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A008298 Triangle of D'Arcais numbers. 8
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

Vincenzo Librandi, Rows n = 1..20, flattened

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

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;

...

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)

# The function bell_matrix is defined in 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

CROSSREFS

Diagonals give A038048, A059356, A059357.

Row sums give A053529.

Cf. A180563.

Sequence in context: A007023 A176103 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 November 21 12:51 EST 2017. Contains 295001 sequences.