A011971 -id:A011971 - 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

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 7, 10, 15, 52, 15, 20, 27, 37, 52, 203, 52, 67, 87, 114, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A046934 Same rule as Aitken triangle (A011971) except a(0,0)=1, a(1,0)=0.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 8, 11, 15, 21, 21, 27, 35, 46, 61, 82, 82, 103, 130, 165, 211, 272, 354, 354, 436, 539, 669, 834, 1045, 1317, 1671, 1671, 2025, 2461, 3000, 3669, 4503, 5548, 6865, 8536, 8536, 10207, 12232, 14693, 17693, 21362, 25865, 31413, 38278, 46814

Offset: 0

N. J. A. Sloane, R. K. Guy

First differences of A046935 = this triangle seen as flattened list without the initial term. - Reinhard Zumkeller, Nov 10 2013

			[0] [   1]
[1] [   0,     1]
[2] [   1,     1,     2]
[3] [   2,     3,     4,     6]
[4] [   6,     8,    11,    15,    21]
[5] [  21,    27,    35,    46,    61,    82]
[6] [  82,   103,   130,   165,   211,   272,   354]
[7] [ 354,   436,   539,   669,   834,  1045,  1317,  1671]
[8] [1671,  2025,  2461,  3000,  3669,  4503,  5548,  6865,  8536]
[9] [8536, 10207, 12232, 14693, 17693, 21362, 25865, 31413, 38278, 46814]

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Borders give A032347 and A032346. Cf. A046935.

a046934 n k = a046934_tabl !! n !! k
a046934_row n = a046934_tabl !! n
a046934_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [0,1]
a046934_list = concat a046934_tabl
-- Reinhard Zumkeller, Nov 10 2013

alias(PS = ListTools:-PartialSums):
A046934Triangle := proc(len) local a, k, P, T; a := 0; P := [1]; T := [];
for k from 1 to len  do T := [op(T), P]; P := PS([a, op(P)]); a := P[-1] od;
ListTools:-Flatten(T) end: A046934Triangle(10);  # Peter Luschny, Mar 29 2022

a[0, 0] = 1; a[1, 0] = 0; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 14 2013 *)

A046936 Same rule as Aitken triangle (A011971) except a(0,0)=0, a(1,0)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 9, 12, 16, 22, 31, 31, 40, 52, 68, 90, 121, 121, 152, 192, 244, 312, 402, 523, 523, 644, 796, 988, 1232, 1544, 1946, 2469, 2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611, 12611, 15080, 18072, 21708, 26140

Offset: 0

N. J. A. Sloane, R. K. Guy

			Triangle starts:
0,
1, 1,
1, 2, 3,
3, 4, 6, 9,
9, 12, 16, 22, 31,
31, 40, 52, 68, 90, 121,
121, 152, 192, 244, 312, 402, 523,
523, 644, 796, 988, 1232, 1544, 1946, 2469,
2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611,
12611, 15080, ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Borders give A040027. Reading across rows gives A007604.

Cf. A121207, A298804.

a046936 n k = a046936_tabl !! n !! k
a046936_row n = a046936_tabl !! n
a046936_tabl = [0] : iterate (\row -> scanl (+) (last row) row) [1,1]
-- Reinhard Zumkeller, Jan 01 2014

a[0, 0] = 0; a[1, 0] = 1; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 15 2013 *)

from itertools import accumulate
def A046936(): # Compare function Gould_diag in A121207.
    yield [0]
    accu = [1, 1]
    while True:
        yield accu
        accu = list(accumulate([accu[-1]] + accu))
g = A046936()
[next(g) for  in range(9)] # _Peter Luschny, Apr 25 2016

A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

Original entry on oeis.org

1, 2, 3, 3, 5, 8, 8, 11, 16, 24, 24, 32, 43, 59, 83, 83, 107, 139, 182, 241, 324, 324, 407, 514, 653, 835, 1076, 1400, 1400, 1724, 2131, 2645, 3298, 4133, 5209, 6609, 6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758

Offset: 0

N. J. A. Sloane, R. K. Guy

			Triangle starts:
[0] [   1]
[1] [   2,    3]
[2] [   3,    5,    8]
[3] [   8,   11,   16,    24]
[4] [  24,   32,   43,    59,    83]
[5] [  83,  107,  139,   182,   241,   324]
[6] [ 324,  407,  514,   653,   835,  1076,  1400]
[7] [1400, 1724, 2131,  2645,  3298,  4133,  5209,  6609]
[8] [6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758]

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Borders give A038561.

