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A082489
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a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k).
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7
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1, 121, 114121, 169417921, 308238414121, 629799991355641, 1387152264043496161, 3220175519103433952161, 7771784978946238318454761, 19326687177288750280293146161, 49215884415076728067274047737961, 127771596843320597524806463425540481
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} (5*n)!/n!^5 * x^n / (1-x)^(5*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^4*(5*n-16)*(5*n-12)*(5*n-11)*(5*n-8)*(5*n-7)*(5*n-6)*a(n) = (5*n-16)*(5*n-12)*(5*n-11)*(5*n-4)*(78250*n^6 - 422550*n^5 + 885665*n^4 - 906704*n^3 + 468906*n^2 - 114379*n + 10086)*a(n-1) - (5*n-16)*(31250*n^9 - 400000*n^8 + 2154375*n^7 - 6337750*n^6 + 11073100*n^5 - 11721380*n^4 + 7379043*n^3 - 2629646*n^2 + 489456*n - 36000)*a(n-2) + (5*n-1)*(31250*n^9 - 556250*n^8 + 4241875*n^7 - 18056500*n^6 + 46858025*n^5 - 76033760*n^4 + 76116292*n^3 - 44628880*n^2 + 13702848*n - 1693440)*a(n-3) - (5*n-6)*(5*n-2)*(5*n-1)*(625*n^7 - 11375*n^6 + 86025*n^5 - 347305*n^4 + 798274*n^3 - 1025292*n^2 + 661408*n - 156480)*a(n-4) + (n-4)^4*(5*n-11)*(5*n-7)*(5*n-6)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-5). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/n^2, where d = 3129.996806129131084... is the root of the equation -1 + 5*d - 10*d^2 + 10*d^3 - 3130*d^4 +d^5 = 0 and c = 0.05674890286773483081841276583916042181... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 61*x^2 + 38101*x^3 + 42394381*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016
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EXAMPLE
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G.f.: A(x) = 1 + 121*x + 114121*x^2 + 169417921*x^3 + 308238414121*x^4 +...
where
A(x) = 1/(1-x) + (5!/1!^5)*x/(1-x)^6 + (10!/2!^5)*x^2/(1-x)^11 + (15!/3!^5)*x^3/(1-x)^16 + (20!/4!^5)*x^4/(1-x)^21 + (25!/5!^5)*x^5/(1-x)^26 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 120*x/(1-x)^6 + 113400*x^2/(1-x)^11 + 168168000*x^3/(1-x)^16 + 305540235000*x^4/(1-x)^21 + 623360743125120*x^5/(1-x)^26 +...+ A008978(n)*x^n/(1-x)^(5*n+1) +...
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MAPLE
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with(combinat):
a:= n-> add(multinomial(n+4*k, n-k, k$5), k=0..n):
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MATHEMATICA
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Table[Sum[Binomial[n, k]*Binomial[n+k, k]*Binomial[n+2*k, k]* Binomial[n+3*k, k]*Binomial[n+4*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 23 2013 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, (5*m)!/m!^5*x^m/(1-x+x*O(x^n))^(5*m+1)), n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k) *binomial(n+3*k, k)*binomial(n+4*k, k))}
for(n=0, 15, print1(a(n), ", "))
(End)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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