0,3

See A115728 for the definition of subpartitions of a partition.

G.f.: 1 = Sum_{n>=1} (1-x)^n * Sum_{k=n*(n-1)/2..n*(n+1)/2-1} a(k)*x^k.

The g.f. is illustrated by:

1 = (1)*(1-x)^1 + (x + 2*x^2)*(1-x)^2 +

(3*x^3 + 7*x^5 + 12*x^6)*(1-x)^3 +

(18*x^6 + 43*x^7 + 76*x^8 + 118*x^9)*(1-x)^4 +

(170*x^10 + 403*x^11 + 711*x^12 + 1107*x^13 + 1605*x^14)*(1-x)^5 + ...

When the sequence is put in the form of a triangle:

1;

1, 2;

3, 7, 12;

18, 43, 76, 118;

170, 403, 711, 1107, 1605;

2220, 5188, 9054, 13986, 20171, 27816;

37149, 85569, 147471, 225363, 322075, 440785, 585046; ...

then the columns of this triangle form column 0 (with offset)

of successive matrix powers of triangle H=A121412.

This sequence is embedded in table A121424 as follows.

Column 0 of successive powers of matrix H begin:

H^1: [1,1,3,18,170,2220,37149,758814,18301950,...];

H^2: 1, [2,7,43,403,5188,85569,1725291,41145705,...];

H^3: 1,3, [12,76,711,9054,147471,2938176,69328365,...];

H^4: 1,4,18, [118,1107,13986,225363,4441557,103755660,...];

H^5: 1,5,25,170, [1605,20171,322075,6285390,145453290,...];

H^6: 1,6,33,233,2220, [27816,440785,8526057,195579123,...];

H^7: 1,7,42,308,2968,37149, [585046,11226958,255436293,...];

H^8: 1,8,52,396,3866,48420,758814, [14459138,326487241,...];

H^9: 1,9,63,498,4932,61902,966477,18301950, [410368743,...];

the terms enclosed in brackets form this sequence.

(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))}

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121424, A121425; column 0 of H^n: A121413, A121417, A121421.

Sequence in context: A049623 A061577 A006488 this_sequence A023606 A157605 A056179

Adjacent sequences: A121427 A121428 A121429 this_sequence A121431 A121432 A121433

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 30 2006