A074664 -id:A074664 - cp's OEIS frontend

A205543 Logarithmic derivative of the Bell numbers (A000110).

1, 3, 10, 39, 171, 822, 4271, 23759, 140518, 878883, 5789015, 40019058, 289513303, 2186421919, 17199606090, 140662816543, 1193865048363, 10499107480518, 95528651305671, 898071593401559, 8712429618413678, 87118795125708283, 896925422648691735

Offset: 1

Paul D. Hanna, Jan 28 2012

nonn

a(n) = number of indecomposable partitions (A074664) of [n+3] in which n+3 lies in a doubleton block (see Link). - David Callan, Oct 08 2014

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 39*x^4/4 + 171*x^5/5 + 822*x^6/6 +...
where exponentiation yields the o.g.f. of the Bell numbers:
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 +...
which equals the series:
exp(L(x)) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) +...

David Callan, A combinatorial interpretation for this sequence

Cf. A000110.

{a(n)=n*polcoeff(log(sum(m=0,n, x^m/prod(k=1,m, 1-k*x +x*O(x^n)))),n)}

L.g.f.: log( Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x) ).

A130167 Another version of triangle in A127743.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 5, 3, 1, 0, 22, 16, 9, 4, 1, 0, 92, 60, 31, 14, 5, 1, 0, 426, 252, 120, 52, 20, 6, 1, 0, 2146, 1160, 510, 209, 80, 27, 7, 1, 0, 11624, 5776, 2348, 904, 335, 116, 35, 8, 1, 0, 67146, 30832, 11610, 4184, 1481, 507, 161, 44, 9, 1

Offset: 0

Philippe Deléham, Aug 03 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,1,2,1,3,1,4,1,5,1,6,1,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008
A154380*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 13 2009

			Triangle begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  6,  5,  3,  1;
  0, 22, 16,  9,  4,  1;
  0, 92, 60, 31, 14,  5,  1; ...

Cf. A074664.

Sum_{k=0..n} T(n,k) = A000110(n).

A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, ....

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 92, 57, 20, 5, 1, 0, 203, 426, 303, 116, 30, 6, 1, 0, 877, 2146, 1752, 744, 205, 42, 7, 1, 0, 4140, 11624, 10845, 5140, 1535, 330, 56, 8, 1, 0, 21147, 67146, 71139, 37676, 12300, 2820, 497, 72, 9, 1

Offset: 0

Peter Luschny, Dec 27 2018

For definitions and comments see A321964.

			First few rows of the array start:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  2,   5,   15,    52,    203,     877,     4140, ... A000110
[2] 1, 2,  6,  22,   92,   426,   2146,   11624,    67146, ... A074664
[3] 1, 3, 12,  57,  303,  1752,  10845,   71139,   491064, ... A321959
[4] 1, 4, 20, 116,  744,  5140,  37676,  290224,  2334300, ...
[5] 1, 5, 30, 205, 1535, 12300, 103975,  918785,  8434740, ...
[6] 1, 6, 42, 330, 2820, 25662, 245358, 2443272, 25188870, ...
[7] 1, 7, 56, 497, 4767, 48496, 516761, 5719399, 65369136, ...
Seen as triangle:
[0] 1;
[1] 0,   1;
[2] 0,   1,    1;
[3] 0,   2,    2,    1;
[4] 0,   5,    6,    3,   1;
[5] 0,  15,   22,   12,   4,   1;
[6] 0,  52,   92,   57,  20,   5,  1;
[7] 0, 203,  426,  303, 116,  30,  6, 1;
[8] 0, 877, 2146, 1752, 744, 205, 42, 7, 1;

Rows of array: A000007, A000110, A074664, A321959.
Columns include: A002378, A033445. Row sums of triangle: A321958.
Cf. A321964.

# The function JacobiSquare is defined in A321964.
s := n -> [seq(n+k, k = 0..9)]: Trow := n -> JacobiSquare(s(n)):
for n from 0 to 7 do lprint(Trow(n)) od;

nmax = 10;
JacobiCF[a_, b_, p_:2] := Module[{m, k}, m = 1; For[k = Length[a], k >= 1, k--, m = 1 - b[[k]]*x - a[[k]]*x^p/m]; 1/m];
JacobiSquare[a_, p_: 2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]];
s[n_] := Table[n + k, {k, 0, nmax}];
row[n_] := row[n] = JacobiSquare[s[n]];
T[, 0] = 1; T[0, ] = 0; T[n_, k_] := row[n][[k + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 13 2019, after Peter Luschny in A321964 *)

def JacobiCF(a, b, dim, p=2):
    m = 1
    for k in range(dim-1, -1, -1):
        m = 1 - b(k)*x - a(k)*x^p/m
    return 1/m
def JacobiGF(a, b, dim, p=2):
    cf = JacobiCF(a, b, dim, p)
    return cf.series(x, dim).list()
def JacobiSquare(a, dim, p=2):
    cf = JacobiCF(a, a, dim, p)
    return cf.series(x, dim).list()
def StieltjesGF(a, dim, p=2):
    return JacobiGF(a, lambda n: 0, dim, p)
def Trow(n): return JacobiSquare(lambda k: n+k, 10)
for n in (0..4): print(Trow(n))

T(n, k) = A(n)[k] where A(n) is the Jacobi square of the sequence s(j) = n + j, j >= 0.

