A124496 -id:A124496 - cp's OEIS frontend

A040027 The Gould numbers.

1, 1, 3, 9, 31, 121, 523, 2469, 12611, 69161, 404663, 2512769, 16485691, 113842301, 824723643, 6249805129, 49416246911, 406754704841, 3478340425563, 30845565317189, 283187362333331, 2687568043654521, 26329932233283223, 265946395403810289, 2766211109503317451

Offset: 0

Henry Gould

Number of permutations beginning with 21 and avoiding 1-23. - Ralf Stephan, Apr 25 2004
Originally defined as main diagonal of an array of binomial recurrence coefficients (see Gould and Quaintance). Also second-from-right diagonal of triangle A121207.
Starting (1, 3, 9, 31, 121, ...) = row sums of triangle A153868. - Gary W. Adamson, Jan 03 2009
Equals eigensequence of triangle A074909 (reflected). - Gary W. Adamson, Apr 10 2009
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m=>-1, is related to the sequence given above. For m=-1 this series dates back to Euler. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A073003 Gompertz's constant and A000110 the Bell numbers, see A163940; A040027(m = -1) = 0. - Johannes W. Meijer, Oct 16 2009
Compare the o.g.f. to the o.g.f. B(x) of the Bell numbers, where B(x) = 1 + x*B(x/(1-x))/(1-x). - Paul D. Hanna, Mar 23 2012
a(n) is the number of set partitions of {1,2,...,n+1} in which the last block is a singleton: the blocks are arranged in order of their least element. An example is given below. - Peter Bala, Dec 17 2014

			a(3) = 9: Arranging the blocks of the 15 set partitions of {1,2,3,4} in order of their least element we find 9 set partitions for which the last block is a singleton, namely, 123|4, 124|3, 134|2, 1|24|3, 1|23|4, 12|3|4, 13|2|4, 14|2|3, and 1|2|3|4. - _Peter Bala_, Dec 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Walaa Asakly, Aubrey Blecher, Charlotte Brennan, Arnold Knowfmacher, Toufik Mansour, and Stephan Wagner, Set partition asymptotics and a conjecture of Gould and Quaintance, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2, 15 August 2014, Pages 672-682.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See pp. 8-9,17.
Robert Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv:2210.13520 [cs.SC], 2022.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 69.
Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079 [math.NT], 2015.
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, October 2014, Volume 6, Issue 4, pp 275-283.
H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials, Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968 [The letter gives the g.f. for this sequence as e^{e^x} Integral_{0..x} e^{e^t-1} dt but the correct g.f. is e^{e^x-1} Integral_0^x e^{1-e^t} dt. - _Don Knuth_, Feb 01 2018]
Sergey Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
Sergey Kitaev and Toufik Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Left-hand border of triangle A046936. Cf. also A011971, A014619, A298804.
Cf. A153868. - Gary W. Adamson, Jan 03 2009
Cf. A074909. - Gary W. Adamson, Apr 10 2009
Row sums of A163940. - Johannes W. Meijer, Oct 16 2009
Cf. A108458 (row sums), A124496 (column 1).

a040027 n = head $ a046936_row (n + 1)  -- Reinhard Zumkeller, Jan 01 2014

A040027 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add(binomial(n,k-1)*procname(n-k),k=1..n) ;
    end if;
end proc: # Johannes W. Meijer, Oct 16 2009

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n, k + 1]*a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]  (* Jean-François Alcover, Jul 02 2013 *)
Rest[CoefficientList[Assuming[Element[x, Reals], Series[E^E^x*(ExpIntegralEi[-E^x] - ExpIntegralEi[-1]), {x, 0, 20}]], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 28 2014 *)

