A000557 -id:A000557 - cp's OEIS frontend

A000556 Expansion of exp(-x) / (1 - exp(x) + exp(-x)).

1, 1, 5, 31, 257, 2671, 33305, 484471, 8054177, 150635551, 3130337705, 71556251911, 1784401334897, 48205833997231, 1402462784186105, 43716593539939351, 1453550100421124417, 51350258701767067711, 1920785418183176050505, 75839622064482770570791

Offset: 0

N. J. A. Sloane

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..401
Gregory Dresden, On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i), arxiv.org:2206.00115 [math.NT], 2022.
Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).
G. Ledin, Jr., On a certain kind of Fibonacci sums, Fib. Quart., 5 (1967), 45-58.
R. L. Ollerton and A. G. Shannon, A Note on Brousseau's Summation Problem, Fibonacci Quart. 58 (2020), no. 5, 190-199.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Golden Ratio.
Eric Weisstein's MathWorld, Lucas Number.

Cf. A005923, A216794.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*(2^j-1), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 05 2019

CoefficientList[Series[E^(-x)/(1-E^x+E^(-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 04 2015 *)
Round@Table[(-1)^(n+1) (PolyLog[-n, 1-GoldenRatio] GoldenRatio + PolyLog[-n, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 30 2015 *)

a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 2)); \\ Michel Marcus, Oct 30 2015

a(n) = Sum_{k=0..n} k!*Fibonacci(k+1)*Stirling2(n,k).
E.g.f.: 1/(1 + U(0)) where U(k) = 1 - 2^k/(1 - x/(x - (k+1)*2^k/U(k+1) )); (continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012
a(n) ~ 2*n! / ((5+sqrt(5)) * log((1+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, May 04 2015
a(n) = (-1)^(n+1)*(Li_{-n}(1-phi)*phi + Li_{-n}(phi)/phi)/sqrt(5), where Li_n(x) is the polylogarithm, phi=(1+sqrt(5))/2 is the golden ratio. - Vladimir Reshetnikov, Oct 30 2015
John W. Layman observes that this is also Sum (-2)^k*binomial(n, k)*b(n-k), where b() = A005923.
From Greg Dresden, May 13 2022 (Start):
For n > 0, a(n) = 1 + 2*Sum_{k=0..floor(n/2-1)} binomial(n,2*k+1) * a(n-2*k-1).
For n > 0, a(n) = Sum_{k=0..n-1} binomial(n,k)*A000557(k).
(End)

A005923 From solution to a difference equation.

Original entry on oeis.org

1, 3, 13, 81, 673, 6993, 87193, 1268361, 21086113, 394368993, 8195330473, 187336699641, 4671623344753, 126204511859793, 3671695236949753, 114451527759954921, 3805443567253430593, 134436722612325267393, 5028681509898733705033, 198550708258762398282201

Offset: 0

N. J. A. Sloane

Binomial transform of A000557. - Vladimir Reshetnikov, Oct 29 2015

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See p. 49.
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.
Prabha Sivaraman Nair, A note on alternating weighted sums of Fibonacci numbers, Math. Montisnigri (2024) Vol. LX, 32-49. See p. 38.

Cf. A000556, A000557.

Round@Table[Sum[Binomial[n, k] (-1)^k (PolyLog[-k, 1-GoldenRatio] - PolyLog[-k, GoldenRatio])/Sqrt[5] , {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

E.g.f.: exp(x)/(1-2*sinh(x)). - Sander Zwegers (s.zwegers(AT)hetnet.nl), Jun 28 2007
E.g.f.: 1/( U(0) -1 ) where U(k) = 1 + 1/(2^k - 2*x*4^k/(2*x*2^k - (k+1)/U(k+1) )); (continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012
a(n) ~ n! * phi  / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio. - Vaclav Kotesovec, Nov 27 2017
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k + 1) * binomial(n,k) * (2^k + 1) * a(n-k). - Ilya Gutkovskiy, Jan 16 2020
a(n) = A000556(n) + A000557(n) for n>0. - Greg Dresden, May 13 2022

More terms from Vladeta Jovovic, Nov 23 2001

A341724 Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.

