A074664 -id:A074664 - cp's OEIS frontend

A122372 Dimension of 8-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 8 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).

1, 7, 55, 438, 3498, 27962, 223604, 1788406, 14305102, 114429193, 915366442, 7322521512, 58577537621, 468602617723, 3748697751384, 29988696932490, 239903055854075, 1919175464438065, 15353030007717639, 122821355074655309

Offset: 0

Mike Zabrocki, Aug 30 2006

nonn

			A122371 a(1) = 7 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7, x7-x8 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6+d_x7.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (28,-316,1845,-5925,10190,-8249,2119).

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122371.

coeffs(convert(series((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/ (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7), q,20),`+`)-O(q^20),q);

n = 8; gf = Sum[n!/(n-d)! q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}]/ Sum[q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}] + O[q]^20;
CoefficientList[gf, q] (* Jean-François Alcover, Dec 03 2018 *)

G.f.: (1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/ (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7) more generally, sum( n!/(n-d)!*q^d/prod((1-r*q),r=1..d), d=0..n) / sum( q^d/prod((1-r*q), r=1..d), d=0..n) where n=8.

A124293 Number of free generators of degree n of symmetric polynomials in 5-noncommuting variables.

Original entry on oeis.org

1, 1, 2, 6, 22, 91, 406, 1896, 9093, 44279, 217500, 1073657, 5314870, 26352107, 130778039, 649352929, 3225196431, 16021584848, 79597062632, 395469296912, 1964908443531, 9762920818182, 48508934285620, 241027326818991, 1197601448443963, 5950578465799856

Offset: 1

Mike Zabrocki, Oct 24 2006

nonn

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=5

Alois P. Heinz, Table of n, a(n) for n = 1..500
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005.
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, Canad. J. Math. 60 (2008), no. 2, 266-296.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (10,-32,37,-11).

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124294, A124295.

I:=[1,1,2,6]; [n le 4 select I[n] else 10*Self(n-1)-32*Self(n-2)+37*Self(n-3)-11*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 09 2016

a:= n-> (Matrix([[6,2,1,1]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [10, -32, 37, -11][i] else 0 fi)^(n-1))[1,4]: seq(a(n), n=1..30); # Alois P. Heinz, Sep 05 2008

LinearRecurrence[{10, -32, 37, -11}, {1, 1, 2, 6}, 30] (* Jean-François Alcover, Jan 08 2016 *)

O.g.f. (1-9q+24q^2-19q^3)/(1-10q+32q^2-37q^3+11q^4) = (1 - 1/(sum_{k=0}^5 q^k/(prod_{i=1}^k (1-i*q))))/q a(n) = add( A055105(n,k), k=1..5) = add(A055106(n,k),k=1..4)

A086329 Triangle T(n,k) read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 11, 1, 0, 1, 16, 48, 26, 1, 0, 1, 25, 140, 202, 57, 1, 0, 1, 36, 325, 916, 747, 120, 1, 0, 1, 49, 651, 3045, 5071, 2559, 247, 1, 0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1, 0, 1, 81, 1968, 19404, 84456, 159736, 117962, 26520, 1013, 1

Offset: 0

Philippe Deléham, Aug 30 2003, Jun 12 2007

See A087903 for another version (transposed). - Philippe Deléham, Jun 13 2004

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  4,    1;
  0, 1,  9,   11,    1;
  0, 1, 16,   48,   26,     1;
  0, 1, 25,  140,  202,    57,     1;
  0, 1, 36,  325,  916,   747,   120,    1;
  0, 1, 49,  651, 3045,  5071,  2559,  247,   1;
  0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000290, A000295, A074664, A084938, A086211, A171151.

T[n_, k_]:= T[n, k]= If[n==0, 1, StirlingS2[n, k] + Sum[(k-m-1)*T[n-j-1, k- m]*StirlingS2[j, m], {m,0,k-1}, {j,0,n-2}]];
A086329[n_, k_]:= T[n,n-k+1];
Table[A086329[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 21 2022 *)

@CachedFunction
def T(n,k): # T=A087903
    if (n==0): return 1
    else: return stirling_number2(n, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )
def A086329(n,k): return T(n, n-k+1)
flatten([[A086329(n, k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Jun 21 2022

Sum_{k=0..n} T(n, k) = A086211(n, 0).
T(n, 1) = 1, n > 0.
T(n, 2) = (n-1)^2, n > 0.
T(k+1, k) = 2^(k+1) - k - 2 = A000295(k+1).
Sum_{k=0..n} T(n, k) = A074664(n+1). - Philippe Deléham, Jun 13 2004
Sum_{k=0..n} T(n,k)*2^k = A171151(n). - Philippe Deléham, Dec 05 2009
T(n, k) = A087903(n, n-k+1). - G. C. Greubel, Jun 21 2022

A095989 INVERTi transform applied to the ordered Bell numbers.

