
  ...

Alois P. Heinz, Rows n = 0..21, flattened
Wikipedia, Permutation

Row sums give A000142.

Cf. A002538, A002620, A124327, A143946, A367955, A368338.

b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
      b(n-1, s)+add(b(n-1, subs(h=h+[0, 1], s)), h=s), 0)+
      `if`(n>k+1, b(n-1, {s[], [n,1]}), 0)+add(h[2]!*expand(
      x^(h[1]-n)*b(n-1, s minus {h})), h=s))(nops(s)))
    end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);

Sum_{k=0..A002620(n)} k * T(n,k) = A002538(n-1) for n >= 1.

cp's OEIS Frontend

A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

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A204856 G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).

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A124327 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).

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A365441 Number of partitions of [2n] whose block maxima sum to 3n.

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A200660 Sum of the number of arcs describing the set partitions of {1,2,...,n}.

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A367969 Number of partitions of [n] whose block maxima sum to k, where k is chosen so as to maximize this number.

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A368675 Number of partitions of [n] whose block maxima sum to 2n.

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A204856 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).