|
%I #7 Nov 28 2019 08:08:45
%S 0,1,126,3078,32800,213750,1008126,3783976,11985408,33297075,83338750,
%T 191592126,410450976,828497488,1589341950,2917620000,5154021376,
%U 8801526501,14585352318,23529456550,37052820000,57089119626,86233820926,127923156648,186649920000,268221484375,380065968126
%N Doubly pentagonal pyramidal numbers.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalPyramidalNumber.html">Pentagonal Pyramidal Number</a>
%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 + 116*x + 1863*x^2 + 7570*x^3 + 9350*x^4 + 3474*x^5 + 304*x^6 + 2*x^7)/(1 - x)^10.
%F a(n) = A002411(A002411(n)).
%F a(n) = Sum_{k=0..A002411(n)} A000326(k).
%F a(n) = n^4 *(n^3+n^2+2) *(n+1)^2 /16. - _R. J. Mathar_, Nov 28 2019
%t A002411[n_] := n^2 (n + 1)/2; a[n_] := A002411[A002411[n]]; Table[a[n], {n, 0, 26}]
%t Table[Sum[k (3 k - 1)/2, {k, 0, n^2 (n + 1)/2}], {n, 0, 26}]
%t nmax = 26; CoefficientList[Series[x (1 + 116 x + 1863 x^2 + 7570 x^3 + 9350 x^4 + 3474 x^5 + 304 x^6 + 2 x^7)/(1 - x)^10, {x, 0, nmax}], x]
%t LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075}, 27]
%Y Cf. A000326, A002411, A140236, A232713, A329753, A329755, A329756, A329757.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Nov 20 2019
|
