|
%I #3 Mar 30 2012 18:36:58
%S 1,1,2,3,7,12,18,43,76,118,170,403,711,1107,1605,2220,5188,9054,13986,
%T 20171,27816,37149,85569,147471,225363,322075,440785,585046,758814,
%U 1725291,2938176,4441557,6285390,8526057,11226958,14459138,18301950
%N Number of subpartitions of partition P=[0,1,1,2,2,2,3,3,3,3,4,...] (A003056).
%C See A115728 for the definition of subpartitions of a partition.
%F G.f.: 1 = Sum_{n>=1} (1-x)^n * Sum_{k=n*(n-1)/2..n*(n+1)/2-1} a(k)*x^k.
%e The g.f. is illustrated by:
%e 1 = (1)*(1-x)^1 + (x + 2*x^2)*(1-x)^2 +
%e (3*x^3 + 7*x^5 + 12*x^6)*(1-x)^3 +
%e (18*x^6 + 43*x^7 + 76*x^8 + 118*x^9)*(1-x)^4 +
%e (170*x^10 + 403*x^11 + 711*x^12 + 1107*x^13 + 1605*x^14)*(1-x)^5 + ...
%e When the sequence is put in the form of a triangle:
%e 1;
%e 1, 2;
%e 3, 7, 12;
%e 18, 43, 76, 118;
%e 170, 403, 711, 1107, 1605;
%e 2220, 5188, 9054, 13986, 20171, 27816;
%e 37149, 85569, 147471, 225363, 322075, 440785, 585046; ...
%e then the columns of this triangle form column 0 (with offset)
%e of successive matrix powers of triangle H=A121412.
%e This sequence is embedded in table A121424 as follows.
%e Column 0 of successive powers of matrix H begin:
%e H^1: [1,1,3,18,170,2220,37149,758814,18301950,...];
%e H^2: 1, [2,7,43,403,5188,85569,1725291,41145705,...];
%e H^3: 1,3, [12,76,711,9054,147471,2938176,69328365,...];
%e H^4: 1,4,18, [118,1107,13986,225363,4441557,103755660,...];
%e H^5: 1,5,25,170, [1605,20171,322075,6285390,145453290,...];
%e H^6: 1,6,33,233,2220, [27816,440785,8526057,195579123,...];
%e H^7: 1,7,42,308,2968,37149, [585046,11226958,255436293,...];
%e H^8: 1,8,52,396,3866,48420,758814, [14459138,326487241,...];
%e H^9: 1,9,63,498,4932,61902,966477,18301950, [410368743,...];
%e the terms enclosed in brackets form this sequence.
%o (PARI) {a(n)=local(A); if(n==0,1,A=x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(1-(1-x)^( (sqrtint(8*k+1)+1)\2 ) )); polcoeff(A, n))}
%Y Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121424, A121425; column 0 of H^n: A121413, A121417, A121421.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 30 2006
|
