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A329523 a(n) = n * (binomial(n + 1, 3) + 1).

%I A329523 #14 Jun 17 2025 18:33:12
%S A329523 0,1,4,15,44,105,216,399,680,1089,1660,2431,3444,4745,6384,8415,10896,
%T A329523 13889,17460,21679,26620,32361,38984,46575,55224,65025,76076,88479,
%U A329523 102340,117769,134880,153791,174624,197505,222564,249935,279756,312169,347320,385359,426440
%N A329523 a(n) = n * (binomial(n + 1, 3) + 1).
%C A329523 The n-th centered n-gonal pyramidal number.
%D A329523 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 142.
%H A329523 Kelvin Voskuijl, <a href="/A329523/b329523.txt">Table of n, a(n) for n = 0..10000</a>
%H A329523 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A329523 G.f.: x * (1 - x + 5*x^2 - x^3) / (1 - x)^5.
%F A329523 E.g.f.: exp(x) * x * (1 + x + x^2 + x^3 / 6).
%F A329523 a(n) = n * (n + 2) * (n^2 - 2*n + 3) / 6.
%F A329523 a(n) = n * (A000292(n-1) + 1).
%F A329523 a(n) = n + 2 * Sum_{k=1..n} A000330(k-1).
%F A329523 a(n) + a(-n) = 4 * A002415(n).
%e A329523 Square array begins:
%e A329523   (0), 1,  2,   3,   4,    5,  ... A001477
%e A329523    0, (1), 3,   7,  14,   25,  ... A004006
%e A329523    0,  1, (4), 11,  24,   45,  ... A006527
%e A329523    0,  1,  5, (15), 34,   65,  ... A006003 (partial sums of A005448)
%e A329523    0,  1,  6,  19, (44),  85,  ... A005900 (partial sums of A001844)
%e A329523    0,  1,  7,  23,  54, (105), ... A004068 (partial sums of A005891)
%e A329523 ...
%e A329523 This sequence is the main diagonal of the array.
%t A329523 Table[n (Binomial[n + 1, 3] + 1), {n, 0, 40}]
%t A329523 nmax = 40; CoefficientList[Series[x (1 - x + 5 x^2 - x^3)/(1 - x)^5, {x, 0, nmax}], x]
%t A329523 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 4, 15, 44}, 41]
%o A329523 (Magma) [ n*(Binomial(n+1,3)+1):n in [0..40]]; // _Marius A. Burtea_, Nov 15 2019
%o A329523 (Magma) R<x>:=PowerSeriesRing(Integers(), 41); [0] cat Coefficients(R!(x*(1-x+5*x^2-x^3)/(1-x)^5)); // _Marius A. Burtea_, Nov 15 2019
%Y A329523 Cf. A000292, A000330, A002415, A002417, A006000, A006484, A008911, A050407, A060354, A100119, A188475 (first differences).
%K A329523 nonn,easy
%O A329523 0,3
%A A329523 _Ilya Gutkovskiy_, Nov 15 2019