A005442 -id:A005442 - cp's OEIS frontend

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

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

A005921 From solution to a difference equation.

Original entry on oeis.org

1, 3, 10, 48, 312, 2520, 24480, 277200, 3588480, 52254720, 845510400, 15048633600, 292190976000, 6146069529600, 139223731046400, 3379030566912000, 87478184521728000, 2406225451069440000, 70080382582898688000

Offset: 0

N. J. A. Sloane

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

P. R. J. Asveld and N. J. A. Sloane, Correspondence, 1987.
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.

Cf. A000045, A000142.

For n >= 1, a(n) = Fibonacci(n+3)*n!. - David W. Wilson, May 15 1997

For n >= 3, a(n) = n*(a(n-1)+a(n-2)*(n-1)). - Christian Krause, Dec 15 2024

More terms from David W. Wilson, May 15 1997

A052568 E.g.f.: (1-x)/(1-3*x+x^2).

Original entry on oeis.org

1, 2, 10, 78, 816, 10680, 167760, 3074400, 64391040, 1517201280, 39720844800, 1143895737600, 35937095040000, 1223098971494400, 44829605505484800, 1760481463732992000, 73744004937867264000, 3282093293695856640000

Offset: 0

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

Laguerre transform of n!Fibonacci(n+1), A005442. - Paul Barry, Aug 08 2008

G. C. Greubel, Table of n, a(n) for n = 0..385
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 510

Apart from signs, row sums of A079461.

Cf. A005442, A122367.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)/(1-3*x+x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a:= n-> n! * (Matrix([[1,1]]). Matrix([[3,1], [ -1,0]])^n)[1,1]: seq(a(n), n=0..20); # Alois P. Heinz, Jun 01 2009

With[{nn=20},CoefficientList[Series[(1-x)/(1-3x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 07 2012 *)
Table[Fibonacci[2n+1] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

x='x+O('x^30); Vec(serlaplace((1-x)/(1-3*x+x^2))) \\ G. C. Greubel, May 23 2018

Recurrence: {a(0)=1, a(1)=2, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0.}
a(n) = Sum(1/5*(1+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^2-3*_Z+1))*n!.
a(n) = Sum_{k=0..n} binomial(n,k)(n!/k!)*k!Fibonacci(k+1). - Paul Barry, Aug 08 2008
a(n) = n!*A122367(n). - R. J. Mathar, Nov 27 2011

Edited by N. J. A. Sloane, May 29 2009

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.

A194938 Triangle read by rows: coefficients of polynomials p(x,n) defined by 1/(1-t-t^2)^x = Sum_{n=1..oo} p(x,n)*t^n/n!.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 8, 9, 1, 0, 42, 59, 18, 1, 0, 264, 450, 215, 30, 1, 0, 2160, 4114, 2475, 565, 45, 1, 0, 20880, 43512, 30814, 9345, 1225, 63, 1, 0, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1, 0, 3064320, 7235568, 6316316, 2673972, 594489, 69552

Offset: 1

Roger L. Bagula, Apr 17 2008

A039692 is a similar triangle but without the leading column.
1/(1-t-t^2) is the g.f. for the Fibonacci numbers (A000045).
Row sums: A005442(n-1).
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. - Peter Luschny, Jan 21 2016

			Triangle begins
1;
0, 1;
0, 3, 1;
0, 8, 9, 1;
0, 42, 59, 18, 1;
0, 264, 450, 215, 30, 1;
0, 2160, 4114, 2475, 565, 45, 1;
0, 20880, 43512, 30814, 9345, 1225, 63, 1;
0, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1;
0, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1;
0, 44634240, 110499696, 103889700, 49087520, 12803175, 1887753, 154350,6630,135,1;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

Cf. A000045, A005442, A039692.

# The function BellMatrix is defined in A264428.
with(combinat): g := n -> factorial(n)*(fibonacci(n)+fibonacci(n+2)):
BellMatrix(g, 10); # Peter Luschny, Jan 21 2016

p[t_] = 1/(1 - t - t^2)^x; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
(* Second program *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, n!*(Fibonacci[n] + Fibonacci[n+2])], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # Peter Luschny, Jan 21 2016

Edited by N. J. A. Sloane, Aug 28 2011

A364422 Expansion of e.g.f. x^2/(1-x-x^2)^2.

Original entry on oeis.org

0, 0, 2, 12, 120, 1200, 14400, 191520, 2862720, 47174400, 852768000, 16765056000, 356377190400, 8144943206400, 199202395392000, 5191467240960000, 143634952581120000, 4204936774950912000, 129865748246986752000, 4219868533182382080000, 143918318293689139200000

Offset: 0

Enrique Navarrete, Sep 01 2023

nonn

a(n) is the number of ways to partition [n] into blocks of size at most 2, order the blocks, order the elements within each block, and choose 2 elements from a block.

			a(6)=14400 since the number of ways to partition [6] into blocks of size at most 2, order the blocks, order the elements within each block, and select 2 elements from a block are the following:
  (i) 12,3,4,5,6: 3600 such orderings, 1 way to choose 2 elements (from the block with 2 elements), hence 3600 ways;
  (ii) 12,34,5,6: 4320 such orderings, 2 ways to choose 2 elements (from one of the two blocks with 2 elements), hence 8640 ways.
  (iii) 12,34,56: 720 such orderings, 3 ways to choose 2 elements (from one of the three blocks with 2 elements), hence 2160 ways.