Original entry on oeis.org

1, -2, 1, 8, -4, 1, -50, 24, -6, 1, 416, -200, 48, -8, 1, -4322, 2080, -500, 80, -10, 1, 53888, -25932, 6240, -1000, 120, -12, 1, -783890, 377216, -90762, 14560, -1750, 168, -14, 1, 13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1

Offset: 0

N. J. A. Sloane, Mar 04 2021

Conjectures from Mélika Tebni, Sep 04 2023: (Start)
For 0 < k < p  and p prime, T(p,k) == 0 (mod p).
For 0 <= k < n  and n = 2^m (m natural number), T(n,k) == 0 (mod n). (End)

			Triangle begins:
         1;
        -2,        1;
         8,       -4,       1;
       -50,       24,      -6,       1;
       416,     -200,      48,      -8,     1;
     -4322,     2080,    -500,      80,   -10,     1;
     53888,   -25932,    6240,   -1000,   120,   -12,   1;
   -783890,   377216,  -90762,   14560, -1750,   168, -14,   1;
  13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1;
  ...

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 3.

Column 0 is a signed version of A000557, column 1 is A341727.

Cf. A000556, A005923, A341723, A341725, A341726.

egf:= k-> x^k / ((1-2*sinh(-x))*k!):
A341724:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341724(n,k), k=0..n)), n=0..8); # Mélika Tebni, Sep 04 2023

From Mélika Tebni, Sep 04 2023: (Start)
E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!).
T(n,k) = (-1)^(n-k)*binomial(n,k)*A000557(n-k).
Recurrence: T(n,0) = (-1)^n*A000557(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)
From Alois P. Heinz, Sep 04 2023: (Start)
|Sum_{k=0..n} T(n,k)| = A000556(n).
Sum_{k=0..n} |T(n,k)| = A005923(n).
Sum_{k=0..n} k * T(n,k) = A341726(n). (End)

A341725 Triangle read by rows: coefficients in expansion of Asveld's polynomials p_j(x).

Original entry on oeis.org

1, 3, 1, 13, 6, 1, 81, 39, 9, 1, 673, 324, 78, 12, 1, 6993, 3365, 810, 130, 15, 1, 87193, 41958, 10095, 1620, 195, 18, 1, 1268361, 610351, 146853, 23555, 2835, 273, 21, 1, 21086113, 10146888, 2441404, 391608, 47110, 4536, 364, 24, 1

Offset: 0

N. J. A. Sloane, Mar 04 2021

			Triangle begins:
      1,
      3,     1,
     13,     6,     1,
     81,    39,     9,    1,
    673,   324,    78,   12,   1,
   6993,  3365,   810,  130,  15,  1,
  87193, 41958, 10095, 1620, 195, 18, 1,
  ...

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 5.

P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.

Cf. A000557, A341723, A341724.

Column 0 is A005923, column 1 is A341728.

egf:= k-> exp(x)*x^k / ((1-2*sinh(x))*k!):
A341725:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341725(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 04 2023

From Mélika Tebni, Sep 04 2023: (Start)
T(n,k) = binomial(n,k)*A005923(n-k).
E.g.f. of column k: exp(x)*x^k / ((1-2*sinh(x))*k!).
T(n,k) = Sum_{j=k..n} binomial(n,j)*A000557(n-j)*binomial(j,k).
Recurrence: T(n,0) = A005923(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)
Sum_{k=0..n} (-1)^k * T(n,k) = A000557(n). - Alois P. Heinz, Sep 04 2023

More terms from Mélika Tebni, Sep 04 2023

A107403 Expansion of e.g.f. 1/(1 - 3*sinh(x)).

Original entry on oeis.org

1, 3, 18, 165, 2016, 30783, 564048, 12057825, 294587136, 8096756763, 247266851328, 8306410495485, 304403359942656, 12085026305182743, 516690458532292608, 23668814542820609145, 1156515067746149400576, 60041982382475841900723, 3300519734382436473765888

Offset: 0

Miklos Kristof, Jun 09 2005

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..379

Cf. A000557, A006154, A136630.

E(x):=1/(1-3*sinh(x)): f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..30);

CoefficientList[Series[1/(1-3*Sinh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

a(n) ~ n!/(sqrt(10)*(log(1/3+sqrt(10)/3))^(n+1)). - Vaclav Kotesovec, Jun 26 2013
a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
a(n) = Sum_{k=0..n} 3^k * k! * A136630(n,k). - Seiichi Manyama, Jun 25 2025

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

A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).

Original entry on oeis.org

1, 2, 4, 10, 32, 114, 448, 1978, 9472, 48738, 270336, 1595114, 9965568, 65852882, 457326592, 3329243546, 25356271616, 201326396098, 1663597019136, 14279558011850, 127044810702848, 1170023757062450, 11136610150121472, 109395885009537402, 1107781178494025728, 11549900930966957346

Offset: 0

Ilya Gutkovskiy, Mar 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..576

Cf. A000557, A000807, A001861.

