A001908 -id:A001908 - cp's OEIS frontend

A001907 Expansion of e.g.f. exp(-x)/(1-4*x).

1, 3, 25, 299, 4785, 95699, 2296777, 64309755, 2057912161, 74084837795, 2963393511801, 130389314519243, 6258687096923665, 325451729040030579, 18225296826241712425, 1093517809574502745499, 69985139812768175711937, 4758989507268235948411715

Offset: 0

N. J. A. Sloane

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harvey P. Dale, Table of n, a(n) for n = 0..350
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000166, A000354, A000180, A001908.

Column k=4 of A320032.

I:=[3,25]; [1] cat [n le 2 select I[n]  else (4*n-1)*Self(n-1)+4*(n-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 08 2015

f:= gfun:-rectoproc({a(n) = (4*n-1)*a(n-1) + 4*(n-1)*a(n-2), a(0)=1, a(1)=3},a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 07 2015

With[{nn=20},CoefficientList[Series[Exp[-x]/(1-4x),{x,0,nn}], x] Range[0,nn]!] (* or *) Table[Sum[(-1)^(n+k) Binomial[n,k]k! 4^k, {k,0,n}], {n,0,20}](* Harvey P. Dale, Oct 25 2011 *)
Join[{1}, RecurrenceTable[{a[1] == 3, a[2] == 25, a[n] == (4 n - 1) a[n-1] + 4(n - 1) a[n-2]}, a, {n, 20}]] (* Vincenzo Librandi, Aug 08 2015 *)

a(n)=sum(k=0,n,(-1)^(n+k)*binomial(n,k)*k!*4^k)

x = 'x+O('x^33); Vec(serlaplace(exp(-x)/(1-4*x))) \\ Gheorghe Coserea, Aug 06 2015

a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n,k)*k!*4^k. - Ralf Stephan, May 22 2004
Recurrence: a(n) = (4*n-1)*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Aug 16 2013
a(n) ~ n! * exp(-1/4)*4^n. - Vaclav Kotesovec, Aug 16 2013
E.g.f. A(x) = exp(-x)/(1-4x) satisfies (1-4x)A' - (3+4x)A = 0. - Gheorghe Coserea, Aug 06 2015
a(n) = exp(-1/4)*4^n*Gamma(n+1,-1/4), where Gamma is the incomplete Gamma function. - Robert Israel, Aug 07 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020

More terms from Ralf Stephan, May 22 2004
Typo fixed by Charles R Greathouse IV, Oct 28 2009
Name clarified by Ilya Gutkovskiy, Jan 17 2020

A320032 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 5, 2, 1, 1, 3, 13, 29, 9, -1, 1, 4, 25, 116, 233, 44, 1, 1, 5, 41, 299, 1393, 2329, 265, -1, 1, 6, 61, 614, 4785, 20894, 27949, 1854, 1, 1, 7, 85, 1097, 12281, 95699, 376093, 391285, 14833, -1, 1, 8, 113, 1784, 26329, 307024, 2296777, 7897952, 6260561, 133496, 1

Offset: 0

Ilya Gutkovskiy, Oct 03 2018

For n > 0 and k > 0, A(n,k) gives the number of derangements of the generalized symmetric group S(k,n), which is the wreath product of Z_k by S_n. - Peter Kagey, Apr 07 2020

			E.g.f. of column k: A_k(x) = 1 + (k - 1)*x/1! + (2*k^2 - 2*k + 1)*x^2/2! + (6*k^3 - 6*k^2 + 3*k - 1)*x^3/3! + (24*k^4 - 24*k^3 + 12*k^2 - 4*k + 1)*x^4/4! + ...
Square array begins:
   1,   1,     1,      1,      1,       1,  ...
  -1,   0,     1,      2,      3,       4,  ...
   1,   1,     5,     13,     25,      41,  ...
  -1,   2,    29,    116,    299,     614,  ...
   1,   9,   233,   1393,   4785,   12281,  ...
  -1,  44,  2329,  20894,  95699,  307024,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Sami H. Assaf, Cyclic derangements, arXiv:1002.3138 [math.CO], 2010.
Hilarion L. M. Faliharimalala and Jiang Zeng, Derangements and Euler's difference table for C_l wr S_n, Electron. J. Combin. 15 (2008), #R65.
Wikipedia, Generalized symmetric group.

