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A220084 a(n) = (n + 1)*(20*n^2 + 19*n + 6)/6.

%I A220084 #55 Aug 07 2025 03:59:06
%S A220084 1,15,62,162,335,601,980,1492,2157,2995,4026,5270,6747,8477,10480,
%T A220084 12776,15385,18327,21622,25290,29351,33825,38732,44092,49925,56251,
%U A220084 63090,70462,78387,86885,95976,105680,116017,127007,138670,151026,164095,177897,192452
%N A220084 a(n) = (n + 1)*(20*n^2 + 19*n + 6)/6.
%C A220084 Sequence related to heptagonal pyramidal numbers (A002413) by a(n) = n*A002413(n) - (n-1)*A002413(n-1).
%C A220084 Other sequences of numbers of the form m*P(k,m)-(m-1)*P(k,m-1), where P(k,m) is the m-th k-gonal pyramidal number:
%C A220084 k=3, A002412(m) = m*A000292(m)-(m-1)*A000292(m-1);
%C A220084 k=4, A051662(m) = (m+1)*A000330(m+1)-m*A000330(m);
%C A220084 k=5, A213772(m) = m*A002411(m)-(m-1)*A002411(m-1);
%C A220084 k=6, A213837(m) = m*A002412(m)-(m-1)*A002412(m-1);
%C A220084 k=7, this sequence;
%C A220084 k=8, A130748(m) = m*A002414(m)-(m-1)*A002414(m-1).
%C A220084 Also, first bisection of A212983.
%C A220084 Binomial transform of (1, 14, 33, 20, 0, 0, 0, ...). - _Gary W. Adamson_, Aug 26 2015
%H A220084 Bruno Berselli, <a href="/A220084/b220084.txt">Table of n, a(n) for n = 0..1000</a>
%H A220084 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A220084 G.f.: (1+11*x+8*x^2)/(1-x)^4.
%F A220084 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3, a(0)=1, a(1)=15, a(2)=62, a(3)=162. - _Harvey P. Dale_, Dec 23 2012
%F A220084 a(n) = (n+1)*A000566(n+1) + Sum_{i=0..n} A000566(i). - _Bruno Berselli_, Dec 18 2013
%F A220084 E.g.f.: exp(x)*(6 + 84*x + 99*x^2 + 20*x^3)/6. - _Elmo R. Oliveira_, Aug 06 2025
%t A220084 Table[(n + 1) (20 n^2 + 19 n + 6)/6, {n, 0, 40}]
%t A220084 LinearRecurrence[{4,-6,4,-1},{1,15,62,162},40] (* _Harvey P. Dale_, Dec 23 2012 *)
%t A220084 CoefficientList[Series[(1 + 11 x + 8 x^2) / (1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o A220084 (Magma) [(n+1)*(20*n^2+19*n+6)/6: n in [0..40]]; // _Bruno Berselli_, Jun 28 2016
%o A220084 (Magma) /* By first comment: */  A002413:=func<n | n*(n+1)*(5*n-2)/6>; [n*A002413(n)-(n-1)*A002413(n-1): n in [1..40]];
%o A220084 (Magma) I:=[1,15,62,162]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%o A220084 (Maxima) makelist((n+1)*(20*n^2+19*n+6)/6, n, 0, 20); /* _Martin Ettl_, Dec 12 2012 */
%o A220084 (PARI) a(n)=(n+1)*(20*n^2+19*n+6)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A220084 Cf. A000292, A000330, A000566, A002411, A002412, A002413, A002414, A051662, A130748, A212983, A213772, A213837.
%K A220084 nonn,easy
%O A220084 0,2
%A A220084 _Bruno Berselli_, Dec 11 2012