Original entry on oeis.org

1, 2, 8, 48, 368, 3376, 35824, 430512, 5773936, 85482032, 1384936688, 24380214960, 463522810736, 9468048895792, 206831329017328, 4812581925690288, 118843801816575088, 3104590192664327216, 85544737118902122224, 2479681575659312797872, 75434373300016828382576

Offset: 1

Mike Zabrocki, Jul 18 2004

nonn

A set composition of n is an ordered sequence [S_1, S_2, ..., S_k] where S_i subset of [n] all disjoint and the union of all S_i is [n] (see A000670). A set composition is atomic if S_1 union ... union S_j does not equal [r] for any r < n and j < k. a(n) is the number of atomic set compositions.

A preference function of n is a word of length n where all the numbers 1 through k occur at least once for some k <= n (see A000670). A preference function is atomic if no strict leading subword contains the only occurrences in the word of the letters 1 through j < k. a(n) is the number of atomic preference functions.

			Atomic set compositions a(1)=1: [{1}]; a(2)=2: [{12}], [{2},{1}]; a(3)=8: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Atomic preference functions a(1) = 1: 1; a(2)=2: 11, 21; a(3)=8: 111, 212, 221, 211, 121, 312, 231, 321.

Alois P. Heinz, Table of n, a(n) for n = 1..400
Hugo Mlodecki, Decompositions of packed words and self duality of Word Quasisymmetric Functions, arXiv:2205.13949 [math.CO], 2022. See Table 2 p. 8.

Cf. A000670, A074664, A095993.

A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n,k)*A000670(n-k),k=1..n); fi; end: add(A000670(k)*x^k,k=0..20): series(1-1/%,x,21): [seq(coeff(%,x,i),i=1..20)];

max = 20; Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; s = 1 - 1/Sum[ Fubini[k, 1] q^k, {k, 0, max}] + O[q]^max; CoefficientList[s/q, q] (* Jean-François Alcover, Mar 31 2016 *)

G.f.: 1 - 1/Sum_{k>=0} A000670(k)*q^k.
G.f.: x/(1-2x/(1-2x/(1-4x/(1-3x/(1-6x/(1-4x/(1-8x/(1-5x/(1-...(continued fraction). - Philippe Deléham, Nov 22 2011
G.f.: (1-T(0))/x, where T(k) = 1 - x*(k+1)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 29 2013
Let A(x) be the g.f. A095989, B(x) the g.f. A000670, then A(x) = (1 - 1/B(x))/x. - Sergei N. Gladkovskii, Nov 29 2013
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019

A098742 Number of indecomposable set partitions of [1..n] without singletons.

Original entry on oeis.org

0, 0, 1, 1, 3, 9, 33, 135, 609, 2985, 15747, 88761, 531561, 3366567, 22462017, 157363329, 1154257683, 8841865833, 70573741857, 585753925047, 5046128460801, 45044554041897, 416005748766771, 3969321053484921, 39077616720410409

Offset: 0

Don Knuth, Oct 01 2004

After a(3) = 1, always divisible by 3. a(n) is 3 times a prime (A001748) when n = 5, 6, 11, 14, 15, 16, 19. - Jonathan Vos Post, Jun 22 2008

			a(5)=9 because of the set partitions 135|24, 134|25, 125|34, 145|23, 15|234, 13|245, 124|35, 12345, 14|235. [Puttenham missed the last of these.]

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26.
George Puttenham, The Arte of English Poesie (1589), page 72, can be said to have stated the problem; but he omitted one case for n=5 and 22 cases for n=6, so he must have had other constraints in mind!

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000296, A074664, A001748, A008585.

