A005442 -id:A005442 - cp's OEIS frontend

A039692 Jabotinsky-triangle related to A039647.

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 264, 450, 215, 30, 1, 2160, 4114, 2475, 565, 45, 1, 20880, 43512, 30814, 9345, 1225, 63, 1, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1

Offset: 1

Wolfdieter Lang

Triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= A(z)/z = 1/(1-z-z^2) where A(z) is the g.f. of the Fibonacci numbers A000045. (Notation of F(z) as in Knuth's paper.)
E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x)=1, are exponential convolution polynomials: E(n,x+y) = Sum_{k=0..n} binomial(n,k)*E(k,x)*E(n-k,y) (cf. Knuth's paper with E(n,x)= n!*F(n,x)).
E.g.f. for E(n,x): (1 - z - z^2)^(-x).
Explicit a(n,m) formula: see Knuth's paper for f(n,m) formula with f(k)= A039647(n).
E.g.f. for the m-th column sequence: ((-log(1 - z - z^2))^m)/m!.
Also the Bell transform of n!*(F(n)+F(n+2)), F(n) the Fibonacci numbers. For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			1;
3, 1;
8, 9, 1;
42, 59, 18, 1;
264, 450, 215, 30, 1;

Vincenzo Librandi, Rows n = 1..50, flattened
D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, The Bell transform

Cf. A039647, A000032, A000045. Another version of this triangle is in A194938.

A000032 := proc(n) option remember; coeftayl( (2-x)/(1-x-x^2),x=0,n) ; end: A039647 := proc(n) (n-1)!*A000032(n) ; end: A039692 := proc(n,m) option remember ; if m = 1 then A039647(n) ; else add( binomial(n-1,j-1)*A039647(j)*procname(n-j,m-1),j=1..n-m+1) ; fi; end: # R. J. Mathar, Jun 01 2009

t[n_, m_] := n!*Sum[StirlingS1[k, m]*Binomial[k, n-k]*(-1)^(k+m)/k!, {k, m, n}]; Table[t[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013, after Vladimir Kruchinin *)

T(n,m) := n!*sum((stirling1(k,m)*binomial(k,n-k))*(-1)^(k+m)/k!,k,m,n); /* Vladimir Kruchinin, Mar 26 2013 */

T(n,m) = n!*sum(k=m,n, (stirling(k,m,1)*binomial(k,n-k))*(-1)^(k+m)/k!);
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print());
/* Joerg Arndt, Mar 27 2013 */

# uses[bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 to the left side of the triangle.
bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # Peter Luschny, Jan 16 2016

a(n, 1)= A039647(n)=(n-1)!*L(n), L(n) := A000032(n) (Lucas); a(n, m) = Sum_{j=1..n-m+1} binomial(n-1, j-1)*A039647(j)*a(n-j, m-1), n >= m >= 2.
Conjectured row sums: sum_{m=1..n} a(n,m) = A005442(n). - R. J. Mathar, Jun 01 2009
T(n,m) = n! * Sum_{k=m..n} stirling1(k,m)*binomial(k,n-k)*(-1)^(k+m)/k!. - Vladimir Kruchinin, Mar 26 2013

A080599 Expansion of e.g.f.: 2/(2-2*x-x^2).

Original entry on oeis.org

1, 1, 3, 12, 66, 450, 3690, 35280, 385560, 4740120, 64751400, 972972000, 15949256400, 283232149200, 5416632421200, 110988861984000, 2425817682288000, 56333385828720000, 1385151050307024000, 35950878932544576000, 982196278209226080000, 28175806418228108640000

