A292870 -id:A292870 - cp's OEIS frontend

A014322 Convolution of Bell numbers with themselves.

1, 2, 5, 14, 44, 154, 595, 2518, 11591, 57672, 308368, 1762500, 10716321, 69011130, 468856113, 3348695194, 25064539520, 196052415230, 1598543907843, 13556379105766, 119332020447219, 1088376385244908, 10268343703117892, 100063762955374568, 1005822726810785809

Offset: 0

N. J. A. Sloane

nonn

Equals row sums of triangle A144155. - Gary W. Adamson, Sep 12 2008

Alois P. Heinz, Table of n, a(n) for n = 0..576
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012

Cf. A000110, A014323, A014325, A144155.

Column k=2 of A292870.

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
[A014322(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

with(combinat):
a:= n-> add(bell(i)*bell(n-i), i=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 13 2014

a[n_]:= Sum[BellB[k]*BellB[n-k], {k,0,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Jan 17 2016 *)

def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
[A014322(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^2, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017

G.f.: ( Sum_{j>=0} A000110(j)*x^j )^2. - G. C. Greubel, Jan 08 2023

A137551 Number of permutations in S_n avoiding {bar 2}413{bar 5} (i.e., every occurrence of 413 is contained in an occurrence of a 24135).

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 144, 525, 2084, 9005, 42288, 215111, 1179738, 6937765, 43504598, 289356385, 2031636826, 14995775647, 115943399636, 936138957225, 7872233481696, 68788474572625, 623323010473012, 5846701373312019, 56677763478164422, 567011396405398185

Offset: 0

Lara Pudwell, Apr 25 2008

nonn

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)
Equals the INVERT transform of the Bell sequence (A000110 with offset 0) [Callan preprint]. - R. J. Mathar, Nov 29 2011

Alois P. Heinz, Table of n, a(n) for n = 0..570
David Callan, The number of bar{2}413bar{5}-avoiding permutations, arXiv:1110.6884 [math.CO], 2011.
Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Row sums of A205574.

Antidiagonal sums of A292870.

read("bVATTER14") ; # http://faculty.valpo.edu/lpudwell/maple/bVATTER14
for n from 1 do f([[2,1],[4,0],[1,0],[3,0],[5,1]], {op(permute(n))} ) ; nops(%) ; print(%) ; od: # R. J. Mathar, May 29 2009
# Another Maple program:
with(combinat):
invtr:= proc(p) local b; b:= proc(n) option remember;
           `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1)) end
        end:
a:= n-> invtr(n-> bell(n))(n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 28 2012

invtr[p_] := Module[{b}, b[n_] := b[n] = If[n<1, 1, Sum[b[n-i]*p[i-1], {i, 1, n+1}]]; b]; a[n_] := invtr[BellB][n-1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

G.f.: ((x^2-4)/(U(0)*(x+1)-x^3+4*x)-1)/(1+x) where U(k)= k*(2*k+3)*x^2 + x - 2 - (2 - x + 2*k*x)*(2 + 3*x + 2*k*x)*(k+1)*x^2/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 28 2012
G.f.: 1/(G(0) - x ) where G(k) =  1 - x/(1 - x*(2*k+1)/(1 - x/(1 - x*(2*k+2)/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 14 2012
G.f.: 1/( G(0) - x ) where G(k) = 1 - x/(1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
G.f.: 1/( Q(0) -x ) where Q(k)= 1 - (k+1)*x - (k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A014323 Three-fold convolution of Bell numbers with themselves.

Original entry on oeis.org

1, 3, 9, 28, 93, 333, 1289, 5394, 24366, 118526, 618924, 3456942, 20573391, 129951231, 867877107, 6106194478, 45109290477, 348836705235, 2816093142803, 23673989688810, 206794355179656, 1873232870155036, 17565534522745008, 170237112831874188

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..576

Cf. A000110, A014322, A014325.

Column k=3 of A292870.

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
A014323:= func< n | (&+[Bell(j)*A014322(n-j): j in [0..n]]) >;
[A014323(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
A014323[n_]:= Sum[BellB[j]*A014322[n-j], {j,0,n}];
Table[A014323[n], {n,0,40}] (* G. C. Greubel, Jan 08 2023 *)

def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
def A014323(n): return sum(bell_number(j)*A014322(n-j) for j in range(n+1))
[A014323(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^3, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017
From G. C. Greubel, Jan 08 2023: (Start)
a(n) = Sum_{j=0..n} A000110(j)*A014322(n-j).
G.f.: ( Sum_{j>=0} A000110(j)*x^j )^3. (End)

A014325 Four-fold convolution of Bell numbers with themselves.