F:= proc(n) option remember; convert(series(1 -1/add(coeff(series(exp(exp(x)-1), x,n+1), x,j)*j!*x^j, j=0..n), x,n+1), polynom) end: a:= n-> coeff(series(x*F(n)/(1+x-F(n)), x,n+1), x,n): seq(a(n), n=0..24); # Alois P. Heinz, Sep 05 2008

f[n_] := f[n] = Normal[ Series[ 1-1/Sum[ SeriesCoefficient[ Series[ Exp[Exp[x] - 1], {x, 0, n + 1}], {x, 0, j}]*j!*x^j, {j, 0, n}], {x, 0, n + 1}]]; a[0] = 0; a[n_] := SeriesCoefficient[ Series[ x*(f[n]/(1 + x - f[n])), {x, 0, n + 1}], {x, 0, n}]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, translated from Maple *)

def A098742_list(dim):
    T = matrix(ZZ,dim,dim)
    for n in range(dim): T[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n,k] = T[n-1,k-1]+(k+1)*T[n-1,k]+(k+2)*T[n-1,k+1]
    return [0,0]+list(T.column(0))
A098742_list(23) # - Peter Luschny, Sep 20 2012

If f(z) is the generating function for A074664, then a(z)=zf(z)/(1+z-f(z)).
Also, if g(z) is the generating function for A000296, then a(z) = 1-1/g(z).
O.g.f.: x^2/(1-x-2*x^2/(1-2*x-3*x^2/(1-3*x-4*x^2/(1-... -n*x-(n+1)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
From Sergei N. Gladkovskii, Sep 20 2012, Nov 04 2012, Feb 04 2013, Feb 23 2013, Apr 18 2013, May 12 2013: (Start) Continued fractions:
G.f.: -x + 2*x/E(0) where E(k)= 1 + 1/(1 + 2*x/(1 - 2*(k+2)*x/E(k+1))).
G.f.: 1 - x*U(0,1/x) where U(k,x)= x - k - (k+1)/U(k+1,x).
G.f.: (1+x)*x/G(0) - x where G(k) = 1 + x - x*(k+1)/(1 - x/G(k+1)).
G.f.: x/Q(0) - x where Q(k)= 1 + x/(x*k-x-1)/Q(k+1).
G.f.: 1 - Q(0) where Q(k)= 1 + x - x/(1 - x*(k+1)/Q(k+1)).
G.f.: 1-x-1/Q(0) where Q(k)= 1 + x/(1 - x - x*(k+1)/(x + 1/Q(k+1))). (End)

More terms from Vladeta Jovovic, Oct 21 2004

A124294 Number of free generators of degree n of symmetric polynomials in 6-noncommuting variables.

Original entry on oeis.org

1, 1, 2, 6, 22, 92, 425, 2119, 11184, 61499, 347980, 2007643, 11734604, 69181578, 410179429, 2441025998, 14562284120, 87012222100, 520458020949, 3115224471290, 18654716694895, 111741999352603, 669466118302169

Offset: 1

Mike Zabrocki, Oct 24 2006

nonn

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=6

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (15, -81, 192, -189, 53).

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124293, A124295.

LinearRecurrence[{15, -81, 192, -189, 53}, {1, 1, 2, 6, 22}, 23] (* Jean-François Alcover, Dec 04 2018 *)

O.g.f.: (1-14*q+68*q^2-135*q^3+91*q^4)/(1-15*q+81*q^2-192*q^3+189*q^4-53*q^5) = (1 - 1/(sum_{k=0}^6 q^k/(prod_{i=1}^k (1-i*q))))/q a(n) = add( A055105(n,k), k=1..6) = add(A055106(n,k),k=1..5)

A124295 Number of free generators of degree n of symmetric polynomials in 7-noncommuting variables.

Original entry on oeis.org

1, 1, 2, 6, 22, 92, 426, 2145, 11589, 66425, 399682, 2500037, 16115347, 106266473, 712602272, 4837372576, 33128183406, 228308233098, 1580495251012, 10976092266889, 76398165848091, 532614662149795, 3717370694711130

Offset: 1

Mike Zabrocki, Oct 24 2006

nonn

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=7

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (21, -170, 669, -1314, 1157, -309).

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124293, A124294.

LinearRecurrence[{21, -170, 669, -1314, 1157, -309}, {1, 1, 2, 6, 22, 92}, 23] (* Jean-François Alcover, Jan 27 2019 *)

O.g.f.: (1-20*q+151*q^2-535*q^3+881*q^4-531*q^5) / (1-21*q+170*q^2 -669*q^3 +1314*q^4-1157*q^5+309*q^6) = (1 - 1/(Sum_{k=0..7} q^k/(prod_{i=1}^k (1-i*q))))/q.

a(n) = add( A055105(n,k), k=1..7) = add(A055106(n,k), k=1..6).

