|
|
COMMENTS
|
This sequence illustrates the following general binomial identity.
Given e.g.f.: Sum_{n>=0} exp((s*p^n + t*q^n)*x) * (u*p^n + v*q^n)^n * x^n/n! = Sum_{n>=0} a(n)*x^n/n!,
then a(n) = Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k = Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.
|
|
|
EXAMPLE
|
E.g.f.: A(x) = 1 + 12*x + 378*x^2/2! + 66324*x^3/3! + 106198818*x^4/4! +...
where
A(x) = exp((5+2)*x) + (2+3)*exp((5*2+2*3)*x)*x + (2^2+3^2)^2*exp((5*2^2+2*3^2)*x)*x^2/2! + (2^3+3^3)^3*exp((5*2^3+2*3^3)*x)*x^3/3! +...
or, equivalently,
A(x) = exp(7*x) + 5*exp(16*x)*x + 13^2*exp(38*x)*x^2/2! + 35^3*exp(94*x)*x^3/3! + 97^4*exp(242*x)*x^4/4! + 275^5*exp(646*x)*x^5/5! +...
Illustrate formula (1):
a(1) = 7 + 5 = 12 ;
a(2) = 7^2 + 2*5*16 + 13^2 = 378 ;
a(3) = 7^3 + 3*5*16^2 + 3*13^2*38 + 35^3 = 66324 ;
a(4) = 7^4 + 4*5*16^3 + 6*13^2*38^2 + 4*35^3*94 + 97^4 = 106198818 ;
a(5) = 7^5 + 5*5*16^4 + 10*13^2*38^3 + 10*35^3*94^2 + 5*97^4*242 + 275^5 = 1683766925772; ...
Illustrate formula (2):
a(1) = 7 + 5 = 12 ;
a(2) = 9^2 + 2*11*8 + 14*11^2 = 378 ;
a(3) = 13^3 + 3*17^2*14 + 3*23*20^2 + 29^3 = 66324 ;
a(4) = 21^4 + 4*29^3*26 + 6*41^2*38^2 + 4*59*56^3 + 83^4 = 106198818 ;
a(5) = 37^5 + 5*53^4*50 + 10*77^3*74^2 + 10*113^2*110^3 + 5*167*164^4 + 1*245^5 = 1683766925772; ...
|
|
|
PROG
|
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, exp((5*2^m+2*3^m+x*O(x^n))*x)*(2^m+3^m)^m*x^m/m!), n)}
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(5*2^k + 2*3^k)^(n-k)*(2^k + 3^k)^k)}
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(5 + 2^(n-k)*3^k)^(n-k)*(2 + 2^(n-k)*3^k)^k)}
(PARI) {a(n)=local(p=2, q=3, s=5, t=2, u=1, v=1); n!*polcoeff(sum(m=0, n, exp((s*p^m+t*q^m+x*O(x^n))*x)*(u*p^m+v*q^m)^m*x^m/m!), n)}
(PARI) {a(n)=local(p=2, q=3, s=5, t=2, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
(PARI) /* right side of the general binomial identity: */
{a(n)=local(p=2, q=3, s=5, t=2, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}
|
