A365668
G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 2 * A(x)).
Original entry on oeis.org
0, 1, 7, 73, 905, 12354, 179305, 2715192, 42414021, 678476755, 11058588574, 182999237590, 3066447596459, 51926183715280, 887204891847960, 15276037569668880, 264797324173666845, 4617195655522976361, 80930337327794271445, 1425171253004955494215, 25202145191953299213490
Offset: 0
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nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 x)/(1 + x)^5, {x, 0, 20}], x], x]
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 2^k, {k, 0, n - 1}], {n, 1, 20}]]
A366035
G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 3 * A(x)).
Original entry on oeis.org
0, 1, 8, 98, 1440, 23389, 404712, 7314724, 136476912, 2608808180, 50828498336, 1005682252458, 20152470321984, 408149824237302, 8341496306085040, 171812412714350280, 3562961488550366480, 74328284438252301996, 1558783863783469298016, 32844108784368485209320, 694957689921176181019520
Offset: 0
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nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^5, {x, 0, 20}], x], x]
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]
A366036
G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 4 * A(x)).
Original entry on oeis.org
0, 1, 9, 127, 2165, 40914, 824859, 17383720, 378373437, 8440227235, 191938302578, 4433259845898, 103716352560119, 2452629475989840, 58529969579982600, 1407775987050271920, 34092047564798908045, 830565580516900384329, 20342106952028722530603, 500573735323751221019425, 12370242700776737398052970
Offset: 0
-
nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 x)/(1 + x)^5, {x, 0, 20}], x], x]
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 4^k, {k, 0, n - 1}], {n, 1, 20}]]
A366038
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k) * n^k.
Original entry on oeis.org
1, 2, 25, 658, 27193, 1548526, 112916830, 10062563610, 1061196371665, 129369938790070, 17909387604206371, 2776290021986848588, 476539253976442601735, 89736215305419802692184, 18395742890606906720656524, 4078527943680251523126851306, 972490249766494185823234587681
Offset: 0
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A366038 := proc(n)
add(binomial(n+k,k)*binomial(n*(n+1),n-k)*n^k,k=0..n) ;
%/(n+1) ;
end proc:
seq(A366038(n),n=0..80) ; # R. J. Mathar, Oct 24 2024
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Unprotect[Power]; 0^0 = 1; Table[1/(n + 1) Sum[Binomial[n + k, k] Binomial[n (n + 1) , n - k] n^k, {k, 0, n}], {n, 0, 16}]
Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]
Showing 1-4 of 4 results.
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