A180563 -id:A180563 - cp's OEIS frontend

A294332 G.f.: exp( Sum_{n>=1} A180563(n) * x^n / n ).

1, 1, -1, 5, -45, 609, -11141, 257281, -7170355, 233936995, -8744103079, 368479396171, -17288353555771, 894005702731735, -50527305282004435, 3099060459670425655, -205028564671300495120, 14554510561318327509610, -1103542106915790217739110, 89009707681627448130203830, -7610129271299704960998906454, 687495658528174987634449288846, -65438091790081511530153327883206, 6545685493719560524729653911676430

Offset: 0

Paul D. Hanna, Oct 28 2017

sign

			G.f.: A(x) = 1 + x - x^2 + 5*x^3 - 45*x^4 + 609*x^5 - 11141*x^6 + 257281*x^7 - 7170355*x^8 + 233936995*x^9 - 8744103079*x^10 +...
such that
log(A(x)) = x - 3*x^2/2 + 19*x^3/3 - 207*x^4/4 + 3331*x^5/5 - 71223*x^6/6 + 1890379*x^7/7 - 59652687*x^8/8 + 2175761971*x^9/9 +...+ A180563(n)*x^n/n +...
where the e.g.f. G(x) of A180563 begins
G(x) = x - 3*x^2/2! + 19*x^3/3! - 207*x^4/4! + 3331*x^5/5! - 71223*x^6/6! + 1890379*x^7/7! +...+ A180563(n)*x^n/n! +...
and satisfies: Product_{n>=1} (1 - G(x)^n) = exp(-x).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A180653, A294331 (variant).

{A180563(n) = my( L = sum(m=1, n, sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(L), n)}
{a(n) = my(A); A = exp( sum(m=1, n+1, A180563(m)*x^m/m +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

A008298 Triangle of D'Arcais numbers.

Original entry on oeis.org

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 144, 450, 215, 30, 1, 1440, 3394, 2475, 565, 45, 1, 5760, 30912, 28294, 9345, 1225, 63, 1, 75600, 293292, 340116, 147889, 27720, 2338, 84, 1, 524160, 3032208, 4335596, 2341332, 579369, 69552, 4074, 108, 1, 6531840, 36290736, 57773700, 38049920, 11744775, 1857513, 154350, 6630, 135, 1

Offset: 1

N. J. A. Sloane

Also the Bell transform of A038048(n+1) and the inverse Bell transform of A180563(n+1) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Named after the Italian mathematician Francesco Flores D'Arcais (1849-1927). - Amiram Eldar, Jun 13 2021

			exp(Sum_{n>0} sigma(n)*u*x^n/n) = 1+u*x/1!+(3*u+u^2)*x^2/2!+(8*u+9*u^2+u^3)*x^3/3!+(42*u+59*u^2+18*u^3+u^4)*x^4/4!+...
Triangle starts:
      1:
      3,      1;
      8,      9,      1;
     42,     59,     18,      1;
    144,    450,    215,     30,     1;
   1440,   3394,   2475,    565,    45,    1;
   5760,  30912,  28294,   9345,  1225,   63,  1;
  75600, 293292, 340116, 147889, 27720, 2338, 84, 1;
  ...
T(4; u) = 4!*(binomial(u+3,4) + binomial(u+1,2)*binomial(u,1) + binomial(u+1,2) + binomial(u,1)^2 + binomial(u,1)) = 42*u+59*u^2+18*u^3+u^4.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.
F. D'Arcais, Développement en série, Intermédiaire Math., Vol. 20 (1913), pp. 233-234.

Seiichi Manyama, Rows n = 1..100, flattened (rows n = 1..20 from Vincenzo Librandi)
Peter Luschny, The Bell transform.

Column k=1..3 give A038048, A059356, A059357.
Row sums give A053529.
Cf. A075525, A180563, A210590.

P := proc(n): if n=0 then 1 else P(n):= (1/n)*(add(x(n-k) * P(k), k=0..n-1)) fi; end: with(numtheory): x := proc(n): sigma(n) * x end: Q := proc(n): n!*P(n) end: T := proc(n, k): coeff(Q(n), x, k) end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, Jul 08 2016

t[0][u_] = 1; t[n_][u_] := t[n][u] = Sum[(n-1)!/(n-k)!*DivisorSigma[1, k]*u*t[n-k][u], {k, 1, n}]; row[n_] := CoefficientList[ t[n][u], u] // Rest; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Oct 03 2012, after Vladeta Jovovic *)

row(n)={local(P(n)=if(n,sum(k=0,n-1,sigma(n-k)*x*P(k))/n,1)); Vecrev(P(n)*n!/x)} \\ T(n,k)=row(n)[k]. - M. F. Hasler, Jul 13 2016

a(n) = if(n<1, 0, (n-1)!*sigma(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1))) \\ Seiichi Manyama, Nov 08 2020 after Peter Luschny

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
print(bell_matrix(lambda n: A038048(n+1), 9)) # Peter Luschny, Jan 19 2016

