A003704 -id:A003704 - cp's OEIS frontend

A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

1, 15, 140, 1050, 6951, 42651, 253660, 1594230, 12463451, 134921787, 1806513072, 25539589530, 355175465191, 4797717669123, 63797550625676, 860468790181686, 12275324511112971, 192498455326842819, 3353266112959628272, 63379650000684213834

Offset: 5

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000481, A000629, A003704, A327506, A346390, A346920, A346954.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ (n-1)! / (log(120^(1/5) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A214622 Triangle read by rows, matrix inverse of [x^(n-k)](skp(n,x)-skp(n,x-1)+x^n) where skp denotes the Swiss-Knife polynomials A153641.

Original entry on oeis.org

1, -1, 1, 3, -2, 1, -10, 9, -3, 1, 45, -40, 18, -4, 1, -256, 225, -100, 30, -5, 1, 1743, -1536, 675, -200, 45, -6, 1, -13840, 12201, -5376, 1575, -350, 63, -7, 1, 125625, -110720, 48804, -14336, 3150, -560, 84, -8, 1, -1282816, 1130625, -498240, 146412, -32256, 5670, -840, 108, -9, 1

Offset: 0

Peter Luschny, Jul 23 2012

			Triangle begins:
     1;
    -1,     1;
     3,    -2,    1;
   -10,     9,   -3,    1;
    45,   -40,   18,   -4,  1;
  -256,   225, -100,   30, -5,  1;
  1743, -1536,  675, -200, 45, -6, 1;
  ...

A214622_row := proc(n) local s,t,k;
  s := series(exp(z*x)/(sech(x)+tanh(x)),x,n+2);
  t := factorial(n)*coeff(s,x,n); seq(coeff(t,z,k), k=(0..n)) end:
for n from 0 to 7 do A214622_row(n) od; # Peter Luschny, Aug 01 2012

A214622row[n_] := Module[{s, t},
   s = Series[Exp[z*x]/(Sech[x] + Tanh[x]), {x, 0, n+2}];
   t = n!*Coefficient[s, x, n];
   Table[Coefficient[t, z, k], {k, 0, n}]];
Table[A214622row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, May 25 2024, after Peter Luschny *)

R = PolynomialRing(ZZ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A109449_signed(n, k) : return 0 if k > n else R(skp(n, x)-skp(n, x-1)+x^n)[k]
T = matrix(ZZ, 9, A109449_signed).inverse(); T

T(n,k) = matrix inverse of A109449(n,k)*(-1)^floor((k-n+5)/2).
T(n,0) = A003704(n+1).
E.g.f.: exp(x*z)/(sech(x)+tanh(x)). - Peter Luschny, Aug 01 2012

cp's OEIS Frontend

A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

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