Cf. A003724, A136630, A352280.

a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[2 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k * Binomial[n, k] * BellB[k, -1] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 27 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

E.g.f.: exp( 2 * sinh(x) ).

a(n) = Sum_{k=0..n} 2^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A341727 Column 1 of A341724.

Original entry on oeis.org

1, -4, 24, -200, 2080, -25932, 377216, -6271120, 117287424, -2437334420, 55714920448, -1389365372760, 37533886128128, -1091981490075868, 34038486791454720, -1131758947520249120, 39982188936149204992, -1495556350390047594276, 59050025742595595436032

Offset: 1

N. J. A. Sloane, Mar 04 2021

sign

Cf. A000556, A000557, A341724 ,A341726, A341728.

A341727 := n -> (-1)^(n-1)*n*add(k!*combinat[fibonacci](k+2)*Stirling2(n-1, k), k=0..n-1):seq(A341727(n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023
# E.g.f. Maple program:
A341727 := series(x / (1 + 2*sinh(x)), x = 0, 20):
seq(n!*coeff(A341727, x, n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023

From Mélika Tebni, Sep 04 2023: (Start)
E.g.f.: x / (1 + 2*sinh(x)).
a(n) = (-1)^(n-1)*n*A000557(n-1).
a(n) = (-1)^(n-1)*Sum_{k=0..n} A000556(k)*(n-k)*binomial(n, k). (End)

More terms from Mélika Tebni, Sep 04 2023

A341728 Column 1 of A341725.

Original entry on oeis.org

1, 6, 39, 324, 3365, 41958, 610351, 10146888, 189775017, 3943689930, 90148635203, 2248040395692, 60731103481789, 1766863166037102, 55075428554246295, 1831224444159278736, 64692540643308320081, 2419861007021854813074, 95544948688075940395627

Offset: 1

N. J. A. Sloane, Mar 04 2021

nonn

Cf. A000557, A005923, A341725, A341726, A341727.

A341728 := n -> add((n-k)*binomial(n, k)add(j!*combinat[fibonacci](j+2)*Stirling2(k,j), j=0..k), k=0..n):seq(A341728(n), n=1.. 19); # Mélika Tebni, Sep 04 2023
# E.g.f. Maple program:
A341728 := series(x*exp(x) / (1 - 2*sinh(x)), x = 0, 20):
seq(n!*coeff(A341728, x, n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023

From Mélika Tebni, Sep 04 2023: (Start)
a(n) = n*A005923(n-1).
E.g.f.: x*exp(x) / (1 - 2*sinh(x)).
a(n) = Sum_{k=0..n} (n-k)*binomial(n, k)*A000557(k). (End)

More terms from Mélika Tebni, Sep 04 2023

A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2j)^n x^(n-k).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 6, 0, 1, 0, 24, 0, 8, 0, 0, 120, 0, 60, 0, 1, 0, 720, 0, 480, 0, 32, 0, 0, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0

Offset: 1

Peter Luschny, Aug 01 2011

See A196776 for a row reversed form of this triangle. - Peter Bala, Oct 06 2011

			The sequence of polynomials P(n, x) begins:
[0]    1;
[1]    1;
[2]    2;
[3]    6 +      x^2;
[4]   24 +    8*x^2;
[5]  120 +   60*x^2 +     x^4;
[6]  720 +  480*x^2 +  32*x^4;
[7] 5040 + 4200*x^2 + 546*x^4 + x^6.

Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.

A193474_polynom := proc(n,x) local k, j;
add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*x^(n-k),j=0..k),k=0..n) end: seq(seq(coeff(A193474_polynom(n,x),x,i),i=0..n),n=0..10);

p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2014 *)

P(n, 0) = A000142(n).
P(n, 1) = A006154(n).
P(n, 2) = A191277(n).
P(n, i) = A000111(n+1), where i is the imaginary unit.
P(n, i)*2^n = A000828(n+1).
P(n, 1/2)*2^n = A000557(n).
P(n, 1/3)*3^n = A107403(n).
P(n, i/2)*2^n = A007289(n).
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
[x^2] P(n+1, x) = A005990(n).

cp's OEIS Frontend

A000556 Expansion of exp(-x) / (1 - exp(x) + exp(-x)).

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A005923 From solution to a difference equation.

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A341724 Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.

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A341725 Triangle read by rows: coefficients in expansion of Asveld's polynomials p_j(x).

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A107403 Expansion of e.g.f. 1/(1 - 3*sinh(x)).

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

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A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

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A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).

A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2j)^n x^(n-k).