A158687 -id:A158687 - cp's OEIS frontend

A100326 Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169).

1, 1, 1, 3, 4, 1, 14, 20, 7, 1, 79, 116, 46, 10, 1, 494, 736, 311, 81, 13, 1, 3294, 4952, 2174, 626, 125, 16, 1, 22952, 34716, 15634, 4798, 1088, 178, 19, 1, 165127, 250868, 115048, 36896, 9094, 1724, 240, 22, 1, 1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25, 1

Offset: 0

Paul D. Hanna, Nov 17 2004

The leftmost column equals A003169 shift one place right.
Each column k > 0 equals the convolution of the prior column and A003169.
Row sums form A100327.
The elements of the matrix inverse are T^(-1)(n,k) = (-1)^(n+k) * A158687(n,k). - R. J. Mathar, Mar 15 2013

			Rows begin:
        1;
        1,       1;
        3,       4,      1;
       14,      20,      7,      1;
       79,     116,     46,     10,     1;
      494,     736,    311,     81,    13,     1;
     3294,    4952,   2174,    626,   125,    16,    1;
    22952,   34716,  15634,   4798,  1088,   178,   19,   1;
   165127,  250868, 115048,  36896,  9094,  1724,  240,  22,  1;
  1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25,  1;
  ...
First column forms A003169 shift right.
Binomial transform of row 3 forms column 3 of square A100324: BINOMIAL([14,20,7,1]) = [14,34,61,96,140,194,259,...].
Binomial transform of row 4 forms column 4 of square A100324: BINOMIAL([79,116,46,10,1]) = [79,195,357,575,860,1224,...].

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A003169, A100324, A100327 (row sums), A158687, A264717 (central terms).

import Data.List (transpose)
a100326 n k = a100326_tabl !! n !! k
a100326_row n = a100326_tabl !! n
a100326_tabl = [1] : f [[1]] where
f xss@(xs:_) = ys : f (ys : xss) where
ys = y : map (sum . zipWith (*) (zs ++ [y])) (map reverse zss)
y = sum $ zipWith (*) [1..] xs
zss@((:zs):) = transpose $ reverse xss
-- Reinhard Zumkeller, Nov 21 2015

A100326 := proc(n,k)
    if k < 0 or k > n then
        0 ;
    elif n = 0 then
        1 ;
    elif k = 0 then
        A003169(n)
    else
        add(procname(i+1,0)*procname(n-i-1,k-1),i=0..n-k) ;
    end if;
end proc: # R. J. Mathar, Mar 15 2013

lim= 9; t[0, 0]=1; t[n_, 0]:= t[n, 0]= Sum[(k+1)*t[n-1,k], {k,0,n-1}]; t[n_, k_]:= t[n, k]= Sum[t[j+1, 0]*t[n-j-1, k-1], {j,0,n-k}]; Flatten[Table[t[n, k], {n,0,lim}, {k,0,n}]] (* Jean-François Alcover, Sep 20 2011 *)

PARI
```
T(n,k)=if(n
				
```

@CachedFunction
def T(n,k): # T = A100326
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==0): return sum((j+1)*T(n-1,j) for j in range(n))
    else: return sum(T(j+1,0)*T(n-j-1,k-1) for j in range(n-k+1))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 30 2023

T(n, 0) = A003169(n) = Sum_{k=0..n-1} (k+1)*T(n-1, k) for n>0, with T(0, 0)=1.
T(n, k) = Sum_{i=0..n-k} T(i+1, 0)*T(n-i-1, k-1) for n > 0.
T(2*n, n) = A264717(n).
Sum_{k=0..n} T(n, k) = A100327(n).
G.f.: A(x, y) = (1 + G(x))/(1 - y*G(x)), where G(x) is the g.f. of A003169.
From G. C. Greubel, Jan 30 2023: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A033999(n). (End)

A129847 a(n) = number of set partitions of {1, 2, ..., n} whose blocks consist only of elements that differ by two or less (that is, have only the forms {i}, {i,i+1}, {i,i+2}, or {i,i+1,i+2}).

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 42, 87, 179, 370, 765, 1580, 3264, 6744, 13933, 28785, 59470, 122865, 253838, 524428, 1083466, 2238435, 4624595, 9554390, 19739321, 40781336, 84254032, 174068400, 359624425, 742982225, 1534997482, 3171296957, 6551883314, 13536157460