{a(n)=local(A=1+x);for(i=1,n,A=1+x*subst(A,x,x/(1-x+x*O(x^n)))/(1-x)^2);polcoeff(A,n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A040027_list = lambda size: Gould_diag(2, size)
print(A040027_list(24)) # Peter Luschny, Apr 24 2016

a(n) = b(n-2), n>1, b(n) = Sum_{k = 1..n} binomial(n, k-1)*b(n-k), b(0) = 1. - Vladeta Jovovic, Apr 28 2001
E.g.f. satisfies A'(x) = exp(x)*A(x)+1. - N. J. A. Sloane
With offset 0, e.g.f.: x + exp(exp(x)) * Integral_{t=0..x} t*exp(-exp(t)+t) dt (fits the recurrence up to n=215). - Ralf Stephan, Apr 25 2004
Recurrence: a(0)=1, a(1)=1, for n > 1, a(n) = n + Sum_{j=1..n-1} binomial(n, j+1)*a(j). - Jon Perry, Apr 26 2005
O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^2. - Paul D. Hanna, Mar 23 2012
From Peter Bala, Dec 17 2014: (Start)
Starting from A(x) = 1 + O(x) (big Oh notation) we can get a series expansion for the o.g.f. by repeatedly applying the above functional equation of Hanna: A(x) = 1 + O(x) = 1 + x/(1-x)^2 + O(x^2) = 1 + x/(1-x)^2 + x^2/((1-x)*(1-2*x)^2) + O(x^3) = ... = 1 + x/(1-x)^2 + x^2/((1-x)*(1-2*x)^2) + x^3/((1-x)*(1-2*x)*(1-3*x)^2) + x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)^2) + ....
a(n) = Sum_{k = 0..n} ( Sum_{j = k..n} Stirling2(j,k)*k^(n-j) ).
Row sums of A108458. First column of A124496. (End)
Conjecture: a(n) = Sum_{k = 0..n} A058006(k)*A048993(n+1, k+1) - Velin Yanev, Aug 31 2021

Entry revised by N. J. A. Sloane, Dec 11 2006
Gould reference updated by Johannes W. Meijer, Aug 02 2009
Don Knuth, Jan 29 2018, suggested that this sequence should be named after H. W. Gould. - N. J. A. Sloane, Jan 30 2018

A121207 Triangle read by rows. The definition is by diagonals. The r-th diagonal from the right, for r >= 0, is given by b(0) = b(1) = 1; b(n+1) = Sum_{k=0..n} binomial(n+2,k+r)*a(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 15, 1, 1, 5, 14, 31, 52, 1, 1, 6, 20, 54, 121, 203, 1, 1, 7, 27, 85, 233, 523, 877, 1, 1, 8, 35, 125, 400, 1101, 2469, 4140, 1, 1, 9, 44, 175, 635, 2046, 5625, 12611, 21147, 1, 1, 10, 54, 236, 952, 3488, 11226, 30846, 69161, 115975

Offset: 0

N. J. A. Sloane, based on email from R. J. Mathar, Dec 11 2006

From Paul D. Hanna, Dec 12 2006: (Start)
Consider the row reversal, which is A124496 with an additional left column (A000110 = Bell numbers). The matrix inverse of this triangle is very simple:
   1;
  -1,   1;
  -1,  -1,   1;
  -1,  -2,  -1,   1;
  -1,  -3,  -3,  -1,   1;
  -1,  -4,  -6,  -4,  -1,   1;
  -1,  -5, -10, -10,  -5,  -1,   1;
  -1,  -6, -15, -20, -15,  -6,  -1,   1;
  -1,  -7, -21, -35, -35, -21,  -7,  -1,   1;
  -1,  -8, -28, -56, -70, -56, -28,  -8,  -1,   1; ...
This gives the recurrence and explains why the Bell numbers appear. (End)
Triangle A160185 = reversal then deletes right border of 1's. - Gary W. Adamson, May 03 2009

			Triangle begins (compare also table 9.2 in the Gould-Quaintance reference):
  1;
  1, 1;
  1, 1,  2;
  1, 1,  3,  5;
  1, 1,  4,  9,  15;
  1, 1,  5, 14,  31, 52;
  1, 1,  6, 20,  54, 121, 203;
  1, 1,  7, 27,  85, 233, 523,  877;
  1, 1,  8, 35, 125, 400,1101, 2469,  4140;
  1, 1,  9, 44, 175, 635,2046, 5625, 12611, 21147;
  1, 1, 10, 54, 236, 952,3488,11226, 30846, 69161, 115975;
  1, 1, 11, 65, 309,1366,5579,20425, 65676,180474, 404663, 678570;
  1, 1, 12, 77, 395,1893,8494,34685,126817,407787,1120666,2512769,4213597;

Alois P. Heinz, Rows n = 0..140, flattened
Robert Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv:2210.13520 [cs.SC], 2022.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See pp. 69-70.
H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials, Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.

