A000556 -id:A000556 - cp's OEIS frontend

A000557 Expansion of e.g.f. 1/(1 - 2*sinh(x)).

1, 2, 8, 50, 416, 4322, 53888, 783890, 13031936, 243733442, 5064992768, 115780447730, 2887222009856, 77998677862562, 2269232452763648, 70734934220015570, 2351893466832306176, 83086463910558199682, 3107896091715557654528, 122711086194279627711410

Offset: 0

N. J. A. Sloane

Inverse binomial transform of A005923. - 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.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.
Prabha Sivaraman Nair, A note on alternating weighted sums of Fibonacci numbers, Math. Montisnigri (2024) Vol. LX, 32-49. See p. 38.
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.
R. L. Ollerton and A. G. Shannon, A Note on Brousseau's Summation Problem, Fibonacci Quart. 58 (2020), no. 5, 190-199.
Daniele Parisse, On hypersequences of an arbitrary sequence and their weighted sums, Integers (2024) Vol. 24, Art. No. A70. See p. 25.
Eric Weisstein's MathWorld, Polylogarithm.

Cf. A000045, A000556, A005923, A136630, A320352, A358031, A358032.

Cf. A006154, A107403.

A000557 := proc(n) local k,j; add(add((-1)^j*binomial(k,j)*(k-2*j)^n,j=0..k),k=0..n) end: # Peter Luschny, Jul 31 2011

f[n_] := Sum[ k!*StirlingS2[n, k]*Fibonacci[k + 2], {k, 0, n}]; Array[f, 20, 0] (* Robert G. Wilson v, Aug 16 2011 *)
With[{nn=20},CoefficientList[Series[1/(1-2*Sinh[x]),{x,0,nn}],x]Range[ 0,nn]!] (* Harvey P. Dale, Mar 11 2012 *)
Round@Table[(-1)^n (PolyLog[-n, 1-GoldenRatio]-PolyLog[-n, GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

my(x='x+O('x^30)); Vec(serlaplace(1/(1-2*sinh(x)))) \\ Michel Marcus, May 18 2022

E.g.f.: 1/(1-2*sinh(x)). - Vladeta Jovovic, Jul 06 2002
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n. - Peter Luschny, Jul 31 2011
a(n) = Sum_{k=0..n} k!*Stirling2(n, k)*Fibonacci(k+2).
a(n) ~ n! / (sqrt(5) * log((1+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, May 04 2015
a(n) = (-1)^n*(Li_{-n}(1-phi)-Li_{-n}(phi))/sqrt(5), where Li_n(x) denotes the polylogarithm, phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 29 2015
a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
Sum_{k=0..n-1} binomial(n,k)*a(k) = A000556(n). - Greg Dresden, Jun 01 2022
a(n) = A000556(n) + A320352(n). - Seiichi Manyama, Oct 26 2022
a(n) = Sum_{k=0..n} 2^k * k! * A136630(n,k). - Seiichi Manyama, Jun 25 2025

More terms from David W. Wilson

A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block.

Original entry on oeis.org

1, 2, 12, 104, 1200, 17312, 299712, 6053504, 139733760, 3628677632, 104701504512, 3323151509504, 115063060869120, 4316023589937152, 174347763227738112, 7545919601962287104, 348366745238330081280, 17087957176042900815872, 887497598764802460352512

Offset: 0

Geoffrey Critzer, Sep 16 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, Generalized Barred Preferential Arrangements, Transactions on Combinatorics (2022).
Sithembele Nkonkobe, Venkat Murali, and Béata Bényi, Generalised Barred Preferential Arrangements, arXiv:1907.08944 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A000556, A000670, A006153, A055882, A080253.

a := n -> 2^(n-1)*(polylog(-n, 1/2)+`if`(n=0,1,0)):
seq(round(evalf(a(n),32)), n=0..18); # Peter Luschny, Nov 03 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 04 2019

nn=25;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[1/(1-a),{x,0,nn}],x]
Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[2]] + PolyLog[-n, -Sqrt[2]])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

a(n) = 2^(n-1)*(polylog(-n, 1/2) + 0^n); \\ Michel Marcus, May 30 2018

def A216794(n):
    return 2^n*add(add((-1)^(j-i)*binomial(j,i)*i^n for i in range(n+1)) for j in range(n+1))
[A216794(n) for n in range(18)] # Peter Luschny, Jul 22 2014

E.g.f.: 1/(2 - exp(2*x)).
E.g.f.: 1 + 2*x/(G(0) - 2*x) where G(k) = 2*k+1 - x*2*(2*k+1)/(2*x + (2*k+2)/(1 + 2*x/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
E.g.f.: 1 + 2*x/( G(0) - 2*x ) where G(k) = 1 - 2*x/(1 + (1*k+1)/G(k+1)); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+2)/( 1 - 4*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * (2/log(2))^n/log(4). - Vaclav Kotesovec, Sep 24 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 8*x^2*(k+1)^2/( 8*x^2*(k+1)^2 - (1-2*x-6*x*k)*(1-8*x-6*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
From Vladimir Reshetnikov, Oct 31 2015: (Start)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) + Li_{-n}(-sqrt(2)))/4, where Li_n(x) is the polylogarithm.
Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*a(n) + A080253(n)*sqrt(2)).
(End)
a(n) = 2^(n-1)*(Li_{-n}(1/2) + 0^n) with 0^0=1. - Peter Luschny, Nov 03 2015
From Peter Bala, Oct 18 2023: (Start)
a(n) = 2^n * A000670(n)
Inverse binomial transform of A080253.
The sequence is the first column of the array (2*I - P^2)^(-1), where P denotes Pascal's triangle A007318. (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)

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

A341723 Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.

