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A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1. 6
1, 1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

First differences of A025192, also second differences of A000244.

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

P. Ribenhoim, The Little Book of Big Primes, Springer-Verlag, N.Y., 1991, p. 53.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = 4*3^(n-2); n >= 2; a(0) = 1; a(1) = 1.

G.f.: (1-x)^2/(1-3*x).

G.f.: 1/(1-sum(j>=1, (2*j-1)*x^j )). - Joerg Arndt, Jul 06 2011

a(n) = 3*a(n-1)+(-1)^n*C(2, 2-n).

a(n) = A003946(n-1), n>0. - R. J. Mathar, Oct 13 2008

a(n) = (-4*n + 9) * a(n-1) + 3 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011

a(n) = Sum_{k, 0<=k<=n} A201780(n,k). - Philippe Deléham, Dec 05 2011

EXAMPLE

1 + x + 4*x^2 + 12*x^3 + 36*x^4 + 108*x^5 + 324*x^6 + 972*x^7 + 2916*x^8 + ...

MATHEMATICA

CoefficientList[Series[(1 - x)^2/(1 - 3 x), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)

PROG

(PARI) {a(n) = local(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-4*k + 9) * A[k-1] + 3 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

CROSSREFS

Cf. A025192, A000244, A003462.

Sequence in context: A170541 A170589 A170637 A170685 A177881 A290899 A290905

Adjacent sequences:  A052153 A052154 A052155 * A052157 A052158 A052159

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, Jan 24 2000

EXTENSIONS

New name from Joerg Arndt, Jul 06 2011

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.