Diagonals, reading from the right, are A000110, A040027, A045501, A045499, A045500.
A124496 is a very similar triangle, obtained by reversing the rows and appending a rightmost diagonal which is A000110, the Bell numbers. See also A046936, A298804, A186020, A160185.
T(2n,n) gives A297924.

function Gould_diag(diag, size)
    size < 1 && return []
    size == 1 && return [1]
    L = [1, 1]
    accu = ones(BigInt, diag)
    for _ in 1:size-2
        accu = cumsum(vcat(accu[end], accu))
        L = vcat(L, accu[end])
    end
L end # Peter Luschny, Mar 30 2022

# This is the Jovovic formula with general index 'd'
# where A040027, A045499, etc. use one explicit integer
# Index n+1 is shifted to n from the original formula.
Gould := proc(n, d) local k;
    if n <= 1 then return 1 else
    return add(binomial(n-1+d, k+d)*Gould(k, d), k=0..n-1);
    fi
end:
# row and col refer to the extrapolated super-table:
# working up to row, not row-1, shows also the Bell numbers
# at the end of each row.
for row from 0 to 13 do
    for col from 0 to row do
       # 'diag' is constant for one of A040027, A045499 etc.
       diag := row - col;
       printf("%4d, ", Gould(col, diag));
    od;
    print();
od; # R. J. Mathar
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, 1,
      add(T(n-j, k-j)*binomial(n-1, j-1), j=1..k))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 08 2018

g[n_ /; n <= 1, ] := 1; g[n, d_] := g[n, d] = Sum[ Binomial[n-1+d, k+d]*g[k, d], {k, 0, n-1}]; Flatten[ Table[ diag = row-col; g[col, diag], {row, 0, 13}, {col, 0, row}]] (* Jean-François Alcover, Nov 25 2011, after R. J. Mathar *)
T[n_, k_] := T[n, k] = If[k == 0, 1, Sum[T[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018, after Alois P. Heinz *)

# Computes the n-th diagonal of the triangle reading from the right.
from itertools import accumulate
def Gould_diag(diag, size):
    if size < 1: return []
    if size == 1: return [1]
    L, accu = [1,1], [1]*diag
    for _ in range(size-2):
        accu = list(accumulate([accu[-1]] + accu))
        L.append(accu[-1])
    return L # Peter Luschny, Apr 24 2016

A045499 Fourth-from-right diagonal of triangle A121207.

Original entry on oeis.org

1, 1, 5, 20, 85, 400, 2046, 11226, 65676, 407787, 2675410, 18475311, 133843405, 1014271763, 8019687099, 66011609670, 564494701167, 5005880952390, 45958055208576, 436161412834300, 4273045478169842, 43160044390231165

Offset: 0

Henry Gould

With leading 0 and offset 3: number of permutations beginning with 4321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

a(n) is the number of set partitions of {1,2,...,n+3} in which the last block has length 3; the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

Seiichi Manyama, Table of n, a(n) for n = 0..573
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.

Cf. A045500, A045501, A346059.

Column k=3 of A124496.

A045499 := proc(n)
    option remember ;
    if n =0  then
        1 ;
    else
        add( binomial(n+2,k+3)*procname(k),k=0..n-1) ;
    end if;
end proc: # R. J. Mathar, Jun 03 2014

a[0] = 1; a[n_] := a[n] = Sum[a[k]*Binomial[n+2, k+3], {k, 0, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 20 2017 *)

{a(n)=local(A=1+x); for(i=1, n, A=1+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^4); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A045499_list = lambda size: Gould_diag(4, size)
print(A045499_list(24)) # Peter Luschny, Apr 24 2016

a(n+1) = Sum_{k=0..n} binomial(n+3, k+3)*a(k). - Vladeta Jovovic, Nov 10 2003
With offset 3, e.g.f.: x^3 + exp(exp(x))/6 * int[0..x, t^3*exp(-exp(t)+t) dt]. - Ralf Stephan, Apr 25 2004
O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^4. [Paul D. Hanna, Mar 23 2012]