Offset: 0

Detlef Pauly (dettodet(AT)yahoo.de), Feb 24 2003

Number of ordered partitions of {1,..,n} with at most 2 elements per block. - Bob Proctor, Apr 18 2005
In other words, number of preferential arrangements of n things (see A000670) in which each clump has size 1 or 2. - N. J. A. Sloane, Apr 13 2014
Recurrences (of the hypergeometric type of the Jovovic formula) mean: multiplying the sequence vector from the left with the associated matrix of the recurrence coefficients (here: an infinite lower triangular matrix with the natural numbers in the main diagonal and the triangular series in the subdiagonal) recovers the sequence up to an index shift. In that sense, this sequence here and many other sequences of the OEIS are eigensequences. - Gary W. Adamson, Feb 14 2011
Number of intervals in the weak (Bruhat) order of S_n that are Boolean algebras. - Richard Stanley, May 09 2011
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A000085, A005442 and A052585. - Peter Bala, Dec 07 2011
From Gus Wiseman, Jul 04 2020: (Start)
Also the number of (1,1,1)-avoiding or cubefree sequences of length n covering an initial interval of positive integers. For example, the a(0) = 1 through a(3) = 12 sequences are:
  ()  (1)  (11)  (112)
           (12)  (121)
           (21)  (122)
                 (123)
                 (132)
                 (211)
                 (212)
                 (213)
                 (221)
                 (231)
                 (312)
                 (321)
(End)

			From _Gus Wiseman_, Jul 04 2020: (Start)
The a(0) = 1 through a(3) = 12 ordered set partitions with block sizes <= 2 are:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}
             {{1},{2}}  {{1,2},{3}}
             {{2},{1}}  {{1,3},{2}}
                        {{2},{1,3}}
                        {{2,3},{1}}
                        {{3},{1,2}}
                        {{1},{2},{3}}
                        {{1},{3},{2}}
                        {{2},{1},{3}}
                        {{2},{3},{1}}
                        {{3},{1},{2}}
                        {{3},{2},{1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 10.
Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, Boolean intervals in the weak order of S_n, arXiv:2306.14734 [math.CO], 2023.
Laura Gellert and Raman Sanyal, On degree sequences of undirected, directed, and bidirected graphs, arXiv preprint arXiv:1512.08448 [math.CO], 2015.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja, Frances Y. Kuo, and Ian H. Sloan, Uncertainty quantification for random domains using periodic random variables, arXiv:2210.17329 [math.NA], 2022.
Dixy Msapato, Counting the number of tau-exceptional sequences over Nakayama algebras, arXiv:2002.12194 [math.RT], 2020.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Index entries for related partition-counting sequences

Cf. A000085, A000670, A002530, A002605, A005442, A052585, A052611.
Cf. A090932, A102726, A106356, A232464, A333755, A335455, A335456.
Column k=2 of A276921.
Cubefree numbers are A004709.
(1,1)-avoiding patterns are A000142.
(1,1,1)-avoiding compositions are A232432.
(1,1,1)-matching patterns are A335508.
(1,1,1)-avoiding permutations of prime indices are A335511.
(1,1,1)-avoiding compositions are ranked by A335513.
(1,1,1,1)-avoiding patterns are A189886.

[n le 2 select 1 else (n-1)*Self(n-1) + Binomial(n-1,2)*Self(n-2): n in [1..31]]; // G. C. Greubel, Jan 31 2023

a:= n-> n! *(Matrix([[1,1], [1/2,0]])^n)[1,1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 01 2009
a:= gfun:-rectoproc({a(n) = n*a(n-1)+(n*(n-1)/2)*a(n-2),a(0)=1,a(1)=1},a(n),remember):
seq(a(n), n=0..40); # Robert Israel, Nov 01 2015

Table[n!*SeriesCoefficient[-2/(-2+2*x+x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[n! ((1+Sqrt[3])^(n+1) - (1-Sqrt[3])^(n+1))/(2^(n+1) Sqrt[3]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

Vec(serlaplace((2/(2-2*x-x^2) + O(x^30)))) \\ Michel Marcus, Nov 02 2015

A002605=BinaryRecurrenceSequence(2,2,0,1)
def A080599(n): return factorial(n)*A002605(n+1)/2^n
[A080599(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

a(n) = n*a(n-1) + (n*(n-1)/2)*a(n-2). - Vladeta Jovovic, Aug 22 2003
E.g.f.: 1/(1-x-x^2/2). - Richard Stanley, May 09 2011
a(n) ~ n!*((1+sqrt(3))/2)^(n+1)/sqrt(3). - Vaclav Kotesovec, Oct 13 2012
a(n) = n!*((1+sqrt(3))^(n+1) - (1-sqrt(3))^(n+1))/(2^(n+1)*sqrt(3)). - Vladimir Reshetnikov, Oct 31 2015
a(n) = A090932(n) * A002530(n+1). - Robert Israel, Nov 01 2015

A005923 From solution to a difference equation.

