OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..60
FORMULA
a(n) = Sum_{k=0..n} Sum_{m=k(k+1)/2..k(k+1)/2+k} [x^m] S(x)^(n-k) for n>=1 where S(x) = Sum_{n>=0} x^(n(n+1)/2).
EXAMPLE
Let coefficients in powers of the series:
S = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
form the following sequences:
S^1: [(1),(1,0),(1,0,0),(1,0,0,0),(1,0,0,0,0),...]
S^2: [(1),(2,1),(2,2,0),(3,2,0,2),(2,2,1,2,0),...]
S^3: [(1),(3,3),(4,6,3),(6,9,3,7),(9,6,9,9,6),...]
S^4: [(1),(4,6),(8,13,12),(14,24,18,20),...]
S^5: [(1),(5,10),(15,25,31),(35,55,60,60),...]
S^6: [(1),(6,15),(26,45,66),(82,120,156,170),...]
...
then the sums of the above grouped terms (in parenthesis)
form the initial terms of the rows of table A162430:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,4,7,7,9,12,11,15,14,19,19,19,22,23,27,26,...
1,6,13,25,39,52,81,97,129,154,187,234,250,321,...
1,10,33,76,157,264,425,626,897,1230,1629,2174,...
1,15,71,210,535,1086,1965,3431,5425,8181,12165,...
1,21,137,528,1622,3921,8254,16396,29136,48773,...
1,28,245,1219,4494,12936,31767,70826,141891,...
1,36,414,2621,11602,39622,112951,283574,637706,...
...
The antidiagonal sums of the above table forms this sequence.
MATHEMATICA
t[n_, k_] := Module[{s = Sum[x^(m*(m+1)/2), {m, 0, k+1}] + O[x]^((k+1)*(k+2)/2)}, Sum[Coefficient[s^n, x, m], {m, k*(k+1)/2, k*(k+1)/2+k}]]; Table[Sum[t[n-k+1, k], {k, 0, n}], {n, 0, 26}] (* Jean-François Alcover, Nov 18 2013 *)
PROG
(PARI) {a(n)=local(S=sum(m=0, n+1, x^(m*(m+1)/2))+O(x^((n+1)*(n+2)/2))); sum(k=0, n, sum(m=k*(k+1)/2, k*(k+1)/2+k, polcoeff(S^(n-k), m)))}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 03 2009
STATUS
approved