A005493 -id:A005493 - cp's OEIS frontend

A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

1, 1, 1, 2, 3, 2, 5, 10, 12, 6, 15, 37, 62, 60, 24, 52, 151, 320, 450, 360, 120, 203, 674, 1712, 3120, 3720, 2520, 720, 877, 3263, 9604, 21336, 33600, 34440, 20160, 5040, 4140, 17007, 56674, 147756, 287784, 394800, 352800, 181440, 40320, 21147, 94828

Offset: 0

Vladeta Jovovic, Jan 02 2001

The transpose of this lower unitriangular array is the U factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20L%20factor%20is%20A049020%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The L factor is A049020 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
  [0] [ 1]
  [1] [ 1,    1]
  [2] [ 2,    3,    2]
  [3] [ 5,   10,   12,    6]
  [4] [15,   37,   62,   60,   24]
  [5] [52,  151,  320,  450,  360,  120]
  [6] [203, 674, 1712, 3120, 3720, 2520, 720]
  ...;
E.g.f. for T(n, 2) = (exp(x)-1)^2*(exp(exp(x)-1)) = x^2 + 2*x^3 + 31/12*x^4 + 8/3*x^5 + 107/45*x^6 + 343/180*x^7 + 28337/20160*x^8 + 349/360*x^9 + ...;
E.g.f. for T(n, 3) = (exp(x)-1)^3*(exp(exp(x)-1)) = x^3 + 5/2*x^4 + 15/4*x^5 + 13/3*x^6 + 127/30*x^7 + 1759/480*x^8 + 34961/12096*x^9 + ...

Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).

Cf. A049020, A001861, A046716.

T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
      `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
    end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # Peter Bala, Oct 15 2023

E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).
n-th row is M^n*[1,0,0,0,...], where M is a tridiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) in the main and subdiagonals; and the rest zeros. - Gary W. Adamson, Jun 23 2011
T(n, k) = k!*A049020(n, k). - R. J. Mathar, May 17 2016
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - Peter Luschny, Dec 06 2023

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

A125098 Values occurring in A123895.

Original entry on oeis.org

0, 1, 10, 11, 12, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1023, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1123, 1200, 1201, 1202, 1203, 1210, 1211, 1212, 1213, 1220, 1221, 1222

Offset: 0

Franklin T. Adams-Watters, Nov 20 2006

Also, numbers n such that A123895(n) = n. Each digit is either 0, has occurred before, or is one more than the largest preceding digit. The number of k-digit numbers in this sequence is A005493(k-1) for 2<=k<=9. The initial 0 prevents a match for k=1; later values fall short because there are no digits beyond 9 available in decimal.

R. J. Mathar, Table of n, a(n) for n = 0..876

Cf. A123895, A005493, A123902.

bag := {} ;
for i from 0 to 99999 do
    bag := bag union {A123895(i)} ;
end do:
sort(convert(bag,list)) ; # R. J. Mathar, Dec 10 2015

f[n_] := Block[{d = Prepend[IntegerDigits@ n, 0], a, b, w}, b = DeleteDuplicates@ d; a = Range[0, Length@ b]; w = FromDigits@ Flatten[Part[a, FirstPosition[b, #]] & /@ d]; w]; Union@ Table[f@n, {n, 0, 10^4}] (* Michael De Vlieger, Dec 09 2015, Version 10 *)

A191099 5th differences of Bell numbers.

Original entry on oeis.org

11, 52, 255, 1335, 7432, 43833, 272947, 1788850, 12303997, 88586135, 666047210, 5218287687, 42518759887, 359651145332, 3152929344235, 28603584325827, 268159523175744, 2594608337127709, 25878365376280647, 265770087291261082, 2807571511844891521

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 25 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A005493, A011965, A011966, A011967, A106436.

Mathematica
```
Differences[BellB[Range[0, 50]], 5]
```

A225591 a(n) = B(n+3) - 6B(n+2) + 8B(n+1)*B(n+1) - B(n), where the B(i) are Bell numbers (A000110).