Original entry on oeis.org

1, 3, 13, 81, 673, 6993, 87193, 1268361, 21086113, 394368993, 8195330473, 187336699641, 4671623344753, 126204511859793, 3671695236949753, 114451527759954921, 3805443567253430593, 134436722612325267393, 5028681509898733705033, 198550708258762398282201

Offset: 0

N. J. A. Sloane

Binomial transform of A000557. - Vladimir Reshetnikov, Oct 29 2015

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See p. 49.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.
Prabha Sivaraman Nair, A note on alternating weighted sums of Fibonacci numbers, Math. Montisnigri (2024) Vol. LX, 32-49. See p. 38.

Cf. A000556, A000557.

Round@Table[Sum[Binomial[n, k] (-1)^k (PolyLog[-k, 1-GoldenRatio] - PolyLog[-k, GoldenRatio])/Sqrt[5] , {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

E.g.f.: exp(x)/(1-2*sinh(x)). - Sander Zwegers (s.zwegers(AT)hetnet.nl), Jun 28 2007
E.g.f.: 1/( U(0) -1 ) where U(k) = 1 + 1/(2^k - 2*x*4^k/(2*x*2^k - (k+1)/U(k+1) )); (continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012
a(n) ~ n! * phi  / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio. - Vaclav Kotesovec, Nov 27 2017
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k + 1) * binomial(n,k) * (2^k + 1) * a(n-k). - Ilya Gutkovskiy, Jan 16 2020
a(n) = A000556(n) + A000557(n) for n>0. - Greg Dresden, May 13 2022

More terms from Vladeta Jovovic, Nov 23 2001

A005443 a(n) = n! * Fibonacci(n).

Original entry on oeis.org

0, 1, 2, 12, 72, 600, 5760, 65520, 846720, 12337920, 199584000, 3552595200, 68976230400, 1450895846400, 32866215782400, 797681364480000, 20650793619456000, 568032822669312000, 16543733655601152000, 508598164809326592000, 16458582085314969600000

Offset: 0

Simon Plouffe

From Enrique Navarrete, Aug 28 2025: (Start)
Number of ways to seat n people on linearly ordered benches placing an odd number of people on each bench.
For example, a(7) = 65520 since the number of ways are (number of people in parentheses):
 1 bench (7): 5040 ways;
 3 benches (5,1,1): 15120 ways;
 3 benches (3,3,1): 15120 ways;
 5 benches (3,1,1,1,1): 25200 ways;
 7 benches (1,1,1,1,1,1,1): 5040 ways.
If the benches are not linearly ordered the number of ways is A088009. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..416
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 511

Cf. A000045, A000142, A005442, A039948, A088009.

[Factorial(n)*Fibonacci(n): n in [0..30]]; // G. C. Greubel, Nov 20 2022

ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 1)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..18); # Zerinvary Lajos, Mar 26 2008

Table[Fibonacci[n]*n!, {n, 0, 25}] (* Zerinvary Lajos, Jul 09 2009 *)

a(n) = n!*fibonacci(n); \\ Michel Marcus, Oct 30 2015

[fibonacci(n)*factorial(n) for n in range(31)] # G. C. Greubel, Nov 20 2022

a(n) = A039948(n, 1).
E.g.f.: x/(1-x-x^2). - Geoffrey Critzer, Sep 01 2013
a(n) = n*a(n-1) + n*(n-1)*a(n-2). - G. C. Greubel, Nov 20 2022

More terms from James Sellers, Dec 24 1999

A005444 From a Fibonacci-like differential equation.

Original entry on oeis.org

1, 1, 3, 8, 50, 214, 2086, 11976, 162816, 1143576, 20472504, 165910128, 3785092032, 33908109936, 967508478192, 9252123203712, 327062428940160, 3236057604910080, 141403289873955840, 1404243298160352000, 76168955916831029760, 735206146073008508160

Offset: 0

Simon Plouffe, N. J. A. Sloane

Sequence is signed: first negative term is a(35) = -230450728485788167742674544892530875760640. - Vladeta Jovovic, Sep 29 2003

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 0..100
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Cf. A000045, A000142, A005445, A048994.

