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Doubly octagonal pyramidal numbers.
4

%I #6 Nov 28 2019 08:02:26

%S 0,1,765,27435,345415,2469420,12352956,48294610,157609530,447989355,

%T 1141711615,2663460261,5775482505,11777133550,22789550070,42150245460,

%U 74946834916,128723876325,214401953745,347453633935,549386792955,849592039296,1287617552320,1915941609990,2803320397950,4038796372975

%N Doubly octagonal pyramidal numbers.

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

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F G.f.: x*(1 + 755*x + 19830*x^2 + 105370*x^3 + 158255*x^4 + 70629*x^5 + 7930*x^6 + 110*x^7)/(1 - x)^10.

%F a(n) = A002414(A002414(n)).

%F a(n) = Sum_{k=0..A002414(n)} A000567(k).

%F a(n) = n *(2*n-1) *(n+1) *(2*n^3+n^2-n+2) *(2*n^3+n^2-n-1) /8 . - _R. J. Mathar_, Nov 28 2019

%t A002414[n_] := n (n + 1) (2 n - 1)/2; a[n_] := A002414[A002414[n]]; Table[a[n], {n, 0, 25}]

%t Table[Sum[k (3 k - 2), {k, 0, n (n + 1) (2 n - 1)/2}], {n, 0, 25}]

%t nmax = 25; CoefficientList[Series[x (1 + 755 x + 19830 x^2 + 105370 x^3 + 158255 x^4 + 70629 x^5 + 7930 x^6 + 110 x^7)/(1 - x)^10, {x, 0, nmax}], x]

%t LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355}, 26]

%Y Cf. A000567, A002414, A140236, A264892, A329753, A329754, A329755, A329756.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Nov 20 2019