A008298 Triangle of D'Arcais numbers.

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

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy

Comments

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

Examples

			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.
		

References

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

Links

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

Crossrefs

Column k=1..3 give A038048, A059356, A059357.

Row sums give A053529.

Cf. A075525, A180563, A210590.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Sage

# 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

Formula

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.

Extensions

More terms from Vladeta Jovovic, Dec 28 2001