A301973 - OEIS

%I #5 Mar 29 2018 15:25:51

%S 0,1,4,15,50,140,336,714,1380,2475,4180,6721,10374,15470,22400,31620,

%T 43656,59109,78660,103075,133210,170016,214544,267950,331500,406575,

%U 494676,597429,716590,854050,1011840,1192136,1397264,1629705,1892100,2187255,2518146,2887924,3299920,3757650,4264820

%N a(n) = (n^2 - 3*n + 6)*binomial(n+2,3)/4.

%C For n > 2, a(n) is the n-th term of the partial sums of n-gonal pyramidal numbers (in other words, a(n) is the n-th 4-dimensional n-gonal number).

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F O.g.f.: x*(1 - 2*x + 6*x^2)/(1 - x)^6.

%F E.g.f.: exp(x)*x*(24 + 24*x + 24*x^2 + 10*x^3 + x^4)/24.

%F a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^5.

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

%t Table[(n^2 - 3 n + 6) Binomial[n + 2, 3]/4, {n, 0, 40}]

%t nmax = 40; CoefficientList[Series[x (1 - 2 x + 6 x^2)/(1 - x)^6, {x, 0, nmax}], x]

%t nmax = 40; CoefficientList[Series[Exp[x] x (24 + 24 x + 24 x^2 + 10 x^3 + x^4)/24, {x, 0, nmax}], x] Range[0, nmax]!

%t Table[SeriesCoefficient[x (1 - 3 x + n x)/(1 - x)^5, {x, 0, n}], {n, 0, 40}]

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 4, 15, 50, 140}, 41]

%Y Cf. A000292, A000332, A001296, A002415, A006484, A060354, A080852, A256859, A291288 (partial sums), A292551.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Mar 29 2018