Columns k=0..5 give A033999, A000166, A000354, A000180, A001907, A001908.
Main diagonal gives A319392.
Cf. A320031.

A:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*A(n-1, k)+(-1)^n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 07 2020

Table[Function[k, n! SeriesCoefficient[Exp[-x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, (-1)^n HypergeometricPFQ[{1, -n}, {}, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(-x)/(1 - k*x).
A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j!*k^j.
A(n,k) = (-1)^n*2F0(1,-n; ; k).

A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)k!*n^k.

Original entry on oeis.org

1, 0, 5, 116, 4785, 307024, 28435285, 3598112580, 596971515329, 125802906617600, 32834740225688901, 10399056510149276980, 3929349957207906673585, 1746371472945523953503376, 901944505258819679842017365, 535692457387043907059336566724, 362573376628272441934460817960705

Offset: 0

Ilya Gutkovskiy, Sep 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..232

Main diagonal of A320032.

Cf. A000166, A000180, A000354, A001907, A001908, A277452.

b:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*b(n-1, k)+(-1)^n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 07 2020

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! n^k, {k, 0, n}], {n, 16}]]
Table[n! SeriesCoefficient[Exp[-x]/(1 - n x), {x, 0, n}], {n, 0, 16}]
Table[(-1)^n HypergeometricPFQ[{1, -n}, {}, n], {n, 0, 16}]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*k!*n^k); \\ Michel Marcus, Sep 18 2018

a(n) = n! * [x^n] exp(-x)/(1 - n*x).
a(n) = exp(-1/n)*n^n*Gamma(n+1,-1/n) for n > 0, where Gamma(a,x) is the incomplete gamma function.
a(n) ~ n! * n^n. - Vaclav Kotesovec, Jun 09 2019

A337155 a(n) = 5^n * (n!)^2 * Sum_{k=0..n} 1 / ((-5)^k * (k!)^2).

Original entry on oeis.org

1, 4, 81, 3644, 291521, 36440124, 6559222321, 1607009468644, 514243029966081, 208268427136262804, 104134213568131402001, 63001199208719498210604, 45360863430278038711634881, 38329929598584942711331474444, 37563331006613243857104844955121, 42258747382439899339242950574511124

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A001908, A073701, A336808, A337152, A337153, A337154.

Table[5^n n!^2 Sum[1/((-5)^k k!^2), {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 5 x), {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = 5^n * (n!)^2 * sum(k=0, n, 1 / ((-5)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 5*x).

a(0) = 1; a(n) = 5 * n^2 * a(n-1) + (-1)^n.

A337554 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5k-4) a(n-k).

Original entry on oeis.org

1, 1, 8, 53, 560, 6961, 105898, 1867393, 37713620, 856269401, 21606253238, 599664843433, 18156702186880, 595557844417441, 21037627605306578, 796218790808110673, 32143778726932363340, 1378765268603813275081, 62619174356163136219918, 3001963660666272082265113

Offset: 0

Ilya Gutkovskiy, Aug 31 2020

nonn

Cf. A000354, A001908, A337552, A337553.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (5 k - 4) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(Exp[x] (4 - 5 x) - 3), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1 / (exp(x + O(x*x^n)) * (4 - 5*x) - 3)))} \\ Andrew Howroyd, Aug 31 2020

E.g.f.: 1 / (exp(x) * (4 - 5*x) - 3).

a(n) ~ n! * c / (3*(1-c) * (4/5 - c)^(n+1)), where c = -LambertW(-3*exp(-4/5)/5). - Vaclav Kotesovec, Aug 31 2020

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A001907 Expansion of e.g.f. exp(-x)/(1-4*x).

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A320032 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).

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A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*k!*n^k.

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A337155 a(n) = 5^n * (n!)^2 * Sum_{k=0..n} 1 / ((-5)^k * (k!)^2).

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A337554 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5*k-4) * a(n-k).

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Formula

A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)k!*n^k.

A337554 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5k-4) a(n-k).