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A270694 Alternating sum of centered heptagonal pyramidal numbers. 1

%I

%S 0,-1,8,-23,51,-94,157,-242,354,-495,670,-881,1133,-1428,1771,-2164,

%T 2612,-3117,3684,-4315,5015,-5786,6633,-7558,8566,-9659,10842,-12117,

%U 13489,-14960,16535,-18216,20008,-21913,23936,-26079,28347,-30742,33269

%N Alternating sum of centered heptagonal pyramidal numbers.

%C More generally, the ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers is -x*(1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^4).

%H Ilya Gutkovskiy, <a href="/A270694/b270694.txt">Table of n, a(n) for n = 0..500</a>

%H OEIS Wiki, <a href="http://oeis.org/wiki/Centered_pyramidal_numbers">Centered pyramidal numbers</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2,2,3,1).

%F G.f.: -x*(1 - 5*x + x^2)/((1 - x)*(1 + x)^4).

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

%F a(n) = ((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48.

%F E.g.f.: (1/48)*(9*exp(x) - (9 + 66*x - 126*x^2 + 28*x^3)*exp(-x)). - _G. C. Greubel_, Mar 28 2016

%t LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -23, 51}, 39]

%t Table[((-1)^n (2 n + 1) (14 n^2 + 14 n - 9) + 9)/48, {n, 0, 38}]

%o (PARI) x='x+O('x^100); concat(0, Vec(-x*(1-5*x+x^2)/((1-x)*(1+x)^4))) \\ _Altug Alkan_, Mar 21 2016

%o (MAGMA) [((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48 : n in [0..40]]; // _Wesley Ivan Hurt_, Mar 21 2016

%Y Cf. A004126 (centered heptagonal pyramidal numbers).

%Y Cf. A000330, A006323 (partial sums of centered heptagonal pyramidal numbers), A019298, A232599.

%K easy,sign

%O 0,3

%A _Ilya Gutkovskiy_, Mar 21 2016

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Last modified February 23 11:30 EST 2018. Contains 299576 sequences. (Running on oeis4.)