A107403 -id:A107403 - 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

A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).

Original entry on oeis.org

1, 3, 9, 30, 117, 516, 2493, 13152, 75177, 460272, 3003921, 20806176, 152114013, 1169842368, 9435180357, 79553524224, 699531782481, 6400932102912, 60820145019801, 599036357936640, 6105903392066373, 64309189153428480, 698936466350352717, 7828833281592926208, 90270159223293364473

Offset: 0

Ilya Gutkovskiy, Mar 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..565

Cf. A027710, A080527, A107403.

Cf. A003724, A136630, A352279.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Exp[3 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

E.g.f.: exp( 3 * sinh(x) ).

a(n) = Sum_{k=0..n} 3^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

Original entry on oeis.org

1, 3, 18, 159, 1872, 27543, 486288, 10016619, 235798272, 6244714443, 183756215808, 5947907121879, 210026879004672, 8034293365747743, 330982609573398528, 14609181655918083939, 687820834029346947072, 34407546247054875367443, 1822450167175258689896448

Offset: 0

Seiichi Manyama, Mar 25 2022

nonn

Cf. A000111, A007289.
Cf. A107403, A352640.
Cf. A136630.

With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*Sin[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

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

a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*binomial(n, 2*k+1)*a(n-2*k-1)));

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / (2^(3/2) * arcsin(1/3)^(n+1)). - Vaclav Kotesovec, Mar 26 2022
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

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).

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

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A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

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A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

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PARI

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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-2*j)^n * x^(n-k).

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A352280 a(0) = 1; a(n) = 3 * 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).