More terms from Vladeta Jovovic, Nov 10 2003

Entry revised by N. J. A. Sloane, Dec 11 2006

A186020 Eigentriangle of the binomial matrix.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 15, 9, 4, 1, 1, 52, 31, 14, 5, 1, 1, 203, 121, 54, 20, 6, 1, 1, 877, 523, 233, 85, 27, 7, 1, 1, 4140, 2469, 1101, 400, 125, 35, 8, 1, 1, 21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1, 115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1

Offset: 0

Paul Barry, Feb 10 2011

Reversal of Gould triangle A121207. First column is A000110. Second column is A040027.
Row sums are A186021. Diagonal sums are A186022.
Construction is described by Paul D. Hanna in A121207. The method of construction is general for this class of eigentriangle.

			Triangle T begins
       1;
       1,     1;
       2,     1,     1;
       5,     3,     1,     1;
      15,     9,     4,     1,    1;
      52,    31,    14,     5,    1,   1;
     203,   121,    54,    20,    6,   1,   1;
     877,   523,   233,    85,   27,   7,   1,  1;
    4140,  2469,  1101,   400,  125,  35,   8,  1,  1;
   21147, 12611,  5625,  2046,  635, 175,  44,  9,  1, 1;
  115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1;
Inverse is the identity matrix I minus binomial matrix B shifted down once, or
T^{-1}(n,k)=if(k=n,1,if(k<n,-binomial(n-1,k),0)). This begins
   1;
  -1,  1;
  -1, -1,   1;
  -1, -2,  -1,   1;
  -1, -3,  -3,  -1,   1;
  -1, -4,  -6,  -4,  -1,   1;
  -1, -5, -10, -10,  -5,  -1,   1;
  -1, -6, -15, -20, -15,  -6,  -1,  1;
  -1, -7, -21, -35, -35, -21,  -7, -1,  1;
  -1, -8, -28, -56, -70, -56, -28, -8, -1, 1;
Production matrix is
      1,     1;
      1,     0,    1;
      2,     1,    0,    1;
      5,     3,    1,    0,   1;
     15,     9,    4,    1,   0,   1;
     52,    31,   14,    5,   1,   0,  1;
    203,   121,   54,   20,   6,   1,  0, 1;
    877,   523,  233,   85,  27,   7,  1, 0, 1;
   4140,  2469, 1101,  400, 125,  35,  8, 1, 0, 1;
  21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 0, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Cf. A000110, A007318, A121207, A124496, A160185, A186021, A186022.

t[n_, k_] := t[n, k] = If[k == 0, 1, Sum[t[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]];
T[n_, k_] := t[n, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 27 2018 *)

Lower triangular (infinite) matrix T = (U - D*P)^{-1} with the unit matrix U, the Pascal matrix P from A007318 and the matrix D with elements delta_{i,j+1}, for i, j >= 0 (row 0 has only 0s). From the Paul Barry paper rewritten in matrix notation. T satisfies P*T = D'*(T - U), with D' the transposed matrix D, that is the diagonal of T has been erased and the row index shifted on the r.h.s. (showing that the name Eigentriangle or -matrix is a misnomer). For finite N X N matrices P*T = D'*(T - U), only up to the last row. - Wolfdieter Lang, Apr 07 2021

A045500 Fifth-from-right diagonal of triangle A121207.

Original entry on oeis.org

1, 1, 6, 27, 125, 635, 3488, 20425, 126817, 831915, 5744784, 41618459, 315388311, 2493721645, 20526285716, 175529425815, 1556577220651, 14290644428279, 135624265589086, 1328702240382589, 13420603191219111, 139592874355534071

Offset: 0

Henry Gould

With leading 0 and offset 4: number of permutations beginning with 54321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

a(n) is the number of set partitions of {1,2,...,n+4} in which the last block has length 4: the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

See also references under sequence A040027.

Seiichi Manyama, Table of n, a(n) for n = 0..573
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2014.

Cf. A040027, A045499, A045501, A121207, A346060.

Column k=4 of A124496.

