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Sequence b_4 (n) arising from homology of partitions with even number of blocks.
2

%I #29 Aug 05 2024 14:01:40

%S 6,138,1452,11444,78642,502846,3089624,18559208,110049502,647720562,

%T 3796113284,22194147996,129581349642,755982695718,4408534120368,

%U 25702339082192,149828229039030,873339353640538,5090437245730652

%N Sequence b_4 (n) arising from homology of partitions with even number of blocks.

%H Ray Chandler, <a href="/A003994/b003994.txt">Table of n, a(n) for n = 3..1000</a>

%H S. Sundaram, <a href="https://citeseerx.ist.psu.edu/pdf/381cddb91c261a5123684d7b056557f0f8c66098">The homology of partitions with an even number of blocks</a>, J. Alg. Comb., 4 (1995), 69-92.

%H S. Sundaram, <a href="http://citeseerx.ist.psu.edu/pdf/08428845a2d7735bf0d5467a3b51ef55de495e3b">Plethysm, partitions with an even number of blocks and Euler numbers</a>, DIMACS Series, Vol. 24 (1996), 171-198, Amer. Math. Soc.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (15,-85,235,-339,253,-87,9).

%F a(n) = 6*a(n-1) - a(n-2) - 8*n^2 + 24*n - 10 + 3^(n-3)*(32*n-80), with a(1)=a(2)=0. - _Sean A. Irvine_, Sep 26 2015

%F From _Colin Barker_, Jun 20 2019: (Start)

%F G.f.: 2*x^3*(3 + 24*x - 54*x^2 - 8*x^3 + 3*x^4) / ((1 - x)^3*(1 - 3*x)^2*(1 - 6*x + x^2)).

%F a(n) = 15*a(n-1) - 85*a(n-2) + 235*a(n-3) - 339*a(n-4) + 253*a(n-5) - 87*a(n-6) + 9*a(n-7) for n>9.

%F a(n) = (-12 + 16*3^n - 3*(3-2*sqrt(2))^n*(-2+sqrt(2)) + 6*(3+2*sqrt(2))^n + 3*sqrt(2)*(3+2*sqrt(2))^n - 16*(3+2*3^n)*n + 48*n^2) / 24.

%F (End)

%F Colin Barker's conjecture confirmed by Sean A. Irvine's formula. - _Ray Chandler_, Jul 06 2023

%Y Cf. A003993.

%K nonn

%O 3,1

%A Sheila Sundaram (sheila(AT)paris-gw.cs.miami.edu)

%E More terms added and incorrect Maple code deleted by _Sean A. Irvine_, Sep 26 2015