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A300842
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Number of 4-cycles in the n-Keller graph.
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4
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0, 40, 157392, 105912000, 45262336000, 15948007170048, 5068043562086400, 1511305636590272512, 431933148480384663552, 119840755593272159109120, 32552253553067741557030912, 8707042875228940141848428544, 2302822994847503802047542591488
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OFFSET
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1,2
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1236, -683808, 225640576, -49921773312, 7890676263936, -926734981152768, 83012813923221504, -5773763810692104192, 315681896111553642496, -13679828285436878389248, 472165430975367085031424, -13008202099499582296686592, 285926854862878421010087936, -4999088578991392797405216768, 69118696176694460893882220544, -748903433664660598884623450112, 6275842948939338662562251145216, -39926474608735258307470565572608, 187801979724765724477135723167744, -628096640746801659752846947516416, 1403466830787796679188111089991680, -1868233417954234085248005360844800, 1115613769753750842093668514398208).
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FORMULA
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Terms satisfy an order-23 linear recurrence (with large coefficients).
a(n) = 2^(2*n - 3)*(21*(-2^(4*n + 1) - 3^n + 4^n + 3^n*4^(n + 1) - 3^n*4^(2*n + 1) + 6^(2*n + 1) - 2*9^n + 21^n - 4*27^n + 64^n) + (-63 - 14*3^n + 21*4^(n + 1) + 7*3^(n + 2)*4^(n + 1) - 21*4^(2*n + 1) + 72*7^n - 28*9^(n + 1))*n + 7*(3 + 9*2^(2*n + 1) - 28*3^n)*n^2 - 84*n^3)/21. - Eric W. Weisstein, Mar 20 2018
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MATHEMATICA
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Table[2^(2 n - 3) (21 (-2^(4 n + 1) - 3^n + 4^n + 3^n 4^(n + 1) - 3^n 4^(2 n + 1) + 6^(2 n + 1) - 2 9^n + 21^n - 4 27^n + 64^n) + (-63 - 14 3^n + 21 4^(n + 1) + 7 3^(n + 2) 4^(n + 1) - 21 4^(2 n + 1) + 72 7^n - 28 9^(n + 1)) n + 7 (3 + 9 2^(2 n + 1) - 28 3^n) n^2 - 84 n^3)/21, {n, 20}] (* Eric W. Weisstein, Mar 20 2018 *)
CoefficientList[Series[-8 x (5 + 13494 x - 7659024 x^2 + 1619423712 x^3 - 136239729408 x^4 - 6166229419008 x^5 + 2751756296921088 x^6 - 323809706190127104 x^7 + 22263967229708795904 x^8 - 1004774859520610533376 x^9 + 30166466190897816010752 x^10 - 557827773596112789700608 x^11 + 3836796315532570883260416 x^12 + 98004685735481941350678528 x^13 - 3439510519824231292370681856 x^14 + 52560369429605424136068268032 x^15 - 472805698361893718926628487168 x^16 + 2514590036498863236378789937152 x^17 - 6697147772007341308881068359680 x^18 + 1543470882179190947000434556928 x^19 + 24429568859168763958151501316096 x^20)/((-1 + 4 x)^4 (-1 + 12 x)^3 (-1 + 16 x)^3 (-1 + 28 x)^2 (-1 + 36 x)^2 (-1 + 48 x)^2 (-1 + 64 x)^2 (-1 + 84 x) (-1 + 108 x) (-1 + 144 x) (-1 + 192 x) (-1 + 256 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 20 2018 *)
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PROG
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(PARI) \\ Requires G function from A300818
seq(n)={my(q2=G(n, 2, [0..3]), q4=G(n, 4, [0..63]));
vector(n, n, q4[n] - 4*q2[n]^2/(4^n) + q2[n]/2)} \\ Andrew Howroyd, Mar 15 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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