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n+3, k+4]*a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)

{a(n)=local(A=1+x); for(i=1, n, A=1+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^5); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A045500_list = lambda size: Gould_diag(5, size)
print(A045500_list(24)) # Peter Luschny, Apr 24 2016

a(n+1) = Sum_{k=0..n} binomial(n+4, k+4)*a(k). - Vladeta Jovovic, Nov 10 2003
With offset 4, e.g.f.: x^4 + exp(exp(x))/24 * int[0..x, t^4*exp(-exp(t)+t) dt]. - Ralf Stephan, Apr 25 2004
O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^5. - Paul D. Hanna, Mar 23 2012

More terms from Vladeta Jovovic, Nov 10 2003

Entry revised by N. J. A. Sloane, Dec 11 2006

A045501 Third-from-right diagonal of triangle A121207.

Original entry on oeis.org

1, 1, 4, 14, 54, 233, 1101, 5625, 30846, 180474, 1120666, 7352471, 50772653, 367819093, 2787354668, 22039186530, 181408823710, 1551307538185, 13756835638385, 126298933271289, 1198630386463990, 11742905240821910

Offset: 1

Henry Gould

With leading 0 and offset 2: number of permutations beginning with 321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004
Second diagonal in table of binomial recurrence coefficients. Related to A040027. - Vladeta Jovovic, Feb 05 2008
Equals eigensequence of triangle A104712. - Gary W. Adamson, Apr 10 2009
a(n) is the number of set partitions of {1,2,...,n+1} in which the last block has length 2; the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.

Cf. A045499, A045500.
Cf. A104712. - Gary W. Adamson, Apr 10 2009
Column k=2 of A124496.

a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n, k+1]*a[k], {k, 0, n-1}];
Array[a, 22] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)

{a(n)=local(A=x+x^2); for(i=1, n, A=x+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^2); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

# The function Gould_diag is defined in A121207.
A045501_list = lambda size: Gould_diag(3, size)
print(A045501_list(24)) # Peter Luschny, Apr 24 2016

a(n+1) = Sum_{k=0..n} binomial(n+2, k+2)*a(k). - Vladeta Jovovic, Nov 10 2003
With offset 2, e.g.f.: x^2 + exp(exp(x))/2 * Integral_{0..x} t^2*exp(-exp(t)+t) dt. - Ralf Stephan, Apr 25 2004
G.f.: A(x) = Sum_{k>=0} x^(k+1)/((1-k*x)^2 * Product_{m=0..k} (1 - m*x)). - Vladeta Jovovic, Feb 05 2008
O.g.f. satisfies: A(x) = x + x*A( x/(1-x) ) / (1-x)^2. - Paul D. Hanna, Mar 23 2012

More terms from Vladeta Jovovic, Nov 10 2003

Entry revised by N. J. A. Sloane, Dec 11 2006

A297924 Number of set partitions of [2n] in which the size of the last block is n.

Original entry on oeis.org

1, 1, 4, 20, 125, 952, 8494, 86025, 969862, 12020580, 162203607, 2363458396, 36930606254, 615302885459, 10878670826170, 203268056115256, 3999642836434361, 82617423216826640, 1786559190116778030, 40344863179696283037, 949348461372003462390

Offset: 0

Alois P. Heinz, Jan 08 2018

nonn

The blocks are ordered with increasing least elements.

a(0) = 1 by convention.

			a(1) = 1: 1|2.
a(2) = 4: 12|34, 13|24, 14|23, 1|2|34.
a(3) = 20: 123|456, 124|356, 125|346, 126|345, 12|3|456, 134|256, 135|246, 136|245, 13|2|456, 145|236, 146|235, 156|234, 1|23|456, 14|2|356, 1|24|356, 15|2|346, 1|25|346, 16|2|345, 1|26|345, 1|2|3|456.

Alois P. Heinz, Table of n, a(n) for n = 0..466
Wikipedia, Partition of a set

Cf. A121207, A124496, A276961, A297926.

b:= proc(n, k) option remember; `if`(n=k, 1,
      add(b(n-j, k)*binomial(n-1, j-1), j=1..n-k))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..25);

b[n_, k_] := b[n, k] = If[n == k, 1, Sum[b[n - j, k]*Binomial[n - 1, j - 1], {j, 1, n - k}]];
a[n_] := b[2*n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

a(n) = A121207(2n,n) = A124496(2n,n).

