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A361678
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Constant term in the expansion of (1 + w + x + y + z + 1/(x*y*z) + 1/(w*y*z) + 1/(w*x*z) + 1/(w*x*y))^n.
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1
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1, 1, 1, 1, 97, 481, 1441, 3361, 77281, 647137, 3195361, 11674081, 116286721, 1147935361, 7611379777, 37451144641, 263670781921, 2456043418081, 19073086806241, 115319128034017, 748239468100417, 6179458007222977, 50636218964639617, 350400618132423937
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OFFSET
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0,5
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/4)} (4*k)!/k!^4 * binomial(4*k,k) * binomial(n,4*k).
Recurrence: 3*n^4*(3*n - 8)*(3*n - 4)*a(n) = 3*(63*n^6 - 405*n^5 + 1015*n^4 - 1355*n^3 + 1049*n^2 - 439*n + 77)*a(n-1) - 3*(n-1)*(189*n^5 - 1485*n^4 + 4685*n^3 - 7575*n^2 + 6313*n - 2163)*a(n-2) + 3*(n-2)*(n-1)*(315*n^4 - 2610*n^3 + 8285*n^2 - 12030*n + 6749)*a(n-3) + (n-3)*(n-2)*(n-1)*(64591*n^3 - 385926*n^2 + 701651*n - 375786)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*(65347*n^2 - 326519*n + 391384)*a(n-5) + 3*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(65473*n - 196383)*a(n-6) - 65509*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
a(n) ~ sqrt(65563/12288 + 3*sqrt(3)/8 + sqrt(3/8 + 65563/(4096*sqrt(3)))) * (1 + 16/3^(3/4))^n / (Pi^2 * n^2). (End)
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MATHEMATICA
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Table[Sum[(4*k)!/k!^4 * Binomial[4*k, k] * Binomial[n, 4*k], {k, 0, n/4}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 22 2023 *)
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PROG
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(PARI) a(n) = sum(k=0, n\4, (4*k)!/k!^4*binomial(4*k, k)*binomial(n, 4*k));
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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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