[(&+[Factorial(j)*Fibonacci(j+1)*StirlingFirst(n,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 21 2022

CoefficientList[Series[1/(1-Log[1+x]-(Log[1+x])^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 1)); \\ Michel Marcus, Oct 30 2015

def A005444(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k+1)* stirling_number1(n,k) for k in (0..n))
[A005444(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

a(n) = Sum_{k=0..n} k!*Fibonacci(k+1)*Stirling1(n, k).
E.g.f.: 1/(1 - log(1+x) - log(1+x)^2). - Vladeta Jovovic, Sep 29 2003
a(n) ~ n! * (-1)^n * exp(n*(1+sqrt(5))/2) / (sqrt(5)*(exp((1+sqrt(5))/2)-1)^(n+1)). - Vaclav Kotesovec, Oct 01 2013

A039948 A triangle related to A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 18, 12, 3, 1, 120, 72, 24, 4, 1, 960, 600, 180, 40, 5, 1, 9360, 5760, 1800, 360, 60, 6, 1, 105840, 65520, 20160, 4200, 630, 84, 7, 1, 1370880, 846720, 262080, 53760, 8400, 1008, 112, 8, 1, 19958400, 12337920, 3810240, 786240, 120960, 15120, 1512, 144, 9, 1

Offset: 0

Wolfdieter Lang

			Triangle begins :
    1;
    1,   1;
    4,   2,   1;
   18,  12,   3,  1;
  120,  72,  24,  4, 1;
  960, 600, 180, 40, 5, 1;
... - _Philippe Deléham_, Nov 08 2011

Seiichi Manyama, Rows n = 0..139, flattened
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Cf. A000045, A000142, A005442, A005443, A110313 (row sums).

Diagonals include: A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288, A010965, A010966.

[(Factorial(n)/Factorial(k))*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2022

T[n_,k_]:= (n!/k!)*Fibonacci[n-k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2022 *)

def A039948(n, k): return factorial(n-k)*binomial(n,k)*fibonacci(n-k+1)
flatten([[A039948(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2022

T(n, m) = n!*Fibonacci(n-m+1)/m!, n >= m >= 0.
T(n, 0) = A005442(n).
T(n, 1) = A005443(n).
E.g.f. for column m: x^m/(m!*(1-x-x^2)), m >= 0.
From G. C. Greubel, Nov 20 2022: (Start)
T(n, n-1) = A000027(n).
T(n, n-2) = 4*A000217(n-1), n >= 2.
T(n, n-3) = 18*A000292(n-2), n >= 3.
T(n, n-4) = 5! * A000332(n), n >= 4.
T(n, n-5) = 8 * 5! * A000389(n), n >= 5.
T(n, n-6) = 13 * 6! * A000579(n), n >= 6.
T(n, n-7) = 21 * 7! * A000580(n), n >= 7.
T(n, n-8) = 34 * 8! * A000581(n), n >= 8.
T(n, n-9) = 55 * 9! * A000582(n), n >= 9.
T(n, n-10) = 89 * 10! * A001287(n), n >= 10.
T(n, n-11) = 12 * 12! * A001288(n), n >= 11.
T(n, n-12) = 233 * 12! * A010965(n), n >= 12.
T(n, n-13) = 89 * 13! * A010966(n), n >= 13.
Sum_{k=0..n} T(n, k) = A110313(n). (End)

A005445 From a Fibonacci-like differential equation.

Original entry on oeis.org

0, 1, 1, 8, 16, 224, 608, 13320, 41760, 1366152, 4440312, 215100192, 655723440, 48242081328, 121651212720, 14627299801728, 24367884018048, 5768946415383552, 2780730890516736, 2872938805170308352, -2941729703083507968, 1764460446550873413120

Offset: 0

Simon Plouffe

sign

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..434
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Cf. A000045, A000142, A005444, A048994, A320352.

[(&+[Factorial(j)*Fibonacci(j)*StirlingFirst(n,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 21 2022

CoefficientList[Series[Log[1+x]/(1-Log[1+x]-(Log[1+x])^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

a(n) = sum(k=0, n, k!*fibonacci(k)*stirling(n, k, 1)); \\ Michel Marcus, Oct 30 2015

def A005445(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k)* stirling_number1(n,k) for k in range(n+1))
[A005445(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

From Vladeta Jovovic, Sep 29 2003: (Start)
a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*Fibonacci(k).
E.g.f.: log(1+x)/(1 - log(1+x) - log(1+x)^2). (End)
a(n) ~ n! * (-1)^(n+1) * (1+1/sqrt(5)) * exp(n*(1+sqrt(5))/2) /(2*(exp((1+sqrt(5))/2)-1)^(n+1)). - Vaclav Kotesovec, Oct 01 2013

More terms from Vladeta Jovovic, Sep 29 2003

A052585 E.g.f. 1/(1-x-2*x^2).