A179488 G.f.: A(x) satisfies A(x) = x/(1 - (1-2x)*A( x/(1-2x) )).

Original entry on oeis.org

1, 1, 2, 7, 32, 172, 1052, 7177, 53792, 437992, 3841772, 36060262, 360234512, 3812425912, 42576007352, 500022862357, 6157034292032, 79278216024592, 1064888929532492, 14890014669234922, 216315676347260912

Offset: 1

Paul D. Hanna, Aug 13 2010

nonn

			G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 172*x^6 + ...
A(x) = x + x*A(x) + x*A(x)*A(x/(1-2x)) + x*A(x)*A(x/(1-2x))*A(x/(1-4x)) + x*A(x)*A(x/(1-2x))*A(x/(1-4x))*A(x/(1-6x)) + ...

Cf. variants: A074664, A179489.

{a(n)=local(A=x+x^2);for(i=1,n,A=x/(1-(1-2*x)*subst(A,x,x/(1-2*x+x^2*O(x^n)))));polcoeff(A,n)}

G.f.: x/(1 - (1-2*x)*x/(1-2*x - (1-4*x)*x/(1-4*x - (1-6*x)*x/(1-6*x - (1-8*x)*x/(1-8*x - ... (continued fraction).
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term of M^(n-1), M = an infinite square production matrix as follows (with the odd integers as the main diagonal):
  1, 1, 0, 0, 0, ...
  1, 3, 1, 0, 0, ...
  1, 1, 5, 1, 0, ...
  1, 1, 1, 7, 1, ...
  1, 1, 1, 1, 9, ...
  ... (End)
G.f.: 2/E(0) where E(k) = 1 + 1/(1 + 2*x/(1 - 2*(2*k+3)*x/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+1)/( 1 - x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 22 2013
G.f.: 1/x - Q(0)/x, where Q(k) = 1 - x/(1 - (2*k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013

A179489 G.f.: A(x) = x/(1 - (1-3x)*A( x/(1-3x) )).

Original entry on oeis.org

1, 1, 2, 8, 44, 288, 2172, 18516, 175352, 1819868, 20491844, 248417128, 3221797252, 44464876996, 650076797232, 10028658649668, 162695157490644, 2767333692834768, 49221196196394252, 913310582666986596

Offset: 1

Paul D. Hanna, Aug 13 2010

nonn

			G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 44*x^5 + 288*x^6 + ...
A(x) = x + x*A(x) + x*A(x)*A(x/(1-3x)) + x*A(x)*A(x/(1-3x))*A(x/(1-6x)) + x*A(x)*A(x/(1-3x))*A(x/(1-6x))*A(x/(1-9x)) +...

Cf. variants: A074664, A179488.

{a(n)=local(A=x+x^2);for(i=1,n,A=x/(1-(1-3*x)*subst(A,x,x/(1-3*x+x^2*O(x^n)))));polcoeff(A,n)}

G.f.: A(x) = x/(1 - (1-3x)*x/(1-3x - (1-6x)*x/(1-6x - (1-9x)*x/(1-9x - (1-12x)*x/(1-12x - ... (continued fraction).
From Gary W. Adamson, Jul 22 2011: (Start)
a(n) = the upper left term in M^(n-1), M = an infinite square production matrix with the series 3*n-2 as the main diagonal:
  1, 1, 0, 0, 0, ...
  1, 4, 1, 0, 0, ...
  1, 1, 7, 1, 0, ...
  1, 1, 1,10, 0, ...
  ... (End)

A273396 Indecomposable collections of multisets with a total of n objects having entries {1,2,...,k} for some k<=n or INVERTi transform of A255906.

Original entry on oeis.org

0, 1, 3, 9, 39, 201, 1227, 8305, 61383, 487761, 4131819, 37072361, 350644047, 3482957945, 36220558835, 393329507169, 4450157382383, 52354044069009, 639307054297779, 8090092395577625, 105935581968131399, 1433456549698679385, 20018656224312123051

Offset: 0

Mike Zabrocki, May 21 2016

nonn

A multiset partition of a multiset S is a set of nonempty multisets whose union is S. The total number of multisets of size n and whose entries have all the values in {1,2,...,k} for some k<=n is given by sequence A255906. A multiset partition is decomposable if there exists a value 1<=dd. A multiset partition is called indecomposable otherwise.

			a(3) = 9 because there are 16 multiset partitions, 9 of them are indecomposable ({{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{1,2}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}, {{2},{1,3}}, {{1,2,3}}) and 7 are decomposable ({{1},{1},{2}}, {{1},{2},{2}}, {{1},{2,2}}, {{2},{1,1}}, {{1},{2},{3}}, {{1},{2,3}}, {{3},{1,2}}).

