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A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x. 4
1, 1, 2, 9, 70, 805, 12480, 245847, 5909338, 168310515, 5556486450, 209003251240, 8835266400450, 415094928861530, 21473740362658640, 1213683089969940075, 74446121738526773490, 4927385997649620215895, 350145746700442604768346 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..18.

FORMULA

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

G.f.: Sum_{n>=0} a(n)*x^n*C(-x)^(n^2+2n) = 1/C(-x) where C(x) is the Catalan function of A000108.

a(n) = number of subpartitions of partition consisting of the first n square numbers starting with zero for n>0; e.g., a(4) = subp([0,1,4,9]) = 70. See A115728 for the definition of subpartitions.

EXAMPLE

1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^6 + 9*x^3/(1+x)^12 + 70*x^4/(1+x)^20 + 805*x^5/(1+x)^30 +...

1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^4 + 9*x^3*(1-x)^9 + 70*x^4*(1-x)^16 + 805*x^5*(1-x)^25 +...

Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 2(n+1) times, starting with a '1' in row 0, as illustrated by:

1;

1, 1;

1, 2,. 2,. 2,. 2;

1, 3,. 5,. 7,. 9,. 9,.. 9,.. 9,.. 9,.. 9;

1, 4,. 9, 16, 25, 34,. 43,. 52,. 61,. 70,. 70,. 70,. 70,. 70,. 70,. 70,. 70;

1, 5, 14, 30, 55, 89, 132, 184, 245, 315, 385, 455, 525, 595, 665, 735, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805; ...

PROG

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

CROSSREFS

Cf. A107877, A177447, A177448, A177449.

Sequence in context: A101482 A099717 A322772 * A193469 A336606 A121879

Adjacent sequences:  A177447 A177448 A177449 * A177451 A177452 A177453

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 09 2010

STATUS

approved

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Last modified November 28 16:25 EST 2021. Contains 349413 sequences. (Running on oeis4.)