A063267 Eighth column (k=7) of septinomial array A063265.
6, 33, 116, 325, 786, 1709, 3424, 6426, 11430, 19437, 31812, 50375, 77506, 116265, 170528, 245140, 346086, 480681, 657780, 888009, 1184018, 1560757, 2035776, 2629550, 3365830, 4272021, 5379588, 6724491, 8347650
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).
Programs
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Maple
[seq((binomial(n+7,n)-binomial(n+1,n)),n=1..29)]; # Zerinvary Lajos, Jun 23 2006
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Mathematica
Table[Binomial[n+7,n]-Binomial[n+1,n],{n,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{6,33,116,325,786,1709,3424,6426},30] (* Harvey P. Dale, Jan 06 2012 *)
Formula
a(n)= A063265(n+2, 7)= (n+1)*(n+2)*(n+10)*(n^4 + 22*n^3 + 193*n^2 + 792*n + 1512)/7!.
G.f.: (2-x)*(1-x+x^2)*(3-3*x+x^2)/(1-x)^8; the numerator polynomial is N7(7, x) = 6 - 15*x + 20*x^2 - 15*x^3 + 6*x^4 - x^5 from row n=7 of array A063266.
a(n) = binomial(n+7,n) - binomial(n+1,n). - Zerinvary Lajos, Jun 23 2006
a(n) = binomial(n+7,n) + binomial(n+6,n) + binomial(n+5,n) + binomial(n+4,n) + binomial(n+3,n) + binomial(n+2,n). - Zerinvary Lajos, Jun 23 2006
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=6, a(1)=33, a(2)=116, a(3)=325, a(4)=786, a(5)=1709, a(6)=3424, a(7)=6426. - Harvey P. Dale, Jan 06 2012
Extensions
More terms from Zerinvary Lajos, Jun 23 2006