P. A. MacMahon, Combinatory Analysis, vol 1, Cambridge, 1915.

Alois P. Heinz, Table of n, a(n) for n = 0..300
R. Orellana, M. Zabrocki, Symmetric group characters as symmetric functions, arXiv:1605.06672 [math.CO], 2016; or extended abstract, arXiv:1510.00438 [math.CO], 2015.

Cf. A074664, A003319.

INVERTi transform of A255906.

A127745 Counts Bell numbers (except for Catalans) associated with the partition number [n].

Original entry on oeis.org

0, 0, 0, 1, 8, 50, 294, 1717, 10194, 62284, 394346, 2597266, 17827166, 127575414, 951411752, 7386583917, 59623674472, 499648882838, 4340548090590, 39033489125836, 362871600781796, 3482858492844510, 34471940635650958, 351444263328831458

Offset: 1

Alford Arnold, Feb 25 2007

A074664 counts the Bell Numbers associated with the partition number [n]. A000108 counts the corresponding Catalan numbers and here we count the remaining Bell numbers associated with the partition number [n].

			There are 15 Bell objects when n = 4, 14 are also Catalans so a(4) = 1.
There are 52 Bell objects when n = 5, 42 are also Catalans; we know that 5 = 4+1 = 1+4 which accounts for two of the non-Catalan Bells so, a(5) = 52 - 42 - 2 = 8.

Cf. A000041, A000108, A000110, A001399, A016098, A035300, A127743, A074664.

a(n) = A074664(n) - A000108(n-1)

A173050 Triangle, read by rows, given by [0,1,1,1,1,1,1,1,...] DELTA [1,0,1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 10, 6, 1, 0, 14, 36, 31, 10, 1, 0, 42, 135, 156, 77, 15, 1, 0, 132, 518, 771, 534, 169, 21, 1, 0, 429, 2015, 3745, 3451, 1610, 345, 28, 1, 0, 1430, 7906, 17897, 21094, 13569, 4537, 676, 36, 1, 0, 4862, 31195, 84278, 123203, 103986

Offset: 0

Philippe Deléham, Feb 08 2010

			Triangle begins:
  1
  0,  1
  0,  1,  1
  0,  2,  3,  1
  0,  5, 10,  6,  1
  0, 14, 36, 31, 10, 1
  ...

Cf. A000108, A074664, A084938.

Sum_{k=0..n} T(n,k) = A074664(n+1).

Corrected by Philippe Deléham, Feb 08 2010

A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - xy - x^2y*f(x, y+1)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 2, 6, 6, 1, 0, 5, 16, 20, 10, 1, 0, 15, 51, 71, 50, 15, 1, 0, 52, 186, 281, 231, 105, 21, 1, 0, 203, 759, 1223, 1114, 616, 196, 28, 1, 0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1, 0, 4140, 16655, 29634, 31011, 21187, 9780

Offset: 1

Michael Somos, Sep 27 2022

Row sums are A000110.

			Triangle starts:
  1,
  0,   1,
  0,   1,    1,
  0,   1,    3,    1,
  0,   2,    6,    6,    1,
  0,   5,   16,   20,   10,    1,
  0,  15,   51,   71,   50,   15,    1,
  0,  52,  186,  281,  231,  105,   21,  1,
  0, 203,  759, 1223, 1114,  616,  196,  28,  1,
  0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1,
  ...

Cf. A000110, A049347, A074664.

T[ n_, k_] := If[n < 0, 0, Coefficient[SeriesCoefficient[ Nest[ 1/(1 - x*y - x^2*y*(#/.y -> y+1))&, 1 + O[x], Ceiling[n/2]], {x, 0, n}], y, k]];

{T(n, k) = if(n < 0, 0, f = 1 + O(x); forstep(i=1, n, 2, f = 1/(1 - x*y - x^2*y*subst(f, y, y+1))); polcoef(polcoef(f, n), k))};

f(x, -1) = 1/(1 + x + x^2).

x + x^2*f(x, 2) = 1 - 1/f(x, 1) is g.f. for A074664.

cp's OEIS Frontend

A205543 Logarithmic derivative of the Bell numbers (A000110).

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A130167 Another version of triangle in A127743.

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A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, ....

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A179488 G.f.: A(x) satisfies A(x) = x/(1 - (1-2x)*A( x/(1-2x) )).

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A179489 G.f.: A(x) = x/(1 - (1-3x)*A( x/(1-3x) )).

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A273396 Indecomposable collections of multisets with a total of n objects having entries {1,2,...,k} for some k<=n or INVERTi transform of A255906.

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A127745 Counts Bell numbers (except for Catalans) associated with the partition number [n].

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A173050 Triangle, read by rows, given by [0,1,1,1,1,1,1,1,...] DELTA [1,0,1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.

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A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - x*y - x^2*y*f(x, y+1)).

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A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - xy - x^2y*f(x, y+1)).