|
| |
|
|
A181069
|
|
Expansion of l.g.f.: Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n.
|
|
3
|
|
|
|
1, 3, 28, 275, 3126, 37632, 475056, 6192531, 82754650, 1127504378, 15603575208, 218727171104, 3099183987004, 44315462038200, 638663235342528, 9267264584278419, 135279095477748642, 1985221072388231742
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * binomial(n,k).
Recurrence: (n-1)^2*n^3*(10*n^2 - 25*n + 16)*a(n) = 2*(n-1)^2*(60*n^5 - 240*n^4 + 341*n^3 - 225*n^2 + 90*n - 16)*a(n-1) + 4*(n-2)^2*n*(4*n - 7)*(4*n - 5)*(10*n^2 - 5*n + 1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014
|
|
|
EXAMPLE
|
L.g.f.: L(x) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 +...
which equals the series:
L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
+ (1 + 2^4*x + 3^4*x^2 + 4^4*x^3 + 5^4*x^4 + 6^4*x^5 + ...)*x^2/2
+ (1 + 3^4*x + 6^4*x^2 + 10^4*x^3 + 15^4*x^4 + 21^4*x^5 + ...)*x^3/3
+ (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 + ...)*x^4/4
+ (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 + 70^4*x^4 + 126^4*x^5 + ...)*x^5/5
+ (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 + ...)*x^6/6
+ (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181068:
exp(L(x)) = 1 + x + 2*x^2 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Table[Sum[Binomial[n-1, k]^3 * Binomial[n, k], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)
|
|
|
PROG
|
(PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^4*n/(n-k))}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1, k)^4*x^k)*x^m/m)+x*O(x^n), n)}
for(n=1, 20, print1(a(n), ", "))
(Magma) [(&+[Binomial(n, k)*Binomial(n-1, k)^3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Apr 05 2021
(Sage) [sum( binomial(n, k)*binomial(n-1, k)^3 for k in (0..n-1) ) for n in (1..20)] # G. C. Greubel, Apr 05 2021
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
