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A186182
Expansion of 1/(1-x*A002294(x)).
1
1, 1, 2, 8, 50, 388, 3363, 31132, 301156, 3007000, 30753169, 320492869, 3391067666, 36331532588, 393353506931, 4296895624750, 47300050998991, 524168531729460, 5842914510975105, 65470405191871331, 737008925038212059, 8331166456981245521
OFFSET
0,3
LINKS
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
a(n) = Sum_{k=1..n} k/(4*n-3*k)*binomial(5*n-4*k-1,n-k), n>0; a(0) = 1.
From Vaclav Kotesovec, Sep 22 2024: (Start)
Recurrence: 8*(n-1)*(2*n-3)*(4*n-5)*(4*n-3)*(302869201*n^9 - 8459245881*n^8 + 104437088286*n^7 - 748013787426*n^6 + 3425181067929*n^5 - 10398368877609*n^4 + 20929298529704*n^3 - 26931712087164*n^2 + 20105202695760*n - 6634504195200)*a(n) = 5*(5*n-21)*(5*n-19)*(5*n-18)*(5*n-17)*(302869201*n^9 - 5733423072*n^8 + 47666412474*n^7 - 228372041208*n^6 + 694720947369*n^5 - 1391357951688*n^4 + 1834379283596*n^3 - 1535202635232*n^2 + 740132688960*n - 156635942400)*a(n-1) + 8*(n-1)*(2*n-3)*(4*n-5)*(4*n-3)*(302869201*n^9 - 8459245881*n^8 + 104437088286*n^7 - 748013787426*n^6 + 3425181067929*n^5 - 10398368877609*n^4 + 20929298529704*n^3 - 26931712087164*n^2 + 20105202695760*n - 6634504195200)*a(n-3) - 3*(341333589527*n^13 - 11542804387321*n^12 + 178153937603069*n^11 - 1661124543043265*n^10 + 10436085419810511*n^9 - 46636299048022863*n^8 + 152460394393134247*n^7 - 369062312013610715*n^6 + 661348648146586462*n^5 - 866258572340414716*n^4 + 805991085205781784*n^3 - 504373614185279520*n^2 + 190252933034572800*n - 32669988422400000)*a(n-4) + 5*(5*n-21)*(5*n-19)*(5*n-18)*(5*n-17)*(302869201*n^9 - 5733423072*n^8 + 47666412474*n^7 - 228372041208*n^6 + 694720947369*n^5 - 1391357951688*n^4 + 1834379283596*n^3 - 1535202635232*n^2 + 740132688960*n - 156635942400)*a(n-5).
a(n) ~ 5^(5*n + 7/2) / (314721 * sqrt(Pi) * n^(3/2) * 2^(8*n - 9/2)). (End)
MATHEMATICA
Join[{1}, Table[Sum[k/(4n-3k) Binomial[5n-4k-1, n-k], {k, n}], {n, 30}]] (* Harvey P. Dale, Feb 05 2012 *)
PROG
(PARI) a(n)=max(1, sum(k=1, n, k/(4*n-3*k)*binomial(5*n-4*k-1, n-k)))
CROSSREFS
Cf. A002294.
Sequence in context: A034491 A360949 A231352 * A274273 A121677 A120956
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Feb 14 2011
STATUS
approved