A127743 Triangular array where T(n,k) is the number of set partitions of n with k atomic parts.

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Feb 24 2007

Triangular array distributing the Bell numbers (A000110). The value associated with each partition is the product of A074664(k) for each part of size k, times the number of compositions associated with the partition (A048996 & A072881). The value for T(n,k) is the total of these values for each partition of n into k parts.
Calculating the appropriate weights can be done by "working backward". Suppose for example we know the weights for 1 through 6 and desire the weight for the partitions of seven: Substitute the weights for each partition value and multiply. For example, 7 = 4+3 so f([4,3]) = 6*2 = 12; adjusting for the number of permutations of [4,3] we now have 2*12 = 24. Continuing in this manner for each partition of seven and summing to 451 we now know all of the values except that associated with the partition [7] which must be 877 - 451 = 426.
From Mike Zabrocki: (Start)
Every set partition can be uniquely split into "atomic" set partitions or is itself already atomic.
  {{1},{2},{3}} = {{1}}|{{1}}|{{1}}
  {{1},{23}} = {{1}}|{{12}}
  {{12},{3}} = {{12}}|{{1}}
  {{13},{2}} is already atomic
  {{123}} is already atomic
where this operation | is defined as {A1,...,Ar}|{B1,...,Bs} = {A1,...,Ar,B1+n,...,Bs+n}
where Bi+n = {bi1+n,bi2+n,...,bik+n} if Bi = {bi1,bi2,...,bik} and n = |A1|+|A2|+...+|Ar|. (End)
Subtriangle (n >= 1 and 1 <= k <= n) of triangle given by [0,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 03 2007
From Peter Bala, Aug 05 2014: (Start)
Let B(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + ... denote the o.g.f. for the Bell numbers A000110. Let f(x) = (B(x) - 1)/(x*B(x)) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + ..., the o.g.f. for the first column of this array. Then this array appears to be the Riordan array (f(x), x*f(x)).
If true, this gives the o.g.f. of the array as (B(x) - 1)/( x*(t + (1 - t)*B(x)) ) = 1 + (1 + t)*x + (2 + 2*t + t^2)*x^2 + ... and also the hockey-stick recurrence: T(n+1,k+1) = T(n,k) + T(n-1,k) + 2*T(n-2,k) + 6*T(n-3,k) + 22*T(n-4,k) + ..., n,k >= 1. (End)

			The partitions of 4 are
  4 31 22 211 1111
and the products are
  1*6 2*2 1*1 3*1 1*1
therefore row 4 of the table is
  6 5 3 1.
From _Philippe Deléham_, Aug 03 2007: (Start)
Triangle begins:
     1;
     1,    1;
     2,    2,   1;
     6,    5,   3,   1;
    22,   16,   9,   4,  1;
    92,   60,  31,  14,  5,  1;
   426,  252, 120,  52, 20,  6, 1;
  2146, 1160, 510, 209, 80, 27, 7, 1; ...
Triangle [0,1,1,2,1,3,1,4,1,...] DELTA [1,0,0,0,0,0,...] 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;
  0,  426,  252, 120,  52, 20,  6, 1;
  0, 2146, 1160, 510, 209, 80, 27, 7, 1; ...
(End)

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A000041, A000110 (row sums), A074664 (1st column), A048996, A072881, A036043, A036042, A084938.

T[n_, m_] := T[n, m] = Sum[Sum[T[k+i, k]*Binomial[n-m-k-1, n-m-k-i], {i, 1, n-m-k}]*Binomial[k+m-1, k], {k, 1, n-m}] + Binomial[n-1, n-m]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Mar 23 2015, after Vladimir Kruchinin *)

T(n,m):=sum((sum(T(k+i,k)*binomial(n-m-k-1,n-m-k-i),i,1,n-m-k))*binomial(k+m-1,k),k,1,n-m)+binomial(n-1,n-m); /* Vladimir Kruchinin, Mar 21 2015 */

{T(n,m) = sum(k=1,n-m, (sum(i=1, n-m-k, (T(k+i, k)*binomial(n-m-k-1, n-m-k-i))*binomial(k+m-1, k)))) + binomial(n-1, n-m)};
for(n=1, 10, for(m=1, n, print1(T(n,m), ", "))) \\ G. C. Greubel, Dec 06 2018

