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A183165 G.f.: Sum_{n>=0} [Sum_{k>=0} C(n+k-1,k)^2*x^k]^n * x^n. 4
1, 1, 2, 10, 63, 521, 5295, 64048, 907199, 14717173, 270429934, 5561319631, 126824201866, 3183127838869, 87328494060529, 2604069098659922, 83975195990867113, 2915521294244073351, 108553405498985038390, 4319110373993534510736, 183057722816741327269600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
G.f.: Sum_{n>=0} [ Sum_{k=0..n-1} C(n-1,k)^2*x^k ]^n * x^n / (1-x)^(2*n^2-n).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 63*x^4 + 521*x^5 + 5295*x^6 +...
which equals the sum of the series:
A(x) = 1 + (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 + 6^2*x^5 +...)^2*x^2
+ (1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 + 21^2*x^5 +...)^3*x^3
+ (1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 35^2*x^4 + 56^2*x^5 +...)^4*x^4
+ (1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 + 126^2*x^5 +...)^5*x^5
+ (1 + 6^2*x + 21^2*x^2 + 56^2*x^3 + 126^2*x^4 + 252^2*x^5 +...)^6*x^6 +...
The g.f. can also be expressed as:
A(x) = 1 + x/(1-x) + (1+x)^2*x^2/(1-x)^6
+ (1 + 2^2*x + x^2)^3*x^3/(1-x)^15
+ (1 + 3^2*x + 3^2*x^2 + x^3)^4*x^4/(1-x)^28
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^5*x^5/(1-x)^45
+ (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^6*x^6/(1-x)^66
+ (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^7*x^7/(1-x)^91 +...
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=0, n, binomial(m+k-1, k)^2*x^k +x*O(x^n))^m*x^m) +x*O(x^n), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=0, m-1, binomial(m-1, k)^2*x^k)^m*x^m/(1-x+x*O(x^n))^(2*m^2-m)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A361829 A361494 A175962 * A129130 A245519 A303483
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 30 2010
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)