Original entry on oeis.org

1, 1, 6, 30, 264, 2520, 30960, 428400, 6894720, 123742080, 2478470400, 54486432000, 1308153369600, 34005760588800, 952248474777600, 28566146568960000, 914137612996608000, 31080323154456576000, 1118898035934142464000, 42518003720397004800000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Laguerre transform is A052563. - Paul Barry, Aug 08 2008

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 530

Cf. A080599, A005442.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1 -x -2*x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

spec := [S,{S=Sequence(Union(Z,Prod(Z,Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{m = 50}, CoefficientList[Series[-1/(-1 + x + 2*x^2), {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 17 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1 -x -2*x^2))) \\ G. C. Greubel, May 17 2018

E.g.f.: 1/(1 -x -2*x^2).
Recurrence: a(1)=1, a(0)=1, (-2*n^2-6*n-4)*a(n)+(-2-n)*a(n+1)+a(n+2)=0.
a(n) = Sum(1/9*(1+4*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+2*_Z^2))*n!.
a(n) = n!*A001045(n+1). - Paul Barry, Aug 08 2008
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A080599 and A005442. - Peter Bala, Dec 07 2011

a(18)-a(19) added by G. C. Greubel, May 17 2018

A364324 a(n) = n!*tribonacci(n+2).

Original entry on oeis.org

1, 1, 4, 24, 168, 1560, 17280, 221760, 3265920, 54069120, 994291200, 20118067200, 444034483200, 10617070464000, 273391121203200, 7542665754624000, 221969877921792000, 6940528784437248000, 229781192298577920000, 8030036368187817984000, 295390797322766745600000

Offset: 0

Enrique Navarrete, Jul 18 2023

nonn

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, and order the elements within each block.

			a(5) = 1560 since the number of ways to partition [5] into blocks of size at most 3, order the blocks, and order the elements within each block are the following:
1) 1,2,3,4,5: 120 ordered blocks; 120 ways;
2) 12,3,4,5: 240 ordered blocks; 480 ways;
3) 12,34,5: 90 ordered blocks; 360 ways;
4) 123,45: 20 ordered blocks; 240 ways;
5) 123,4,5: 60 ordered blocks; 360 ways.

Cf. A000073, A000142, A002866, A005442, A189886.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-i)*binomial(n, i)*i!, i=1..min(n, 3)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 18 2023

With[{m = 21}, Range[0, m - 1]! * LinearRecurrence[{1, 1, 1}, {1, 1, 2}, m]] (* Amiram Eldar, Jul 28 2023 *)

E.g.f.: 1/(1-x-x^2-x^3).

a(n) = A000142(n) * A000073(n+2).

A005922 a(1)=1; a(n) = n!*Fibonacci(n+2), n > 1.

Original entry on oeis.org

1, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320, 522547200, 9300614400, 180583603200, 3798482688000, 86044973414400, 2088355965696000, 54064489070592000, 1487129136869376000, 43312058119249920000

Offset: 1

N. J. A. Sloane

From solution to a difference equation.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.

Cf. A005921, A078700.

[1] cat [Factorial(n)*Fibonacci(n+2): n in [2..20]]; // Vincenzo Librandi, Jul 10 2012

Join[{1},Table[n!Fibonacci[n+2],{n,2,20}]] (* Harvey P. Dale, May 26 2024 *)

a(n)=if(n>1, n!*fibonacci(n+2), 1) \\ Charles R Greathouse IV, Jun 30 2017

More terms and better description from Vladeta Jovovic, Jan 23 2005

cp's OEIS Frontend

A039692 Jabotinsky-triangle related to A039647.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Sage

Formula

A080599 Expansion of e.g.f.: 2/(2-2*x-x^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A005923 From solution to a difference equation.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A005443 a(n) = n! * Fibonacci(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A005444 From a Fibonacci-like differential equation.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A039948 A triangle related to A000045 (Fibonacci numbers).

Views

Author

Keywords

Examples

Links

Crossrefs