Original entry on oeis.org

0, 16, 160, 1686, 21276, 328498, 6149136, 137105016, 3577543452, 107601726030, 3683660206080, 142035221781402, 6113719409724768, 291540411275223912, 15300594717301253800, 878667035554110785662, 54932693182800769213284

Offset: 0

Michel Marcus, Jun 19 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
B. Chern, P. Diaconis, D. M. Kane, R. C. Rhoades, Closed expressions for averages of set partition statistics, 2013.

Cf. A005493, A226506 (see Prop 3.1 (i) in Chern et al. link).

[Bell(n+3)-6*Bell(n+2)+8*Bell(n+1)*Bell(n+1)-Bell(n): n in [0..20]]; // Vincenzo Librandi, Jul 16 2013

Table[BellB[n+3] - 6 BellB[n+2] + 8 BellB[n+1] BellB[n+1] - BellB[n], {n, 0, 20}] (* Vincenzo Librandi, Jul 16 2013 *)
#[[4]]-6#[[3]]+8#[[2]]^2-#[[1]]&/@Partition[BellB[Range[0,20]],4,1] (* Harvey P. Dale, Nov 01 2016 *)

B(n) = if (n<=1, return (1), return (sum(i=0, n-1, binomial(n-1, i)*B(n-1-i))))
a(n) = B(n+3) - 6*B(n+2) + 8*B(n+1)*B(n+1) - B(n)

A226506 a(n) = B(n+2)-3*B(n+1)+B(n), where B(i) are the Bell numbers A000110.

Original entry on oeis.org

0, 0, 2, 12, 62, 320, 1712, 9604, 56674, 351792, 2293862, 15682216, 112179608, 837905016, 6522165834, 52807401908, 443962338894, 3869376656384, 34908008426360, 325530083655692, 3133830448212442, 31106728455899128, 318009567467999574, 3344865730200667832, 36161434396223563504

Offset: 0

N. J. A. Sloane, Jun 10 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
B. Chern, P. Diaconis, D. M. Kane, and R. C. Rhoades, Closed expressions for averages of set partition statistics, arXiv:1304.4309 [math.CO], 2013.

Cf. A000110.

Cf. A005493, A225591 (see Prop 3.1 (i) in Chern et al. link).

[Bell(n+2)-3*Bell(n+1)+Bell(n): n in [0..30]]; // Vincenzo Librandi, Jul 16 2013

Table[BellB[n+2] - 3 BellB[n+1] + BellB[n], {n, 0, 30}] (* Vincenzo Librandi, Jul 16 2013 *)
#[[3]]-3#[[2]]+#[[1]]&/@Partition[BellB[Range[0,30]],3,1] (* Harvey P. Dale, Aug 26 2021 *)

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1

Offset: 0

Peter Luschny, Apr 10 2016

			1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.

Peter Luschny, Extensions of the binomial

Variant: A143494 (the main entry for this triangle).
A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
A049444, A136124, A143491 (matrix inverse).
Cf. A048993, A269951.

A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n,k), k=0..n), n=0..9);

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]

T(n, k) = S2(n+1, k+1) - S2(n, k+1).

A339030 T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 8, 1, 0, 5, 85, 50, 10, 1, 0, 6, 300, 280, 75, 12, 1, 0, 7, 1071, 1540, 525, 105, 14, 1, 0, 8, 3976, 8456, 3570, 840, 140, 16, 1, 0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1

Offset: 0

Peter Luschny, Nov 22 2020

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 2, 1]
3: [0, 3, 6,     1]
4: [0, 4, 24,    8,     1]
5: [0, 5, 85,    50,    10,    1]
6: [0, 6, 300,   280,   75,    12,   1]
7: [0, 7, 1071,  1540,  525,   105,  14,   1]
8: [0, 8, 3976,  8456,  3570,  840,  140,  16,  1]
9: [0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1]
.
T(4,0) = 0  = 0*card({})
T(4,1) = 4  = 4*card({1|2|3|4}).
T(4,2) = 24 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
            + 2*card({12|34, 13|24, 14|23}).
T(4,3) = 8  = 2*card({123|4, 124|3, 134|2, 1|234}).
T(4,4) = 1  = 1*card({1234}).
.
Seen as the projection of a 2-dimensional statistic this is, for n = 6:
[  0   0    0     0     0    0   0]
[  0   0    0     0     0    0   6]
[  0   0    0    45   180   75   0]
[  0   0   20   180    80    0   0]
[  0   0   30    45     0    0   0]
[  0   0   12     0     0    0   0]
[  0   1    0     0     0    0   0]
The row sum projection gives row 6 of this triangle, and the column sum projection gives [0, 1, 62, 270, 260, 75, 6], which appears in a decapitated version as row 5 in A321331.

