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A008298 Triangle of D'Arcais numbers. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.

LINKS

Seiichi Manyama, Rows n = 1..100, flattened (rows n = 1..20 from Vincenzo Librandi)

Peter Luschny, The Bell transform.

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.

EXAMPLE

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.

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-Fran├žois Alcover, Oct 03 2012, after Vladeta Jovovic *)

PROG

(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

(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

CROSSREFS

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

Row sums give A053529.

Cf. A075525, A180563, A210590.

Sequence in context: A176103 A308666 A076238 * A039692 A071815 A178301

Adjacent sequences:  A008295 A008296 A008297 * A008299 A008300 A008301

KEYWORD

nonn,tabl,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Dec 28 2001

STATUS

approved

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Last modified March 1 17:42 EST 2021. Contains 341737 sequences. (Running on oeis4.)