A039948 -id:A039948 - cp's OEIS frontend

A005442 a(n) = n!*Fibonacci(n+1).

1, 1, 4, 18, 120, 960, 9360, 105840, 1370880, 19958400, 322963200, 5748019200, 111607372800, 2347586841600, 53178757632000, 1290674601216000, 33413695451136000, 919096314200064000, 26768324463648768000

Offset: 0

Simon Plouffe

Number of ways to use the elements of {1,...,n} once each to form a sequence of lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005
Number of pairs (p,S) where p is a permutation of {1,2,...,n} and S is a subset of {1,2,...,n} such that if s is in S then p(s) is not in S.  For example a(2) = 4 because we have (p=(1)(2), s={}); (p=(1,2), s={}); (p=(1,2), s={1}); (p=(1,2), s={2}) where p is given in cycle notation. - Geoffrey Critzer, Jul 01 2013
Another way to state the first comment: a(n) is the number of ways to partition [n] into blocks of size at most 2, order the blocks, and order the elements within each block. For example, a(5)=960 since there are 3 cases: 1) 1,2,3,4,5: 120 such ordered blocks, 1 way to order elements within blocks, hence 120 ways; 2) 12,3,4,5: 240 such ordered blocks, 2 ways to order elements within blocks, hence 480 ways; 3) 12,34,5: 90 such ordered blocks, 4 ways to order elements within blocks, hence 360 ways. - Enrique Navarrete, Sep 01 2023

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, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.
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 494
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Index entries for related partition-counting sequences

Row sums of Fibonacci Jabotinsky-triangle A039692.

Cf. A000045, A000142, A039948, A052585, A080599.

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

Table[Fibonacci[n + 1]*n!, {n, 0, 20}] (* Zerinvary Lajos, Jul 09 2009 *)

a(n) = n!*fibonacci(n+1) \\ Charles R Greathouse IV, Oct 03 2016

[fibonacci(n+1)*factorial(n) for n in range(21)] # G. C. Greubel, Nov 20 2022

a(n) = A039948(n,0).
E.g.f.: 1/(1-x-x^2).
D-finite with recurrence a(n) = n*a(n-1)+n*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A080599 and A052585. - Peter Bala, Dec 07 2011

Comments from Wolfdieter Lang

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

A110313 Expansion of e.g.f. exp(x)/(1-x-x^2).

Original entry on oeis.org

1, 2, 7, 34, 221, 1786, 17347, 196442, 2542969, 37030546, 599172671, 10664259442, 207061905877, 4355429249354, 98661276360571, 2394559287772906, 61991654930903537, 1705178260099590562, 49662655090649112439, 1526761411676393108546, 49407037167974524897741

Offset: 0

Paul Barry, Jul 19 2005

Binomial transform of A005442.

Rows sums of A039948.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000045, A005442, A039948 (row sums).

[(&+[Binomial(n,k)*Factorial(k)*Fibonacci(k+1): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 30 2021

Table[Sum[Binomial[n, k]k!Fibonacci[k+1], {k,0,n}], {n,0,30}] (* Emanuele Munarini, May 27 2011 *)
With[{nn=20},CoefficientList[Series[Exp[x]/(1-x-x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2023 *)

makelist(sum(binomial(n,k)*k!*fib(k+1), k,0,n), n,0,30); /* Emanuele Munarini, May 27 2011 */

[sum(factorial(n)*fibonacci(n-k+1)/factorial(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 30 2021

E.g.f.: exp(x)/(1-x-x^2).
a(n) = Sum_{k=0..n} binomial(n,k)*k!*Fibonacci(k+1).
a(n) = Sum_{k=0..n} n!*Fibonacci(n-k+1)/k!.
Recurrence: a(n+3) = (n+4)*a(n+2) + (n+2)^2*a(n+1) - (n+2)*(n+1)*a(n). - Emanuele Munarini, May 27 2011
a(n) ~ n!*exp((sqrt(5)-1)/2)*((1+sqrt(5))/2)^(n+1)/sqrt(5). - Vaclav Kotesovec, Oct 18 2012

A110314 Inverse of number triangle related to Fibonacci numbers.

Original entry on oeis.org

1, -1, 1, -2, -2, 1, 0, -6, -3, 1, 0, 0, -12, -4, 1, 0, 0, 0, -20, -5, 1, 0, 0, 0, 0, -30, -6, 1, 0, 0, 0, 0, 0, -42, -7, 1, 0, 0, 0, 0, 0, 0, -56, -8, 1, 0, 0, 0, 0, 0, 0, 0, -72, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, -90, -10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -110, -11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -132, -12, 1

Offset: 0

Paul Barry, Jul 19 2005

Row sums are 1-n^2 with g.f. (1-3x)/(1-x)^3. Diagonal sums are A110315. Inverse of A039948.