Offset: 0

Rhodes Peele (rpeele(AT)mail.aum.edu), May 22 2007

(1) Bryce Duncan and I found this sequence while studying a graph invariant we call the Bell number. For a simple graph G = (V,E), we define B(G) to be the number of partitions P of V in which each block of P is an independent set of G. The sequence considered here results from choosing V = {1, 2, ..., n} and E = {(i,j) : |i - j| > 2}. (The classical Bell numbers B(n) come from the edgeless graph on n vertices.)
(2) The constant r in the formula is the dominant root of the characteristic equation of a linear homogeneous recurrence relation that also defines {a(n)}. (This recurrence relation, along with initial conditions, appears in the Mathematica program given below. The formula for a(n) is analogous to one version of the Binet formula for the Fibonacci numbers, namely F(n) = the integer nearest to (1/sqrt(5)) p^n where p is the golden mean. The shifted Fibonacci numbers F(n+1) are also graphical Bell numbers: replace the condition |i - j| > 2 with |i - j| > 1 to obtain them.
(3) Bell numbers for simple graphs satisfy the deletion-contraction identity B(G) = B(G\e) - B(G/e), but not the product identity B(G union H) = B(G)B(H) for disjoint graphs G and H. However, we do have B(B join H) = B(G)B(H) for the join of graphs G and H. The join graph G join H results from adding to their disjoint union, all possible edges that join a vertex of G to a vertex of H.
Diagonal sums of triangle A158687. - Paul Barry, Mar 24 2009
a(n) is the number of compositions (ordered partitions) of n into parts 1, 2, 3, and 4 where there are two kinds of part 3. - Joerg Arndt, Sep 16 2016
a(n) is the number of ways to tile a skew double-strip of n cells using squares, "double", and "triangular triple" tiles. Here is the skew double-strip corresponding to n=12, with 12 cells:
   _ ___ _ ___ _ ___
  |   |   |   |   |   |   |
 |__|___|_|___| |___|
|   |   |   |   |   |   |
|_|___|_|___|_|___|,
and here are the three possible "double" tiles and two possible "triangular triple" tiles:
   _    _                     _      _____
  |   |  |   |                   |   |    |       |
 |  |  |  |     _____     |   |   |     |
|   |      |   |   |       |   |       |    |   |
|_|,     |_|,  |_____|,  |_____|,   |_|
As an example, here is one of the b(12) = 3264 ways to tile the skew double-strip of 12 cells:
   _ ___ _____ _______
  |   |   |       |   |   |
 |__|_  |    |   | _|
|       |   |   |       |
|_____|___|_|___ _|. - Greg Dresden and Ruotong Li, Jun 12 2024

			a(4) = 10, with set partitions {{1},{2},{3}, {4}}; {{1,2}, {3}, {4}}; {{1,3}, {2}, {4}}; {{1}, {2,3}, {4}}; {{1}, {2,4}, {3}}; {{1}, {2}, {3,4}}; {{1,2,3}, {4}}; {{1}, {2,3,4}}; {{1,2}, {3,4}} and {{1,3}, {2,4}}.

Herbert S. Wilf, Generatingfunctiononogy (2nd Edition), Academic Press 1990, pp. 1 - 10 and pp. 20 - 23.
Arthur T. Benjamin and Jennifer J. Quinn, Proofs that Really Count, Dolciani Mathematical Expositions (MAA), (2003), p. 1. (A relevant combinatorial interpretation of Fibonacci numbers.)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
B. Duncan and R. Peele, Bell and Stirling numbers for graphs, JIS 12 (2009), Article 09.7.1.
W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015.
W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, Mathematics of Computation 87 (2018), 987-1011.

Cf. A000110, A000045, A141073, A158687, A276893.

Column k=3 of A276719.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|2|1|1>>^n)[4, 4]:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 15 2016

a[1] = 1; a[2] = 2; a[3] = 5; a[4] = 10
a[n_] := a[n] = a[n-1] + a[n-2] + 2 a[n-3] + a[n-4]
Table[a[n],{n,1,30}]

a(n) = the integer nearest to s r^n, where r = 2.0659948920 ... and s = 0.53979687305... . (More information in comment (2).)
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} C(n-j-k,k)*C(2k,j). - Paul Barry, Mar 24 2009
G.f.: 1/(1 - x - x^2 - 2*x^3 - x^4). - Colin Barker, May 02 2012
a(n) = a(n-1) + a(n-2) + 2*(n-3) + a(n-4) with a(0) = a(1) = 1, a(2) = 2, a(3) = 5. - Taras Goy, Aug 04 2017
a(2*n) = a(n)^2 - a(n-1)^2 + a(n-2)^2 + 2*a(n-1)*(a(n+1)-a(n)). - Greg Dresden, Jul 03 2024

a(0)=1 prepended by Alois P. Heinz, Sep 15 2016

A077936 Expansion of 1/(1 - 2x - 2x^2 - x^3).

Original entry on oeis.org

1, 2, 6, 17, 48, 136, 385, 1090, 3086, 8737, 24736, 70032, 198273, 561346, 1589270, 4499505, 12738896, 36066072, 102109441, 289089922, 818464798, 2317218881, 6560457280, 18573817120, 52585767681, 148879626882, 421504606246, 1193354233937, 3378597307248

Offset: 0

N. J. A. Sloane, Nov 17 2002

Row sum of A158687. - Paul Barry, Mar 24 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,2,1).

I:=[1,2,6]; [n le 3 select I[n] else 2*Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 06 2015

CoefficientList[Series[1/(1-2*x-2*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,1},{1,2,6},40] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

Vec(1/(1-2*x-2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Jan 31 2012

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-j,k)*C(2k,j). - Paul Barry, Mar 24 2009

a(n) = 2*a(n-1) + 2*a(n-2) + a(n-3) with a(0) = 1, a(1) = 2, a(2) = 6. - Taras Goy, Aug 04 2017

cp's OEIS Frontend

A100326 Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

SageMath

Formula

A129847 a(n) = number of set partitions of {1, 2, ..., n} whose blocks consist only of elements that differ by two or less (that is, have only the forms {i}, {i,i+1}, {i,i+2}, or {i,i+1,i+2}).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A077936 Expansion of 1/(1 - 2*x - 2*x^2 - x^3).

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A077936 Expansion of 1/(1 - 2x - 2x^2 - x^3).