A005444 -id:A005444 - cp's OEIS frontend

A005445 From a Fibonacci-like differential equation.

0, 1, 1, 8, 16, 224, 608, 13320, 41760, 1366152, 4440312, 215100192, 655723440, 48242081328, 121651212720, 14627299801728, 24367884018048, 5768946415383552, 2780730890516736, 2872938805170308352, -2941729703083507968, 1764460446550873413120

Offset: 0

Simon Plouffe

sign

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..434
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, A005444, A048994, A320352.

[(&+[Factorial(j)*Fibonacci(j)*StirlingFirst(n,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 21 2022

CoefficientList[Series[Log[1+x]/(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)*stirling(n, k, 1)); \\ Michel Marcus, Oct 30 2015

def A005445(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k)* stirling_number1(n,k) for k in range(n+1))
[A005445(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

From Vladeta Jovovic, Sep 29 2003: (Start)
a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*Fibonacci(k).
E.g.f.: log(1+x)/(1 - log(1+x) - log(1+x)^2). (End)
a(n) ~ n! * (-1)^(n+1) * (1+1/sqrt(5)) * exp(n*(1+sqrt(5))/2) /(2*(exp((1+sqrt(5))/2)-1)^(n+1)). - Vaclav Kotesovec, Oct 01 2013

More terms from Vladeta Jovovic, Sep 29 2003

A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

Original entry on oeis.org

1, 2, 4, 10, 30, 108, 444, 2112, 11040, 65712, 414816, 2992944, 21876816, 188936928, 1527813216, 15991733376, 133364903040, 1794144752640, 13329036288000, 270750383400960, 1167153128110080, 57074973648030720, -103080839984916480, 17319631144046423040, -171982551742151685120

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

Exponential self-convolution of A006252.

Seiichi Manyama, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms

Cf. A005444, A005649, A006252, A052801, A089064.

Cf. A006252, A354120, A354121.

a:=series(1/(1 - log(1 + x))^2, x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[1/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (k + 1)!, {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*(k + 1)!.
a(n) ~ n! * 2 * (-1)^(n+1) / (n * log(n)^3) * (1 - 3*(gamma+1) / log(n) + (6*gamma^2 + 12*gamma + 6 - Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 15 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354013 Expansion of e.g.f. 1/(1 + log(1-x) * (1 - log(1-x))).

Original entry on oeis.org

1, 1, 5, 32, 278, 3014, 39226, 595608, 10335888, 201785688, 4377151464, 104444584848, 2718748442208, 76668029954736, 2328328726108368, 75759574181169792, 2629417097250852480, 96963968323279825920, 3786037089608099128320, 156041617540423798782720

Offset: 0

Seiichi Manyama, May 14 2022

nonn

Cf. A000045, A000556, A000776, A005444, A320352, A354015, A354018.

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*(1+2*sum(k=1, j-1, 1/k))*binomial(i, j)*v[i-j+1])); v;

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

a(0) = 1; a(n) = Sum_{k=1..n} A000776(k-1) * binomial(n,k) * a(n-k).
a(n) = Sum_{k=0..n} k! * Fibonacci(k+1) * |Stirling1(n,k)|.
a(n) ~ n! / (sqrt(5) * exp((sqrt(5)-1)/2) * (1 - exp((1-sqrt(5))/2))^(n+1)). - Vaclav Kotesovec, May 15 2022

A189423 Expansion of e.g.f. exp(log(1+x) + log(1+x)^2).

Original entry on oeis.org

1, 1, 2, 0, 10, -50, 368, -3052, 28740, -302220, 3508152, -44532048, 613399752, -9109006920, 145029146208, -2463935369040, 44482964644368, -850291412311152, 17153458120885152, -364163960169826944, 8114899768747511712, -189364681355153357088, 4617713773733245962240

Offset: 0

Vladimir Kruchinin, Apr 21 2011

sign

Seiichi Manyama, Table of n, a(n) for n = 0..447
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A005444, A047974.

a(n):=sum(sum(k!*binomial(m,k-m)*stirling1(n,k),k,m,n)/m!,m,1,n);

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)*(1+log(1+x))))) \\ Seiichi Manyama, May 14 2022

a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*stirling(n, k, 1)); \\ Seiichi Manyama, May 14 2022

a(0) = 1; a(n) = Sum_{m=1..n} Sum_{k=m..n} k!*binomial(m,k-m)*stirling1(n,k)/m! for n>0.

a(n) = Sum_{k=0..n} A047974(k) * Stirling1(n,k). - Seiichi Manyama, May 14 2022

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).

A305923 Expansion of e.g.f. exp(x)/(1 - log(1 + x) - log(1 + x)^2).

Original entry on oeis.org

1, 2, 6, 21, 105, 580, 4332, 33173, 333057, 3249334, 41175698, 485901669, 7470988137, 102962077608, 1870375878472, 29342124588357, 617978798588225, 10818920340476010, 260570216908845406, 5009431835664474101, 136578252867673635369, 2844357524328057280332, 87134882338620095240484

Offset: 0

Ilya Gutkovskiy, Jun 14 2018

sign

Binomial transform of A005444.

Sequence is signed: first negative term is a(61).

			E.g.f.: A(x) = 1 + 2*x/1! + 6*x^2/2! + 21*x^3/3! + 105*x^4/4! + 580*x^5/5! + 4332*x^6/6! + ...

N. J. A. Sloane, Transforms

Cf. A000045, A000556, A005444, A005923, A291981.

a:=series(exp(x)/(1-log(1+x)-log(1+x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Exp[x]/(1 - Log[1 + x] - Log[1 + x]^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS1[k, j] j! Fibonacci[j + 1], {j, 0, k}], {k, 0, n}], {n, 0, 22}]

a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k)*Stirling1(k,j)*j!*Fibonacci(j+1).

a(n) ~ (-1)^n * n! * exp(exp(-phi) - phi^2) / (sqrt(5) * (1 - exp(-phi))^(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 26 2019

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

A005445 From a Fibonacci-like differential equation.

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A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

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A354013 Expansion of e.g.f. 1/(1 + log(1-x) * (1 - log(1-x))).

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A189423 Expansion of e.g.f. exp(log(1+x) + log(1+x)^2).

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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

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A305923 Expansion of e.g.f. exp(x)/(1 - log(1 + x) - log(1 + x)^2).

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

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