Cf. A005493 with 1 prepended are the row sums.

Cf. A080510, A262071, A321331.

def A339030Row(n):
    if n == 0: return [1]
    M = matrix(n + 1)
    for k in (1..n):
        for p in SetPartitions(n):
            if p.max_block_size() == k:
                M[k, len(p)] += p.cardinality()
    return [sum(M[k, j] for j in (0..n)) for k in (0..n)]
for n in (0..9): print(A339030Row(n))

A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

Original entry on oeis.org

1, 10, 75, 520, 3556, 24626, 174805, 1279240, 9677151, 75750752, 613656836, 5142797660, 44557627661, 398786697398, 3683575764083, 35084121263136, 344242894197456, 3476490965903174, 36104281709286841, 385257741260565844, 4220537246457019687, 47432055430482106880

Offset: 3

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000292, A000392, A003128, A005493, A049020, A189924, A346843, A346844.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 3), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=3..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^3/3!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[StirlingS2[n, k] Binomial[k, 3], {k, 0, n}], {n, 3, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 3] BellB[n - k], {k, 0, n}], {n, 3, 24}]
Table[(BellB[n+3] - 6*BellB[n+2] + 8*BellB[n+1] - BellB[n])/6, {n, 3, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^3/3!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,3).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,3) * Bell(n-k).
a(n) = (Bell(n+3) - 6*Bell(n+2) + 8*Bell(n+1) - Bell(n))/6. - Vaclav Kotesovec, Aug 06 2021
a(n) ~ exp(-1 - n + n/LambertW(n)) * (n - LambertW(n))^3 * n^n / (6 * sqrt(1 + LambertW(n)) * LambertW(n)^(n+3)). - Vaclav Kotesovec, Jun 28 2022

A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.

Original entry on oeis.org

1, 15, 155, 1400, 11991, 101031, 853315, 7300260, 63641006, 567304452, 5181338526, 48538121450, 466611951261, 4603782469653, 46613101232933, 484188586821376, 5157850655391981, 56321812548867229, 630125374420189131, 7219368394888423554, 84658119388335562972

Offset: 4

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000332, A000453, A003128, A005493, A049020, A346842, A346844.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 4), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=4..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^4/4!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
Table[Sum[StirlingS2[n, k] Binomial[k, 4], {k, 0, n}], {n, 4, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 4] BellB[n - k], {k, 0, n}], {n, 4, 24}]
Table[(BellB[n] - 24*BellB[n+1] + 29*BellB[n+2] - 10*BellB[n+3] + BellB[n+4])/24, {n, 4, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)
With[{nn=30},Drop[CoefficientList[Series[(Exp[Exp[x]-1](Exp[x]-1)^4)/4!,{x,0,nn}],x] Range[0,nn]!,4]] (* Harvey P. Dale, Oct 03 2024 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^4/4!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,4).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,4) * Bell(n-k).
a(n) = (Bell(n) - 24*Bell(n+1) + 29*Bell(n+2) - 10*Bell(n+3) + Bell(n+4))/24. - Vaclav Kotesovec, Aug 06 2021

cp's OEIS Frontend

A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A125098 Values occurring in A123895.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A191099 5th differences of Bell numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A225591 a(n) = B(n+3) - 6*B(n+2) + 8*B(n+1)*B(n+1) - B(n), where the B(i) are Bell numbers (A000110).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A226506 a(n) = B(n+2)-3*B(n+1)+B(n), where B(i) are the Bell numbers A000110.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A339030 T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.

Views

Author

Keywords

Examples

Crossrefs

Programs

SageMath

A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

Views

A225591 a(n) = B(n+3) - 6B(n+2) + 8B(n+1)*B(n+1) - B(n), where the B(i) are Bell numbers (A000110).

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.