A293037
E.g.f.: exp(1 + x - exp(x)).
Original entry on oeis.org
1, 0, -1, -1, 2, 9, 9, -50, -267, -413, 2180, 17731, 50533, -110176, -1966797, -9938669, -8638718, 278475061, 2540956509, 9816860358, -27172288399, -725503033401, -5592543175252, -15823587507881, 168392610536153, 2848115497132448, 20819319685262839
Offset: 0
-
f:= series(exp(1 + x - exp(x)), x= 0, 101): seq(factorial(n) * coeff(f, x, n), n = 0..30); # Muniru A Asiru, Oct 31 2017
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n+1, 1):
seq(a(n), n=0..35); # Alois P. Heinz, Dec 01 2021
-
m = 26; Range[0, m]! * CoefficientList[Series[Exp[1 + x - Exp[x]], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -1], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)
-
my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-exp(x)+1+x)))
-
a(n) = if(n==0, 1, -sum(k=0, n-2, binomial(n-1, k)*a(k))); \\ Seiichi Manyama, Aug 02 2021
A293024
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(x) - Sum_{i=0..k} x^i/i!).
Original entry on oeis.org
1, 1, 1, 1, 0, 2, 1, 0, 1, 5, 1, 0, 0, 1, 15, 1, 0, 0, 1, 4, 52, 1, 0, 0, 0, 1, 11, 203, 1, 0, 0, 0, 1, 1, 41, 877, 1, 0, 0, 0, 0, 1, 11, 162, 4140, 1, 0, 0, 0, 0, 1, 1, 36, 715, 21147, 1, 0, 0, 0, 0, 0, 1, 1, 92, 3425, 115975, 1, 0, 0, 0, 0, 0, 1, 1, 36, 491, 17722, 678570
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, ...
5, 1, 1, 0, 0, 0, 0, 0, ...
15, 4, 1, 1, 0, 0, 0, 0, ...
52, 11, 1, 1, 1, 0, 0, 0, ...
203, 41, 11, 1, 1, 1, 0, 0, ...
877, 162, 36, 1, 1, 1, 1, 0, ...
-
A:= proc(n, k) option remember; `if`(n=0, 1, add(
A(n-j, k)*binomial(n-1, j-1), j=1+k..n))
end:
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Sep 28 2017
-
A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}];
A[, ] = 0;
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)
-
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
ary = [1]
(1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
ary
end
def A293024(n)
a = []
(0..n).each{|i| a << A(i, n - i)}
ary = []
(0..n).each{|i|
(0..i).each{|j|
ary << a[i - j][j]
}
}
ary
end
p A293024(20)
A293038
E.g.f.: exp(1 + x + x^2/2! - exp(x)).
Original entry on oeis.org
1, 0, 0, -1, -1, -1, 9, 34, 90, -71, -1645, -9439, -25367, 45902, 1070146, 7122361, 24063637, -54352333, -1501032375, -12319959348, -53177369044, 80189626539, 3910291080509, 40317032441401, 228707685648269, 38882013140648, -16392939262378536
Offset: 0
-
seq(factorial(n) * coeftayl(exp(1+x+x^2/2!-exp(x)), x = 0, n),n = 0..50); # Muniru A Asiru, Oct 05 2017
-
my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2)))
A293039
E.g.f.: exp(1 + x + x^2/2! + x^3/3! - exp(x)).
Original entry on oeis.org
1, 0, 0, 0, -1, -1, -1, -1, 34, 125, 335, 791, -4027, -41328, -223510, -966174, -1082043, 22493107, 255137121, 1853859145, 8611832136, 6734302429, -364364045001, -4974309134233, -41550393316275, -223452696895652, -173393115915136, 14282249293678744
Offset: 0
-
With[{nn=30},CoefficientList[Series[Exp[1+x+x^2/2!+x^3/3!-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2022 *)
-
my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2+x^3/6)))
A293040
E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).
Original entry on oeis.org
1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 125, 461, 1253, 3002, 6720, -111684, -978758, -5246983, -22948029, -89534309, 164027151, 5722510249, 55413784239, 393256686307, 2377996545081, 7807749195198, -46231762188586, -1125536160278906, -12849721017510166
Offset: 0
-
seq(factorial(n)*coeftayl(exp(1+x+x^2/2!+x^3/3!+x^4/4!-exp(x)), x = 0, n),n=0..50); # Muniru A Asiru, Oct 06 2017
-
my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2+x^3/6+x^4/24)))
Showing 1-5 of 5 results.
Comments