G.f.: Sum_{1<=k<=n} T(n, k)*u^k*t^n/n! = ((1-t)*(1-t^2)*(1-t^3)...)^(-u).
Recurrence for degree n D'Arcais polynomials T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = Sum_{k=1..n} (n-1)!/(n-k)!*sigma(k)*u*T(n-k; u), T(0; u) = 1. - Vladeta Jovovic, Oct 11 2002
T(n; u) = n!*Sum_{pi} Product_{i=1..n} binomial(u+k(i)-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Oct 11 2002
E.g.f.: exp(Sum_{n>0} sigma(n)*u*x^n/n), where sigma(n)=A000203(n). - Vladeta Jovovic, Jan 10 2003
T(n, k) = coeff(n!*P(n), x^k), n >= 1 and 1 <= k <= n, with P(n) = (1/n)*Sum_{k=0..n-1} sigma(n-k)*P(k)*x for n >= 1 and P(n=0) = 1. See A036039. - Johannes W. Meijer, Jul 08 2016
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.

More terms from Vladeta Jovovic, Dec 28 2001

A294330 E.g.f. A(x) satisfies: Product_{n>=1} (1 - (-A(x))^n) = exp(x).

Original entry on oeis.org

1, 3, 19, 207, 3331, 71223, 1890379, 59652687, 2175761971, 89953773543, 4155502117339, 212122704251967, 11857607972675011, 720435277883199063, 47273215180877201899, 3331797538738820992047, 251025685429022007354451, 20133640365773761748643783, 1712740622904757368673592059

Offset: 1

Paul D. Hanna, Oct 28 2017

nonn

Unsigned version of A180563.

			E.g.f.: A(x) = x + 3*x^2/2! + 19*x^3/3! + 207*x^4/4! + 3331*x^5/5! + 71223*x^6/6! + 1890379*x^7/7! + 59652687*x^8/8! + 2175761971*x^9/9! + 89953773543*x^10/10! +...
such that  A( log(Q(x)) ) = x, where:
Q(x) = Product_{n>=1} (1 - (-x)^n);
log(Q(x)) = x - 3*x^2/2 + 4*x^3/3 - 7*x^4/4 + 6*x^5/5 - 12*x^6/6 + 8*x^7/7 - 15*x^8/8 + 13*x^9/9 - 18*x^10/10 +...+ (-1)^(n-1)*sigma(n)*x^n/n +...
and Q(x) = 1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 - x^40 - x^51 - x^57 - x^70 + x^77 + x^92 + x^100 +...+ A121373(n)*x^n +...
Also,
exp(3*x) = 1 + 3*A(x) - 5*A(x)^3 - 7*A(x)^6 + 9*A(x)^10 + 11*A(x)^15 - 13*A(x)^21 - 15*A(x)^28 + 17*A(x)^36 +...+ (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) +...
ALTERNATE GENERATING FUNCTION.
L.g.f.: L(x) = x + 3*x^2/2 + 19*x^3/3 + 207*x^4/4 + 3331*x^5/5 + 71223*x^6/6 + 1890379*x^7/7 + 59652687*x^8/8 + 2175761971*x^9/9 + 89953773543*x^10/10 +...
such that
exp(L(x)) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 732*x^5 + 12672*x^6 + 283704*x^7 + 7757526*x^8 + 249885110*x^9 + 9255184676*x^10 +...+ A294331(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A294331, A010815, A180563 (variant).

(* Calculation of constants {d,c}: *) eq = FindRoot[{E^r == QPochhammer[-s], (E^r*(Log[1 + s] + QPolyGamma[0, 1, -s]))/(s*Log[-s]) + Derivative[0, 1][QPochhammer][-s, -s] == 0}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 400]; {N[1/r/E /. eq, 120], val = s*E^r*Sqrt[-r*(1 + s) * (Log[-s]^2/(E^(2*r)*(1 + s)*QPolyGamma[1, 1, -s] + s*Log[-s]*(-s*(1 + s) * Log[-s] * Derivative[0, 1][QPochhammer][-s, -s]^2 + E^r*(1 + s)*((-2 - Log[-s]) * Derivative[0, 1][QPochhammer][-s, -s] + s*Log[-s] * Derivative[0, 2][QPochhammer][-s, -s]) + 2*E^(2*r)*(-1 + (1 + s) * Derivative[0, 0, 1][QPolyGamma][0, 1, -s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Sep 28 2023 *)

{a(n) = local( L = sum(m=1, n, (-1)^(m-1) * sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(L), n)}
for(n=1, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) Sum_{n>=1} (-1)^(n-1) * sigma(n) * A(x)^n / n = x.
(2) Sum_{n>=0} (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) = exp(3*x).
(3) A(x) = Series_Reversion( log(Q(x)) ) where Q(x) = Product_{n>=1} (1 - (-x)^n).
a(n) ~ c * d^n * n^(n-1), where d = 1.788680223969315995... and c = 0.254472375755339325... - Vaclav Kotesovec, Oct 29 2017

cp's OEIS Frontend

A294332 G.f.: exp( Sum_{n>=1} A180563(n) * x^n / n ).

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A008298 Triangle of D'Arcais numbers.

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A294330 E.g.f. A(x) satisfies: Product_{n>=1} (1 - (-A(x))^n) = exp(x).

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