T(n, m) = Sum_{k=1..n-m}( Sum_{i=1..n-m-k}(T(k+i, k)*C(n-m-k-1, n-m-k-i))*C(k+m-1, k) ) + C(n-1, n-m). - Vladimir Kruchinin, Mar 21 2015

Edited by Franklin T. Adams-Watters, Jan 25 2010

A086211 Triangle related to Bell numbers; T(n,k) read by rows, n>=0, 0<=k<=n: T(n,k) = k*T(n-1,k) + Sum(0<=j, T(n-1,k-1+j)); T(0,0)=1, T(0,k)=0 if k>0.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 9, 6, 1, 22, 31, 28, 10, 1, 92, 123, 126, 69, 15, 1, 426, 549, 586, 418, 145, 21, 1, 2146, 2695, 2892, 2425, 1165, 272, 28, 1, 11624, 14319, 15262, 14058, 8551, 2826, 469, 36, 1

Offset: 0

Philippe Deléham, Aug 27 2003, Jun 16 2007

With offset 1 for k, T(n,k) is the number of indecomposable set partitions of [n+2] in which 1 is in the k-th block when the blocks are arranged in order of increasing largest entry. For example, T(2,2)=3 counts 2/134, 23/14, 3/124; see Link. - David Callan, Aug 30 2014

			Triangle begins:
1;
1, 1;
2, 3, 1;
6, 9, 6, 1;
22, 31, 28, 10, 1;
92, 123, 126, 69, 15, 1;
426, 549, 586, 418, 145, 21, 1;
2146, 2695, 2892, 2425, 1165, 272, 28, 1;
11624, 14319, 15262, 14058, 8551, 2826, 469, 36, 1 ;
...

David Callan, A combinatorial interpretation for this sequence
Chunyan Yan, Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

Cf. A000110.

Sum(k=0..n, A000110(k)*T(n-k,0)) = A000110(n+1).

Sum_{k=0..n} T(n, k) = A074664(n+2). - Philippe Deléham, May 10 2005

A152431 Eigentriangle, row sums = A000110, the Bell numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 2, 2, 5, 22, 6, 4, 5, 15, 92, 22, 12, 10, 15, 52, 426, 92, 44, 30, 30, 52, 203, 2146, 426, 184, 110, 90, 104, 203, 877, 11624, 2146, 852, 460, 330, 312, 406, 877, 4140, 67146, 11624, 4292, 2130, 1380, 1144, 1218, 1754, 4140, 21147

Offset: 1

Gary W. Adamson, Dec 04 2008

Row sums = the Bell numbers, A000110, starting with offset 1: (1, 2, 5, 15, 52,...).
Left border = A074664 (1, 1, 2, 6, 22 92, 426,...), the INVERTi transform of (1, 2, 5, 15, 52,...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
2, 1, 2;
6, 2, 2, 5;
22, 6, 4, 5, 15;
92, 22, 12, 10, 15, 52;
426, 92, 44, 30, 30, 52, 203;
2146, 426, 184, 110, 90, 104, 203, 877;
11624, 2146, 852, 460, 330, 312, 406, 877, 4140;
67146, 11624, 4292, 2130, 1380, 1144, 1218, 1754, 4140, 21147;
411142, 67146, 23248, 10730, 6390, 4784, 4466, 5262, 8280, 21147, 115975;
...
Row 4 = (6, 2, 2, 5) = termwise products of (6, 2, 1, 1) and (1, 1, 2, 5).

A000110, A074664

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A074664 in every column: (1, 1, 2, 6, 22, 92, 426,...). Q = a matrix with the Bell numbers (1, 1, 2, 5, 15,...) as the main diagonal and the rest zeros.

cp's OEIS Frontend

A122372 Dimension of 8-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 8 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).

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A124293 Number of free generators of degree n of symmetric polynomials in 5-noncommuting variables.

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A086329 Triangle T(n,k) read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] where DELTA is the operator defined in A084938.

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A095989 INVERTi transform applied to the ordered Bell numbers.

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A098742 Number of indecomposable set partitions of [1..n] without singletons.

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A124294 Number of free generators of degree n of symmetric polynomials in 6-noncommuting variables.

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A124295 Number of free generators of degree n of symmetric polynomials in 7-noncommuting variables.

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