			Rows begin
1;
-1,1;
-2,-2,1;
0,-6,-3,1;
0,0,-12,-4,1;
0,0,0,-20,-5,1;
0,0,0,0,-30,-6,1;

T(n, k)=if(n=k, 1, if(n-k=1, -binomial(n, 1), if(n-k=2, -2*binomial(n, 2), 0)))

A365962 Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 48, 30, 9, 1, 312, 192, 60, 12, 1, 2520, 1560, 480, 100, 15, 1, 24480, 15120, 4680, 960, 150, 18, 1, 277200, 171360, 52920, 10920, 1680, 210, 21, 1, 3588480, 2217600, 685440, 141120, 21840, 2688, 280, 24, 1, 52254720, 32296320, 9979200, 2056320, 317520, 39312, 4032, 360, 27, 1

Offset: 0

Mélika Tebni, Sep 23 2023

			Triangle begins:
      1,
      3,     1,
     10,     6,    1,
     48,    30,    9,   1,
    312,   192,   60,  12,   1,
   2520,  1560,  480, 100,  15,  1,
  24480, 15120, 4680, 960, 150, 18, 1,
  ...

Winston de Greef, Table of n, a(n) for n = 0..11324
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.

Cf. A000045, A005921 (col 0), A005922 (col 1), A039948, A341725.

T := proc(n, k) option remember;if k = n then 1  else (n!/k!*combinat[fibonacci](n-k+3)) fi end: seq(print(seq(T(n, k), k = 0..n)), n=0..9);
# second Maple program:
T := (n,k) -> add(Stirling2(j, k)*add(Stirling1(n, i)*A341725(i, j), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);

T(n,k) = n!/k!*sum(j=k, n, fibonacci(j-k+1)*binomial(2,n-j)) \\ Winston de Greef, Oct 21 2023

T(n,k) = if(n == k, 1, n!/k!*fibonacci(n-k+3)) \\ Winston de Greef, Oct 21 2023

E.g.f. of column k: (1+x)^2*x^k / ((1-x-x^2)*k!), k >= 0.
T(n,n) = 1 and T(n,k) = n!/k!*Fibonacci(n-k+3), n > k >= 0.
T(n,k) = n!/k!*Sum_{j=k..n} Fibonacci(j-k+1)*binomial(2,n-j).
T(n,k) = n!/k!*Sum_{j=k..n} (Fibonacci(j-k)+(-1)^(j-k))*binomial(3,n-j).
Recurrence: T(n,0) = A005921(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A341725(i,j)).
Sum_{j=k..n} (-1)^(n-j)*(n-j+1)!*binomial(n,j)*T(j,k) = A039948(n,k).

A366133 Triangle read by rows: coefficients in expansion of another Asveld's polynomials Pi_j(x).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 9, 3, 1, 50, 32, 18, 4, 1, 214, 250, 80, 30, 5, 1, 2086, 1284, 750, 160, 45, 6, 1, 11976, 14602, 4494, 1750, 280, 63, 7, 1, 162816, 95808, 58408, 11984, 3500, 448, 84, 8, 1, 1143576, 1465344, 431136, 175224, 26964, 6300, 672, 108, 9, 1, 20472504, 11435760, 7326720, 1437120, 438060, 53928, 10500, 960, 135, 10, 1

Offset: 0

Mélika Tebni, Sep 30 2023

First negative term is T(35,0) = -230450728485788167742674544892530875760640.
Conjectures: For 0 < k < p and p prime, T(p,k) == 0 (mod p).
For 0 < k < n (k odd) and n = 2^m (m natural number), T(n,k) == 0 (mod n).

			Triangle begins:
      1,
      1,     1,
      3,     2,    1,
      8,     9,    3,    1,
     50,    32,   18,    4,   1,
    214,   250,   80,   30,   5,  1,
   2086,  1284,  750,  160,  45,  6,  1,
  11976, 14602, 4494, 1750, 280, 63,  7,  1,
  ...

P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous part, Fib. Quart. 27 (1989), 303-309. See Table 2 p. 308.

Cf. A000045, A005444 (col 0), A005445, A039948, A048994, A305923 (row sums).

T := (n, k) -> binomial(n,k)*add(j!*combinat[fibonacci](j+1)*Stirling1(n-k,j), j=0 .. n-k): seq(print(seq(T(n, k), k = 0 .. n)), n=0 .. 9);
# second Maple program:
T := (n, k) -> add(Stirling2(j, k)/j!*add(i!*combinat[fibonacci](i-j+1)*Stirling1(n, i), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);

T(n,k) = binomial(n,k)*A005444(n-k).
Sum_{k=1..n} (-1)^(k-1)*(k-1)!*T(n, k) = A005445(n).
E.g.f. of column k: x^k / ((1-log(1+x)-log(1+x)^2)*k!), k >= 0.
Recurrence: T(n,0) = A005444(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A039948(i,j)).

cp's OEIS Frontend

A005442 a(n) = n!*Fibonacci(n+1).

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

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A110313 Expansion of e.g.f. exp(x)/(1-x-x^2).

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A110314 Inverse of number triangle related to Fibonacci numbers.

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A365962 Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).

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A366133 Triangle read by rows: coefficients in expansion of another Asveld's polynomials Pi_j(x).

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