A160185 Triangle read by rows, (1 / ((-1)A129184 A007318 + I)) - I, I = Identity matrix.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 15, 9, 4, 1, 52, 31, 14, 5, 1, 203, 121, 54, 20, 6, 1, 877, 523, 233, 85, 27, 7, 1, 4140, 2469, 1101, 400, 125, 35, 8, 1, 21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1

Offset: 0

Gary W. Adamson, May 03 2009

Inverse binomial transform of the triangle shifts to left (= adding I as right border, I = Identity matrix); resulting in reversed rows of A121207.
Left border = Bell numbers, A000110 = eigensequence of Pascal's triangle.
Successive columns from left to right = eigensequences of Pascal's triangle deleting columns one at a time.
Row sums of the triangle = A060719: (1, 3, 9, 29, 103, ...). - Gary W. Adamson, May 20 2013
From Gary W. Adamson, Jul 18 2019: (Start)
Rows are eigensequences of triangles exemplified by the following arrangement of binomial sequences. Example: row 5 is (1, 5, 14, 31, 52, 0, 0, 0, ...), the eigensequence of:
  1;
  4, 1;
  6, 3, 1;
  4, 3, 2, 1;
  1, 1, 1, 1, 1;
  ... and the rest zeros.
Similarly, the production matrix for (1, 6, 20, 54, 121, 203, 0, 0, 0, ...) is:
   1;
   5, 1;
  10, 4, 1;
  10, 6, 3, 1;
   5, 4, 3, 2, 1;
   1, 1, 1, 1, 1, 1;
  ... and the rest zeros. (End)

			First few rows of the triangle:
       1;
       2,     1;
       5,     3,     1;
      15,     9,     4,     1;
      52,    31,    14,     5,    1;
     203,   121,    54,    20,    6,   1;
     877,   523,   233,    85,   27,   7,   1;
    4140,  2469,  1101,   400,  125,  35,   8,  1;
   21147, 12611,  5625,  2046,  635, 175,  44,  9,  1;
  115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1;
  ...

Cf. A121207, A124496, A186020.

Cf. A060719.

Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix.

Equals reversal by rows of triangle A121207, then delete right border. A121207 begins: 1; 1, 1; 1, 1, 2 1, 1, 3, 5; ...

Corrected by Alois P. Heinz, Apr 18 2013

A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 15, 0, 1, 9, 22, 37, 52, 0, 1, 17, 52, 99, 151, 203, 0, 1, 33, 130, 283, 471, 674, 877, 0, 1, 65, 340, 855, 1561, 2386, 3263, 4140, 0, 1, 129, 922, 2707, 5451, 8930, 12867, 17007, 21147, 0, 1, 257, 2572, 8919, 19921, 35098, 53411, 73681, 94828, 115975

Offset: 1

Christian G. Bower, Jun 03 2005; Emeric Deutsch, Nov 14 2006

Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.

Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - Augustine O. Munagi, Jul 16 2007

			Triangle T(n,k) starts:
  1;
  0,1;
  0,1,2;
  0,1,3,5;
  0,1,5,10,15;
T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.
The arrays U(n,k) starts:
   1  0  0   0   0 ...
   1  1  1   1   1 ...
   2  3  5   9  17 ...
   5 10 22  52 130 ...
  15 37 99 283 855 ...

Alois P. Heinz, Rows n = 1..141, flattened

Row sums of T(n, k) yield A124496(n, 1).
Cf. A108461.
Columns of U(n, k): A000110, A005493, A033452.
Rows of U(n, k): A000007, A000012, A000051.
Main diagonal: A108459.

T[n_, k_] := Sum[If[n == k, 1, i^(n-k)]*StirlingS2[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2024, after Vladeta Jovovic *)

T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).
Double e.g.f.: exp(exp(x)*(exp(y)-1)).
U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - Vladeta Jovovic, Jul 12 2007

Edited by N. J. A. Sloane, May 22 2008, at the suggestion of Vladeta Jovovic. This entry is a composite of two entries submitted independently by Christian G. Bower and Emeric Deutsch, with additional comments from Augustine O. Munagi.

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