Original entry on oeis.org

1, -1, 1, 5, -2, 1, -31, 15, -3, 1, 257, -124, 30, -4, 1, -2671, 1285, -310, 50, -5, 1, 33305, -16026, 3855, -620, 75, -6, 1, -484471, 233135, -56091, 8995, -1085, 105, -7, 1, 8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 1

Offset: 0

N. J. A. Sloane, Mar 04 2021

Conjectures from Mélika Tebni, Sep 09 2023: (Start)
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). (End)

			Triangle begins:
        1;
       -1,        1;
        5,       -2,      1;
      -31,       15,     -3,       1;
      257,     -124,     30,      -4,     1;
    -2671,     1285,   -310,      50,    -5,     1;
    33305,   -16026,   3855,    -620,    75,    -6,   1;
  -484471,   233135, -56091,    8995, -1085,   105,  -7,  1;
  8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 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 2.

Column 0 is a signed version of A000556, column 1 is A341726.

Cf. A341724, A341725.

egf:= k-> exp(x)*x^k / ((1+2*sinh(x))*k!):
A341723:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 09 2023
second Maple program:
A341723:= (n, k)-> (-1)^(n-k)*binomial(n, k)*add(j!*combinat[fibonacci](j+1)*Stirling2(n-k,j), j=0.. n-k):
seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 09 2023

From Mélika Tebni, Sep 09 2023: (Start)
E.g.f. of column k: exp(x)*x^k / ((1+2*sinh(x))*k!).
T(n,k) = (-1)^(n-k)*binomial(n,k)*A000556(n-k).
Recurrence: T(n,0) = (-1)^n*A000556(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)

A355408 Expansion of e.g.f. 1/(1 + exp(x) - exp(3*x)).

Original entry on oeis.org

1, 2, 16, 170, 2416, 42962, 916696, 22819610, 649207456, 20778364322, 738918769576, 28905116527850, 1233506128752496, 57025618592932082, 2839117599033828856, 151446758367400488890, 8617182795217834505536, 520954229292164353554242

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Cf. A000556, A355378, A355409.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(x)-exp(3*x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j-1)*binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n,k) * a(n-k).

a(n) ~ n! / ((3 + 2*r) * log(r)^(n+1)), where r = 2*cosh(log((25 + 3*sqrt(69)) / 2) / 6)/sqrt(3). - Vaclav Kotesovec, Jul 01 2022

A341726 Column 1 of A341723.

Original entry on oeis.org

1, -2, 15, -124, 1285, -16026, 233135, -3875768, 72487593, -1506355510, 34433714755, -858675022932, 23197217353661, -674881675961234, 21036941762791575, -699465496639029616, 24710351707159115089, -924304656631807218798, 36494922945480344959595

Offset: 1

N. J. A. Sloane, Mar 04 2021

sign

Cf. A000556, A341723.

a := n -> (-1)^(n+1)*n*add(k!*combinat:-fibonacci(k+1)*Stirling2(n-1, k), k = 0..n-1): seq(a(n), n = 0..20); # Peter Luschny, May 13 2022

a(n) = (-1)^(n+1)*n*Sum_{k=0..n-1} k!*Fibonacci(k+1)*Stirling2(n-1, k). - Peter Luschny, May 13 2022

More terms from Peter Luschny, May 13 2022

A337556 a(0) = 1; a(n) = (1/3) * Sum_{k=1..n} binomial(n,k) * (4^k-1) * a(n-k).

Original entry on oeis.org

1, 1, 7, 57, 607, 8121, 130527, 2447257, 52435327, 1263925881, 33851510047, 997303255257, 32052722909247, 1116001351341241, 41845542004844767, 1681112968022124057, 72039936723424794367, 3280036569708658302201, 158127582939120607830687, 8046697501049910668173657

Offset: 0

Ilya Gutkovskiy, Aug 31 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..387

Cf. A000556, A002450, A337555.

E:= 3 / (3 + exp(x) - exp(4*x)):
S:= series(E,x,41):
seq(n!*coeff(S,x,n),n=0..40); # Robert Israel, Oct 13 2020

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

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

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

a(n) ~ n! * (r^3 - 1) * (4*r^3 - 16*r^2 + 64*r - 3) / (771 * log(r)^(n+1)), where r = 1.452626878833844... is the positive real root of the equation r*(r^3 - 1) = 3. - Vaclav Kotesovec, Aug 31 2020

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

cp's OEIS Frontend

A000557 Expansion of e.g.f. 1/(1 - 2*sinh(x)).

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A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block.

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

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A341723 Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.

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A355408 Expansion of e.g.f. 1/(1 + exp(x) - exp(3*x)).

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