Original entry on oeis.org

1, 4, 14, 48, 169, 624, 2442, 10188, 45452, 217100, 1109914, 6064584, 35330715, 218788432, 1435302930, 9940062428, 72422364227, 553338786504, 4420324121772, 36820875272488, 319053830821880, 2869645346679368, 26739383194844404, 257682847299543248

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..575

Cf. A000110, A014322, A014323.

Column k=4 of A292870.

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
A014325:= func< n | (&+[A014322(j)*A014322(n-j): j in [0..n]]) >;
[A014325(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
A014325[n_]:= Sum[A014322[j]*A014322[n-j], {j,0,n}];
Table[A014325[n], {n,0,40}] (* G. C. Greubel, Jan 08 2023 *)

def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
def A014325(n): return sum(A014322(j)*A014322(n-j) for j in range(n+1))
[A014325(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^4, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017

G.f.: ( Sum_{j>=0} A000110(j)*x^j )^4. - G. C. Greubel, Jan 08 2023

A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 15, 14, 9, 4, 1, 0, 52, 44, 28, 14, 5, 1, 0, 203, 154, 93, 48, 20, 6, 1, 0, 877, 595, 333, 169, 75, 27, 7, 1, 0, 4140, 2518, 1289, 624, 280, 110, 35, 8, 1, 0, 21147, 11591, 5394, 2442, 1071, 435, 154, 44, 9, 1

Offset: 0

Philippe Deléham, Jan 29 2012

Bell convolution triangle ; g.f. for column k : (x*B(x))^k with  B(x) g.f. for A000110 (Bell numbers).
Riordan array (1, x*B(x)), when B(x) the g.f. of A000110.
Row sums are in A137551.

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,  1;
  0,   5,   5,  3,  1;
  0,  15,  14,  9,  4,  1;
  0,  52,  44, 28, 14,  5, 1;
  0, 203, 154, 93, 48, 20, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. Columns : A000007, A000110, A014322, A014323, A014325 ; Diagonals : A000012, A001477, A000096, A005586.
Another version: A292870.
T(2n,n) gives: A292871.

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-bell(n-1)); # Peter Luschny, Oct 19 2022

Sum_{k=0..n} T(n,k) = A137551(n), n>0.

A292871 a(n) = [x^n] (1/(1 - x - x^2/(1 - 2x - 2x^2/(1 - 3x - 3x^2/(1 - 4x - 4x^2/(1 - ...))))))^n.

Original entry on oeis.org

1, 1, 5, 28, 169, 1071, 7034, 47538, 329249, 2331424, 16856915, 124387286, 936799582, 7204759238, 56634639780, 455560907508, 3755017488657, 31763254337955, 276141607672244, 2470749459597450, 22777862470135279, 216542289861590847, 2123786397875045480, 21490054470340915524, 224275454800219674782

Offset: 0

Ilya Gutkovskiy, Sep 25 2017

nonn

a(n) is the n-th term of the n-fold convolution of Bell numbers with themselves. - Alois P. Heinz, Feb 12 2019

Alois P. Heinz, Table of n, a(n) for n = 0..573

Main diagonal of A292870.

Cf. A000110, A205574.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
     `if`(k=1, add(b(n-j, k)*binomial(n-1, j-1), j=1..n),
     (h-> add(b(j, h)*b(n-j, k-h), j=0..n))(iquo(k,2)))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, May 31 2018

Table[SeriesCoefficient[1/(1 - x + ContinuedFractionK[-k x^2, 1 - (k + 1) x, {k, 1, n}])^n, {x, 0, n}], {n, 0, 24}]

a(n) = A292870(n,n).

a(n) = A205574(2n,n).

cp's OEIS Frontend

A014322 Convolution of Bell numbers with themselves.

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A137551 Number of permutations in S_n avoiding {bar 2}413{bar 5} (i.e., every occurrence of 413 is contained in an occurrence of a 24135).

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A014323 Three-fold convolution of Bell numbers with themselves.

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Magma

Mathematica

SageMath

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A014325 Four-fold convolution of Bell numbers with themselves.

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A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A292871 a(n) = [x^n] (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^n.

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Maple

Mathematica

Formula

A292871 a(n) = [x^n] (1/(1 - x - x^2/(1 - 2x - 2x^2/(1 - 3x - 3x^2/(1 - 4x - 4x^2/(1 - ...))))))^n.