A005443 -id:A005443 - cp's OEIS frontend

A039948 A triangle related to A000045 (Fibonacci numbers).

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)

A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

Original entry on oeis.org

0, 1, 3, 19, 159, 1651, 20583, 299419, 4977759, 93097891, 1934655063, 44224195819, 1102820674959, 29792843865331, 866769668577543, 27018340680076219, 898343366411181759, 31736205208791131971, 1187110673532381604023, 46871464129796857140619, 1948059531745350527058159

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

From Peter Bala, Aug 19 2025: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, ...] with an apparent period of 6 = phi(9) beginning at n = 2. Cf. A004123.(End)

Seiichi Manyama, Table of n, a(n) for n = 0..401

Cf. A000045, A000556, A000557, A005443, A005445, A048993, A263576.

seq(n!*coeff(series((exp(x) - 1)/(exp(x) - exp(2*x) + 1), x=0, 22), x, n), n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[(Exp[x] - 1)/(Exp[x] - Exp[2 x] + 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Fibonacci[k] k!, {k, 0, n}], {n, 0, 20}]

E.g.f.: (1 + sinh(x) - cosh(x))/(1 - 2*sinh(x)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*Fibonacci(k)*k!.
a(n) ~ n! / (sqrt(5) * phi^2 * (log(phi))^(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2018

A065140 a(n) = 2^n(2n)!.

Original entry on oeis.org

1, 4, 96, 5760, 645120, 116121600, 30656102400, 11158821273600, 5356234211328000, 3278015337332736000, 2491291656372879360000, 2301953490488540528640000, 2541356653499348743618560000, 3303763649549153366704128000000, 4995290638118319890456641536000000

Offset: 0

Karol A. Penson, Oct 16 2001

From Enrique Navarrete, Aug 29 2025: (Start)
For n > 0, 1/2*a(n) is the number of ways to seat 2*n people on linearly ordered benches placing an even number of people (>=2) on each bench.
For example, 1/2*a(4)=322560 since the number of ways are (number of people in parentheses):
  1 bench (8): 40320 ways;
  2 benches (6,2): 80640 ways;
  2 benches (4,4): 40320 ways;
  3 benches (4,2,2): 120960 ways;
  4 benches (2,2,2,2): 40320 ways.
If the benches are not linearly ordered the number of ways is A088026.
If we seat an odd number of people on linearly ordered benches the number of ways is A005443. (End)

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000680, A088026.

Table[2^n (2n)!,{n,0,15}] (* Harvey P. Dale, Nov 28 2011 *)

{ for (n=0, 100, write("b065140.txt", n, " ", 2^n*(2*n)!) ) } \\ Harry J. Smith, Oct 11 2009

Hypergeometric generating function, in Maple notation: 1/sqrt(1-8*x), i.e., a(0)=1, a(n)=round(evalf(subs(x=0, n!*diff(1/(sqrt(1-8*x)), x$n)))), for n>=1.
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x>=0} x^n*exp(-sqrt(x/2))/(2*sqrt(2*x)) dx, for n>=0.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*x*(k+1)*(2*k+1)/(4*x*(k+1)*(2*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)/2).
Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)/2). (End)
From Alexandre Herrera, Apr 18 2025: (Start)
Sum_{n>=0} x^(4*n)*(-1)^(n)/a(2n) = cos(x/2)*cosh(x/2).
Sum_{n>=0} x^(4*n+2)*(-1)^(n)/a(2n+1) = sin(x/2)*sinh(x/2).
Sum_{n>=0} x^(2*n)*(-1)^(n)/a(n) = cos(x*sqrt(2)/2).
Sum_{n>=0} x^(2*n)/a(n) = cosh(x*sqrt(2)/2). (End)

A052567 E.g.f.: (1-x)^2/(1-3*x+x^2).

Original entry on oeis.org

1, 1, 6, 48, 504, 6600, 103680, 1900080, 39795840, 937681920, 24548832000, 706966444800, 22210346188800, 755916735974400, 27706219904563200, 1088037381150720000, 45576301518139392000, 2028445209752113152000, 95589693062063456256000, 4754884242802394308608000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 509

Cf. A000142, A001906, A005443, A088305.

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

With[{nn=20},CoefficientList[Series[(1-x)^2/(1-3x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 06 2021 *)

E.g.f.: (-1+x)^2/(1-3*x+x^2).
Recurrence: {a(1)=1, a(0)=1, a(2)=6, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0}
Sum(-1/5*(3*_alpha-2)*_alpha^(-1-n), _alpha=RootOf(_Z^2-3*_Z+1))*n!
a(n) = n! * Fibonacci(2*n) for n > 0. - Ilya Gutkovskiy, Jul 17 2021

A080833 E.g.f.: exp( x/(1 - x - x^2) ).

Original entry on oeis.org

1, 1, 3, 19, 145, 1401, 16051, 213403, 3223809, 54514225, 1019601091, 20890209891, 465156779473, 11181638663209, 288536019179955, 7953590111627371, 233211718410856321, 7246720953253750113, 237849724555558441219, 8221578401608012672435, 298505383888840158941841

Offset: 0

Emanuele Munarini, Mar 28 2003

From Peter Bala, Mar 25 2022: (Start)
The sequence terms are odd. 3 divides a(3*n-1), 9 divides a(9*n-1) and 27 divides a(27*n-1); 5 divides a(5*n+4), 25 divides a(25*n+9) and 125 divides a(125*n+34); 7 divides a(7*n+6), 49 divides a(49*n+34) and 343 divides a(343*n + 83); 15 divides a(15*n+14) and 17 divides a(17*n+13).
More generally, the congruence a(n+k) == a(n) (mod k) holds for all n and k. It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, taken modulo 7 the sequence becomes [1, 1, 3, 5, 5, 1, 0, 1, 1, 3, 3, 5, 5, 1, 0, ...], a purely periodic sequence with period 7. (End)

Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A005443.

