A005445 -id:A005445 - cp's OEIS frontend

A005444 From a Fibonacci-like differential equation.

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

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

A354018 Expansion of e.g.f. -log(1-x)/(1 + log(1-x) - log(1-x)^2).

Original entry on oeis.org

0, 1, 3, 20, 172, 1864, 24248, 368136, 6388128, 124711944, 2705241672, 64550432352, 1680280323984, 47383464508080, 1438986494794704, 46821994627363968, 1625069178022566528, 59927028756823323648, 2339899614887520358656, 96439023491479275172608

Offset: 0

Seiichi Manyama, May 14 2022

nonn

Cf. A000045, A005444, A005445, A265165, A320352, A354013.

Table[Sum[k! * Fibonacci[k] * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 15 2022 *)
With[{nn=20},CoefficientList[Series[-Log[1-x]/(1+Log[1-x]-Log[1-x]^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 22 2024 *)

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-x)/(1+log(1-x)-log(1-x)^2))))

a(n) = sum(k=0, n, k!*fibonacci(k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} k! * Fibonacci(k) * |Stirling1(n,k)|.

a(n) ~ n! * (sqrt(5) - 1) / (2 * sqrt(5) * exp((sqrt(5) - 1)/2) * (1 - exp((1 - sqrt(5))/2))^(n+1)). - Vaclav Kotesovec, May 15 2022

A358032 Expansion of e.g.f. (1 + log(1+x))/(1 - log(1+x) * (1 + log(1+x))).

Original entry on oeis.org

1, 2, 4, 16, 66, 438, 2694, 25296, 204576, 2509728, 24912816, 381010320, 4440815472, 82150191264, 1089159690912, 23879423005440, 351430312958208, 9005004020293632, 144184020764472576, 4277182103330660352, 73227226213747521792, 2499666592623881921280

Offset: 0

Seiichi Manyama, Oct 25 2022

nonn

Cf. A005444, A005445.

Cf. A000557, A358031.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+log(1+x))/(1-log(1+x)*(1+log(1+x)))))

a(n) = sum(k=0, n, k!*fibonacci(k+2)*stirling(n, k, 1));

a(n) = Sum_{k=0..n} k! * Fibonacci(k+2) * Stirling1(n,k).

a(n) = A005444(n) + A005445(n).

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

A005444 From a Fibonacci-like differential equation.

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

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A354018 Expansion of e.g.f. -log(1-x)/(1 + log(1-x) - log(1-x)^2).

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A358032 Expansion of e.g.f. (1 + log(1+x))/(1 - log(1+x) * (1 + log(1+x))).

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PARI

PARI

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

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Programs

Maple

Formula