0,3

Also number of subpartitions of partition consisting of first n-1 triangular numbers; e.g. a(4) = subp([1,3,6]) = 37. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 26 2006

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(1 + k*(k+1)/2). G.f.: 1 = Sum_{k>=0} a(k)*x^k/(1+x)^((k+1)*(k+2)/2).

1 = 1*(1-x)^1 + 1*x*(1-x)^2 + 2*x^2*(1-x)^4 + 7*x^3*(1-x)^7 + 37*x^4*(1-x)^11 + 268*x^5*(1-x)^16 + 2496*x^6*(1-x)^22 +...

Also equals the final term in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated n+1 times, starting with a '1' in row 0, as illustrated by:

1;

1, 1;

1, 2,. 2,. 2;

1, 3,. 5,. 7,. 7,. 7,.. 7;

1, 4,. 9, 16, 23, 30,. 37,. 37,. 37,. 37,. 37;

1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268; ...

(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(1+k*(k+1)/2)), n)}

Cf. A107876, A107878, A107879.

Cf. A115728, A115729.

Sequence in context: A001515 A083659 A036247 this_sequence A001028 A116481 A102743

Adjacent sequences: A107874 A107875 A107876 this_sequence A107878 A107879 A107880

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2005, Apr 10 2007