CoefficientList[Series[E^(x/(1-x-x^2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1-x-x^2)))) \\ Michel Marcus, Jun 07 2021

E.g.f.: exp( x/(1 - x - x^2) ).
a(n) = n!*sum{i=0..n, sum{j=0..n, C(i+j-1, j)*C(j, n-i-j)/i!}}. - Paul Barry, Aug 29 2005
E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) =  1 + 1/(k+1)/(1-x-x^2)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 27 2013
Recurrence: a(n) = (2*n-1)*a(n-1) + (n-2)*(n-1)*a(n-2) - (n-2)*(n-1)*(2*n-7)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Jun 27 2013
a(n) ~ ((1+sqrt(5))/2)^n*exp(2*sqrt(n)/5^(1/4)-n-1/10)*n^(n-1/4)/(sqrt(2)*5^(1/8)). - Vaclav Kotesovec, Jun 27 2013
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * k! * Fibonacci(k) * a(n-k). - Ilya Gutkovskiy, Jun 07 2021

A100404 a(n) = L(n) * n! where L(n) are the Lucas numbers.

Original entry on oeis.org

2, 1, 6, 24, 168, 1320, 12960, 146160, 1895040, 27578880, 446342400, 7943443200, 154238515200, 3244277836800, 73491299481600, 1783667837952000, 46176597282816000, 1270159805730816000, 36992915271696384000, 1137260043722170368000, 36802508677688033280000

Offset: 0

Parthasarathy Nambi, Jan 11 2005

nonn

			If n=3, L(3) * 3! = 24.
If n=4, L(4) * 4! = 168.

Cf. A000032, A005443.

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

E.g.f.: (2 - x) / (1 - x - x^2). - Ilya Gutkovskiy, Jun 07 2021

Extended by Zerinvary Lajos, Jul 09 2009

Edited by R. J. Mathar, Jul 31 2009

A192253 1-sequence of reduction of (n!) by x^2 -> x+1.

Original entry on oeis.org

0, 1, 3, 15, 87, 687, 6447, 71967, 918687, 13256607, 212840607, 3765435807, 72741666207, 1523637512607, 34389853295007, 832071217775007, 21482864837231007, 589515687506543007, 17133249343107695007, 525731414152434287007, 16984313499467403887007

Offset: 1

Clark Kimberling, Jun 27 2011

nonn

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]". After the tenth term, the final digit is 7, for terms in both A192253 and A192252. After the 100th term, the final 6 digits of terms in A192253 are 5,6,7,0,0,7.

Cf. A192252, A192232.

Mathematica
```
(See A192252.)
```

Conjecture: a(n) -n*a(n-1) -(n-1)*(n-3)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 04 2014

Conjecture: partial sums of A005443. - Sean A. Irvine, Jul 12 2022

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

A328055 Expansion of e.g.f. -log(1 - x / (1 - x)^2).

Original entry on oeis.org

0, 1, 5, 32, 270, 2904, 38400, 605520, 11113200, 232888320, 5488560000, 143704108800, 4138573824000, 130020673305600, 4425201196416000, 162194862064435200, 6369479157000960000, 266808274486161408000, 11874724379464826880000, 559591797303082672128000

Offset: 0

Ilya Gutkovskiy, Oct 03 2019

nonn

a(n) is the number of ways to choose one element from each branch of labeled octupi with n nodes (cf. A029767 and example below). - Enrique Navarrete, Oct 29 2023

			For n=2, the 3 labeled octupi are the following, and there are 2+2+1 ways to choose one element from each branch:
O-1-2;
O-2-1;
1-O-2. - _Enrique Navarrete_, Oct 29 2023

Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.

[0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // Marius A. Burtea, Oct 03 2019

nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]

my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ Michel Marcus, Oct 03 2019

E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).

a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.

A103740 Numbers k such that k! * F(k) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 15, 41, 98, 149, 193, 233, 265, 403, 898, 935, 1291, 2079

Offset: 1

Jason Earls, Mar 28 2005

All values through 2079 have been proved prime with WinPFGW. No more terms up to 6700. Primality testing 2079!*F(2079)+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2087 Running N-1 test using base 2099 Calling Brillhart-Lehmer-Selfridge with factored part 33.88% 2079!*F(2079)+1 is prime! (15.2535s+0.0043s)
No more terms < 6729. - David Wasserman, Apr 24 2008
No more terms < 12000. - Michael S. Branicky, Jun 30 2024

			a(6)=7 because 7!*fibonacci(7)+1 = 65521, a prime.

Cf. A005443.

Select[Range[410],PrimeQ[#!Fibonacci[#]+1]&] (* The program generates the first 17 terms of the sequence. *) (* Harvey P. Dale, Jan 30 2024 *)

cp's OEIS Frontend

A039948 A triangle related to A000045 (Fibonacci numbers).

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A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

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A065140 a(n) = 2^n*(2*n)!.

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A052567 E.g.f.: (1-x)^2/(1-3*x+x^2).

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A080833 E.g.f.: exp( x/(1 - x - x^2) ).

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A100404 a(n) = L(n) * n! where L(n) are the Lucas numbers.

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A192253 1-sequence of reduction of (n!) by x^2 -> x+1.

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A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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A065140 a(n) = 2^n(2n)!.