Cf. A011971.

a046937 n k = a046937_tabl !! n !! k
a046937_row n = a046937_tabl !! n
a046937_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [2,3]
-- Reinhard Zumkeller, Jan 13 2013

# Compare the analogue algorithm for the Catalan triangle in A350584.
A046937Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := P; T := [op(T), P] od; T end:
A046937Triangle(9): ListTools:-Flatten(%); # Peter Luschny, Mar 27 2022

a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 06 2013 *)

A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 52, 37, 27, 20, 15, 203, 151, 114, 87, 67, 52, 877, 674, 523, 409, 322, 255, 203, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 115975, 94828, 77821, 64077, 52922, 43833, 36401, 30304, 25287, 21147

Offset: 0

N. J. A. Sloane, Oct 14 2006

a(n,k) is k-th difference of Bell numbers, with a(n,1) = A000110(n) for  n>0, a(n,k) = a(n,k-1) - a(n-1, k-1), k<=n, with diagonal (k=n) also equal to Bell numbers (n>=0). - Richard R. Forberg, Jul 13 2013
From Don Knuth, Jan 29 2018: (Start)
If the offset here is changed from 0 to 1, then we can say:
a(n,k) is the number of equivalence classes of [n] in which 1 not equiv to 2, ..., 1 not equiv to k.
In Volume 4A, page 418, I pointed out that a(n,k) is the number of set partitions in which k is the smallest of its block.
And in exercise 7.2.1.5--33, I pointed out that a(n,k) is the number of equivalence relations in which 1 not equiv to 2, 2 not equiv to 3, ..., k-1 not equiv to k. (End)

			Triangle begins:
    1
    2   1
    5   3   2
   15  10   7  5
   52  37  27 20 15
  203 151 114 87 67 52
  ...

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
A. Dil, Veli Kurt, Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices, J. Int. Seq. 14 (2011) # 11.4.6
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018
Eric Weisstein's World of Mathematics, Bell Triangle.

Cf. A011971. Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, A011968, A011969, A046934, A011972, A094577, A095149, A106436, A108041, A108042, A108043.

a123346 n k = a123346_tabl !! n !! k
a123346_row n = a123346_tabl !! n
a123346_tabl = map reverse a011971_tabl
-- Reinhard Zumkeller, Dec 09 2012

a[n_, k_] := Sum[Binomial[n - k, i - k] BellB[i], {i, k, n}];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A123346_list = blist = [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A123346_list += reversed(blist)
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

a(n,k) = Sum_{i=k..n} binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

More terms from Alexander Adamchuk and Vladeta Jovovic, Oct 14 2006

A011972 Sequence formed by reading rows of triangle defined in A011971.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 15, 20, 27, 37, 52, 67, 87, 114, 151, 203, 255, 322, 409, 523, 674, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828

Offset: 0

N. J. A. Sloane and J. H. Conway

Terms that are repeated in A011971 are included only once. In other words, dropping the elements on the diagonal and reading by rows gives this sequence. [Joerg Arndt, May 31 2013]

			Triangle T(n, k) begins:
[0]   1;
[1]   2,    3;
[2]   5,    7,   10;
[3]  15,   20,   27,   37;
[4]  52,   67,   87,  114,  151;
[5] 203,  255,  322,  409,  523,  674;
[6] 877, 1080, 1335, 1657, 2066, 2589, 3263;
...

Chai Wah Wu, Rows n = 0..200, flattened

T := (n, k) -> local i; add(binomial(k, i)*combinat:-bell(n - k + i + 1), i = 0..k): seq(seq(T(n, k), k=0..n), n = 0..9);  # Peter Luschny, Dec 02 2023

T[n_, k_] := Sum[Binomial[k, i] BellB[n - k + i + 1], {i, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

from itertools import accumulate
A011972_list = blist = [1]
for _ in range(10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A011972_list += blist[1:]
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

A095675 Triangle read by rows, formed from product of Aitken's (or Bell's) triangle (A011971) and Pascal's triangle (A007318).

Original entry on oeis.org

1, 3, 2, 10, 13, 5, 37, 72, 55, 15, 151, 393, 450, 245, 52, 674, 2202, 3365, 2748, 1166, 203, 3263, 12850, 24582, 26781, 17048, 5936, 877, 17007, 78488, 180477, 245971, 208856, 109107, 32243, 4140, 94828, 502327, 1349900, 2209695, 2346559, 1634998

Offset: 0

N. J. A. Sloane, based on a suggestion from Gary W. Adamson, Jun 22 2004

These triangles are to be thought of as infinite lower-triangular matrices.

			Triangle begins:
1
3 2
10 13 5
37 72 55 15
151 393 450 245 52

Cf. A007318, A011971, A095674. Row sums give A095676. First column is A005493.

a[0, 0] = 1; a[n_, 0] := a[n - 1, n - 1]; a[n_, k_] := a[n, k] = If[k < n + 1, a[n, k - 1] + a[n - 1, k - 1], 0]; p[n_, r_] := If[r <= n + 1, Binomial[n, r], 0]; am = Table[ a[n, r], {n, 0, 9}, {r, 0, 9}]; pm = Table[p[n, r], {n, 0, 9}, {r, 0, 9}]; t = Flatten[am.pm]; Delete[ t, Position[t, 0]] (* Robert G. Wilson v, Jul 12 2004 *)

More terms from Robert G. Wilson v, Jul 13 2004

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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A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

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A046934 Same rule as Aitken triangle (A011971) except a(0,0)=1, a(1,0)=0.

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A046936 Same rule as Aitken triangle (A011971) except a(0,0)=0, a(1,0)=1.

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A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

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A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

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A011972 Sequence formed by reading rows of triangle defined in A011971.

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