A005923 -id:A005923 - 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

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

Original entry on oeis.org

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)

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

A366298 Expansion of e.g.f. 1 / (-2 + Sum_{k=1..3} exp(-k*x)).

Original entry on oeis.org

1, 6, 58, 828, 15766, 375276, 10719118, 357202068, 13603819126, 582854637276, 27747071520478, 1453003753611108, 83005119616449286, 5136947527401250476, 342365553703113120238, 24447711909762202272948, 1862151878019906517540246, 150702660087903415402794876, 12913688931657425188926182398

Offset: 0

Ilya Gutkovskiy, Oct 06 2023

nonn

Cf. A001550, A004701, A005923, A319509, A366299, A366300, A366301, A366302.

nmax = 18; CoefficientList[Series[1/(-2 + Sum[Exp[-k x], {k, 1, 3}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + 3^k) * a(n-k).

A366299 Expansion of e.g.f. 1 / (-3 + Sum_{k=1..4} exp(-k*x)).

Original entry on oeis.org

1, 10, 170, 4300, 145046, 6115900, 309453710, 18267444100, 1232400398966, 93535914320620, 7887919177776350, 731710341934820500, 74046493229735962886, 8117679564133907097340, 958393800813241073719790, 121232569802975799394430500, 16357741845227058108680934806, 2345072789674603792983906178060

Offset: 0

Ilya Gutkovskiy, Oct 06 2023

nonn

Cf. A001551, A004702, A005923, A319509, A366298, A366300, A366301, A366302.

nmax = 17; CoefficientList[Series[1/(-3 + Sum[Exp[-k x], {k, 1, 4}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k + 4^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + 3^k + 4^k) * a(n-k).

A366300 Expansion of e.g.f. 1 / (-4 + Sum_{k=1..5} exp(-k*x)).

Original entry on oeis.org

1, 15, 395, 15525, 813671, 53306325, 4190730335, 384368222925, 40289992211591, 4751157347330085, 622528350091484975, 89724601853904952125, 14107579506569655343511, 2403010007367884873188245, 440801776092151383251034815, 86635186648455606881413582125, 18162432724968339044562784395431

Offset: 0

Ilya Gutkovskiy, Oct 06 2023

nonn

Cf. A001552, A004703, A005923, A319509, A366298, A366299, A366301, A366302.

nmax = 16; CoefficientList[Series[1/(-4 + Sum[Exp[-k x], {k, 1, 5}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k + 4^k + 5^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + ... + 5^k) * a(n-k).

A366301 Expansion of e.g.f. 1 / (-5 + Sum_{k=1..6} exp(-k*x)).

Original entry on oeis.org

1, 21, 791, 44541, 3344327, 313883661, 35351663831, 4645129190541, 697553757742247, 117844709608925901, 22120757207544654071, 4567542244067740041741, 1028853921587420129556167, 251065459281889114259025741, 65978874409961267115296383511, 18577448234544937135538443584141

Offset: 0

Ilya Gutkovskiy, Oct 06 2023

nonn

Cf. A001553, A004704, A005923, A319509, A366298, A366299, A366300, A366302.

nmax = 15; CoefficientList[Series[1/(-5 + Sum[Exp[-k x], {k, 1, 6}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k + 4^k + 5^k + 6^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + ... + 6^k) * a(n-k).

A366302 Expansion of e.g.f. 1 / (-6 + Sum_{k=1..7} exp(-k*x)).

Original entry on oeis.org

1, 28, 1428, 108976, 11088924, 1410452848, 215282610348, 38335940184976, 7801807561068444, 1786227911508713008, 454397569178386774668, 127153351764004535348176, 38815768300684586111354364, 12836619471891836987050169968, 4571701128215207034965181098988, 1744488930796462320024115801858576

Offset: 0

Ilya Gutkovskiy, Oct 06 2023

nonn

Cf. A001554, A004705, A005923, A319509, A366298, A366299, A366300, A366301.

nmax = 15; CoefficientList[Series[1/(-6 + Sum[Exp[-k x], {k, 1, 7}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k + 4^k + 5^k + 6^k + 7^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + ... + 7^k) * a(n-k).

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

cp's OEIS Frontend

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

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

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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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A366298 Expansion of e.g.f. 1 / (-2 + Sum_{k=1..3} exp(-k*x)).

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A366299 Expansion of e.g.f. 1 / (-3 + Sum_{k=1..4} exp(-k*x)).

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A366300 Expansion of e.g.f. 1 / (-4 + Sum_{k=1..5} exp(-k*x)).

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A366301 Expansion of e.g.f. 1 / (-5 + Sum_{k=1..6} exp(-k*x)).

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