This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A229049 #23 Apr 23 2025 04:50:01
%S A229049 1,721,7489441,137322405361,3249278494922881,88913838899388373921,
%T A229049 2672932604015450235911761,85821373873727899097881489201,
%U A229049 2892941442791065880984595547275841,101239016762863657789924022384195015281,3649717311362112161867915447690005522733041
%N A229049 G.f.: Sum_{n >= 0} (6*n)!/n!^6 * x^n / (1-x)^(6*n+1).
%C A229049 Diagonal of the rational function 1 / (1 - x - y - z - u - v - w - x*y*z*u*v*w). - _Ilya Gutkovskiy_, Apr 22 2025
%H A229049 Alois P. Heinz, <a href="/A229049/b229049.txt">Table of n, a(n) for n = 0..100</a>
%F A229049 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) * C(n+5*k,k).
%F A229049 Recurrence: n^5*(2*n - 5)*(2*n - 3)*(3*n - 10)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 7)*a(n) = (2*n - 5)*(3*n - 10)*(3*n - 7)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 5)*(839916*n^8 - 6159384*n^7 + 18804591*n^6 - 30967129*n^5 + 29803190*n^4 - 16984623*n^3 + 5534242*n^2 - 929843*n + 60482)*a(n-1) - (3*n - 10)*(6*n - 25)*(6*n - 19)*(58320*n^12 - 1030320*n^11 + 7973640*n^10 - 35550360*n^9 + 101096973*n^8 - 191892891*n^7 + 247426961*n^6 - 216687345*n^5 + 127127767*n^4 - 48662719*n^3 + 11593839*n^2 - 1535715*n + 84350)*a(n-2) + 10*(6*n - 25)*(6*n - 1)*(23328*n^13 - 618192*n^12 + 7344432*n^11 - 51616440*n^10 + 238504338*n^9 - 761904909*n^8 + 1722993100*n^7 - 2778206390*n^6 + 3175831572*n^5 - 2526793076*n^4 + 1352618106*n^3 - 459806772*n^2 + 89082095*n - 7435050)*a(n-3) - 5*(3*n - 1)*(6*n - 7)*(6*n - 1)*(11664*n^12 - 369360*n^11 + 5235192*n^10 - 43777800*n^9 + 239670873*n^8 - 901183065*n^7 + 2374616540*n^6 - 4392523494*n^5 + 5622136222*n^4 - 4816276070*n^3 + 2596763070*n^2 - 784074950*n + 100205500)*a(n-4) + (2*n - 1)*(3*n - 4)*(3*n - 1)*(6*n - 13)*(6*n - 7)*(6*n - 1)*(648*n^9 - 19548*n^8 + 256338*n^7 - 1909293*n^6 + 8851093*n^5 - 26285080*n^4 + 49492875*n^3 - 56141750*n^2 + 34024625*n - 8063750)*a(n-5) - (n-5)^5*(2*n - 3)*(2*n - 1)*(3*n - 7)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(6*n - 19)*(6*n - 13)*(6*n - 7)*(6*n - 1)*a(n-6). - _Vaclav Kotesovec_, Sep 23 2013
%F A229049 a(n) ~ c*d^n/(n^(5/2)), where d = 46661.9996785484656481246... is the root of the equation 1 - 6*d + 15*d^2 - 20*d^3 + 15*d^4 - 46662*d^5 + d^6 = 0 and c = 0.024758197509539176365175770882978221... - _Vaclav Kotesovec_, Sep 23 2013
%F A229049 exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 361*x^2 + 2496841*x^3 + 34333162981*x^4 + ... appears to have integer coefficients. - _Peter Bala_, Jan 13 2016
%e A229049 G.f.: A(x) = 1 + 721*x + 7489441*x^2 + 137322405361*x^3 + 3249278494922881*x^4 +...
%e A229049 where
%e A229049 A(x) = 1/(1-x) + (6!/1!^6)*x/(1-x)^7 + (12!/2!^6)*x^2/(1-x)^13 + (18!/3!^6)*x^3/(1-x)^19 + (24!/4!^6)*x^4/(1-x)^25 + (30!/5!^6)*x^5/(1-x)^31 +...
%e A229049 Equivalently,
%e A229049 A(x) = 1/(1-x) + 720*x/(1-x)^7 + 7484400*x^2/(1-x)^13 + 137225088000*x^3/(1-x)^19 + 3246670537110000*x^4/(1-x)^25 + 88832646059788350720*x^5/(1-x)^31 +...+ A008979(n)*x^n/(1-x)^(6*n+1) +...
%p A229049 with(combinat):
%p A229049 a:= n-> add(multinomial(n+5*k, n-k, k$6), k=0..n):
%p A229049 seq(a(n), n=0..15); # _Alois P. Heinz_, Sep 23 2013
%t A229049 Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k] *Binomial[n+3*k,k] *Binomial[n+4*k,k]*Binomial[n+5*k,k], {k, 0, n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 23 2013 *)
%o A229049 (PARI) {a(n)=polcoeff(sum(m=0, n, (6*m)!/m!^6*x^m/(1-x+x*O(x^n))^(6*m+1)), n)}
%o A229049 for(n=0,15,print1(a(n),", "))
%o A229049 (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)*binomial(n+5*k,k))}
%o A229049 for(n=0,15,print1(a(n),", "))
%Y A229049 Cf. A008979, A001850, A081798, A082488, A082489, A229050.
%Y A229049 Column k = 6 of A229142.
%K A229049 nonn,easy
%O A229049 0,2
%A A229049 _Paul D. Hanna_, Sep 22 2013