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A253678
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Number of perfect matchings in the graph C_8 X C_n.
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8
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1058, 39952, 155682, 3113860, 19681538, 311853312, 2415542018, 33898728836, 294554220578, 3827188349968, 35866638601250, 442299574618756, 4365923647238658, 51942700201804032, 531410627302657538, 6169093269471927940, 64681086501382749218, 738453913359765339152, 7872683691901209561122, 88873260229652630182276
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OFFSET
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3,1
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REFERENCES
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S. N. Perepechko, Combinatorial properties of dimer problem on tori (in Russian). Mathematical physics and its applications, The fourth int. conf. Samara, 2014, 280-281.
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LINKS
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Eric Weisstein's World of Mathematics, Matching
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FORMULA
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a(n) = 14*a(n-1) + 145*a(n-2) - 2492*a(n-3) - 5832*a(n-4) + 164332*a(n-5) + 6360*a(n-6) - 5592188*a(n-7) + 5575094*a(n-8) + 111829704*a(n-9) - 176471286*a(n-10) - 1404071060*a(n-11) + 2757391176*a(n-12) + 11493707876*a(n-13) - 26094214040*a(n-14) - 62666476628*a(n-15) + 161092194209*a(n-16) + 229194775110*a(n-17) - 673504262865*a(n-18) - 556186915928*a(n-19) + 1946775340976*a(n-20) + 855365272888*a(n-21) - 3933950269712*a(n-22) - 705783359960*a(n-23) + 5586898052980*a(n-24) - 5586898052980*a(n-26) + 705783359960*a(n-27) + 3933950269712*a(n-28) - 855365272888*a(n-29) - 1946775340976*a(n-30) + 556186915928*a(n-31) + 673504262865*a(n-32) - 229194775110*a(n-33) - 161092194209*a(n-34) + 62666476628*a(n-35) + 26094214040*a(n-36) - 11493707876*a(n-37) - 2757391176*a(n-38) + 1404071060*a(n-39) + 176471286*a(n-40) - 111829704*a(n-41) - 5575094*a(n-42) + 5592188*a(n-43) - 6360*a(n-44) - 164332*a(n-45) + 5832*a(n-46) + 2492*a(n-47) - 145*a(n-48) - 14*a(n-49) + a(n-50).
G.f.: 2*x^3*(529 + 12570*x - 278528*x^2 - 1111096*x^3 + 29622124*x^4 + 15949216*x^5 - 1354335880*x^6 + 1073870160*x^7 + 33231636934*x^8 - 49093408612*x^9 - 484852497568*x^10 + 922702092728*x^11 + 4448623050276*x^12 - 9889298009728*x^13 - 26519860399096*x^14 + 66909591407824*x^15 + 104242913448099*x^16 - 300153880511538*x^17 - 268804327853184*x^18 + 917127529551440*x^19 + 437177534552376*x^20 - 1937370697752896*x^21 - 386856893695952*x^22 + 2851262465341600*x^23 + 31463729114724*x^24 - 2933939639544920*x^25 + 353114911609152*x^26 + 2113468417316080*x^27 - 452714140134072*x^28 - 1064902306141568*x^29 + 302352881352848*x^30 + 373692292484128*x^31 - 126783009087417*x^32 - 90391126093930*x^33 + 35100066280832*x^34 + 14772327002472*x^35 - 6497628908516*x^36 - 1572040067936*x^37 + 799287715544*x^38 + 101192826896*x^39 - 63992712074*x^40 - 3215530756*x^41 + 3212411488*x^42 - 3162664*x^43 - 94666796*x^44 + 3355392*x^45 + 1438440*x^46 - 83696*x^47 - 8091*x^48 + 578*x^49)/((1-x)*(1+x)*(1+4*x+x^2)*(1-4*x+x^2)*(1-2*x-x^2)*(1+2*x-x^2)*(1+8*x+16*x^2+8*x^3+x^4)* (1-14*x+34*x^2-14*x^3+x^4)*(1-8*x+16*x^2-8*x^3+x^4)*(1-4*x^2+x^4)*(1+4*x-4*x^2-4*x^3+x^4)*(1+8*x-10*x^2-8*x^3+x^4)*(1-4*x-4*x^2+4*x^3+x^4)*(1-8*x-10*x^2+8*x^3+x^4)*(1-14*x^2+34*x^4-14*x^6+x^8)).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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