Cf. A000045, A000142, A001629, A005442.

With[{m = 20}, Range[0, m]! * CoefficientList[Series[x^2/(1 - x - x^2)^2, {x, 0, m}], x]] (* Amiram Eldar, Sep 02 2023 *)

a(n) = A000142(n)*A001629(n).

A005924 From solution to a difference equation.

Original entry on oeis.org

1, 7, 49, 415, 4321, 53887, 783889, 13031935

Offset: 1

N. J. A. Sloane

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.

Numbers so far satisfy a(n) = A000557(n) - 1. - Ralf Stephan, May 23 2004

A039929 Second column of Fibonacci Jabotinsky-triangle A039692.

Original entry on oeis.org

0, 1, 9, 59, 450, 4114, 43512, 528492, 7235568, 110499696, 1862118720, 34342356960, 688092312960, 14886351037440, 345878769358080, 8590707803462400, 227153424885811200, 6371121297516595200

Offset: 1

Wolfdieter Lang

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

Cf. A039692, A039647, A000032, A005442.

With[{nn=20},CoefficientList[Series[Log[1-x-x^2]^2/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 11 2017 *)

a(n) = A039692(n, 2); a(n) = (n-1)!*sum(L(j)*L(n-j)/(n-j), j=1..n-1), n >= 2, L(n)= A000032(n) (Lucas).

A247504 Square array read by antidiagonals, A(n,k) = k!*[x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0,k>=0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 6, 0, 1, 1, 4, 18, 24, 0, 1, 1, 4, 24, 120, 120, 0, 1, 1, 4, 24, 168, 960, 720, 0, 1, 1, 4, 24, 192, 1560, 9360, 5040, 0, 1, 1, 4, 24, 192, 1800, 17280, 105840, 40320, 0, 1, 1, 4, 24, 192, 1920, 20880, 221760, 1370880, 362880, 0

Offset: 0

Peter Luschny, Nov 03 2014

			n\k[0][1][2] [3]  [4]   [5]    [6]     [7]      [8]       [9]
[0] 1, 0, 0,  0,   0,    0,     0,      0,       0,        0
[1] 1, 1, 2,  6,  24,  120,   720,   5040,   40320,   362880 [A000142]
[2] 1, 1, 4, 18, 120,  960,  9360, 105840, 1370880, 19958400 [A005442]
[3] 1, 1, 4, 24, 168, 1560, 17280, 221760, 3265920, 54069120
[.] .  .  .  .  .   .   .   .    .    .    .     .     .
oo] 1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280 [A002866]
'
As a triangular array, starts:
1,
1, 0,
1, 1, 0,
1, 1, 2, 0,
1, 1, 4, 6,  0,
1, 1, 4, 18, 24,  0,
1, 1, 4, 24, 120, 120, 0,
1, 1, 4, 24, 168, 960, 720, 0.

Cf. A247505, A247506, A002866, A000142, A005442.

A := (n,k) -> k!*coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k): seq(print(seq(A(n,k), k=0..9)), n=0..7);

A365293 a(n) = n!*tetranacci(n+3).

Original entry on oeis.org

1, 1, 4, 24, 192, 1800, 20880, 282240, 4354560, 75479040, 1455148800, 30855686400, 713712384000, 17884003737600, 482619020083200, 13954193180928000, 430360865206272000, 14102295149150208000, 489295008086556672000, 17919783031425859584000

Offset: 0

Enrique Navarrete, Aug 31 2023

nonn

a(n) is the number of ways to partition [n] into blocks of size at most 4, order the blocks, and order the elements within each block.

			a(5) = 1800 since the number of ways to partition [5] into blocks of size at most 4, order the blocks, and order the elements within each block are the following:
1) 1234,5: 10 such ordered blocks; 240 ways;
2) 123,4,5: 60 such ordered blocks; 360 ways;
3) 123,45: 20 such ordered blocks; 240 ways;
4) 12,34,5: 90 such ordered blocks; 360 ways;
5) 12,3,4,5: 240 such ordered blocks; 480 ways;
6) 1,2,3,4,5: 120 such ordered blocks; 120 ways.

Cf. A000078, A000142, A002866, A005442, A276924, A364324.

Table[n! SeriesCoefficient[1/(1-x-x^2-x^3-x^4),{x,0,n}],{n,0,19}] (* Stefano Spezia, Aug 31 2023 *)

E.g.f.: 1/(1-x-x^2-x^3-x^4).

a(n) = A000142(n) * A000078(n+3).

cp's OEIS Frontend

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

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A005921 From solution to a difference equation.

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A052568 E.g.f.: (1-x)/(1-3*x+x^2).

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

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A194938 Triangle read by rows: coefficients of polynomials p(x,n) defined by 1/(1-t-t^2)^x = Sum_{n=1..oo} p(x,n)*t^n/n!.

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A364422 Expansion of e.g.f. x^2/(1-x-x^2)^2.

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A005924 From solution to a difference equation.

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