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}]
A386769
Expansion of (1/x) * Series_Reversion( x * (1-2*x) / (1+3*x)^2 ).
Original entry on oeis.org
1, 8, 89, 1162, 16646, 253218, 4016769, 65713602, 1100773166, 18786755128, 325518195674, 5711193510092, 101260078423336, 1811480526001238, 32657053453306929, 592701233703282882, 10820725155122336406, 198584549759713158048, 3661487133197990007534
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)/(1+3*x)^2)/x)
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a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(2*(n+1), k)*binomial(2*n-k, n-k))/(n+1);
A115189
Third diagonal (M=3) of triangle A115154 (called Y(3,1)).
Original entry on oeis.org
1, 40, 466, 4870, 50053, 516256, 5371210, 56422444, 598180570, 6395471992, 68897550064, 747270112390, 8154207352045, 89462667739360, 986322436120690, 10922076949690180, 121428345789834430, 1354895520307913680
Offset: 0
466 = a(2) = A064063(5) - 6*A064063(4) = 1606 - 6*190.
A115190
Fourth diagonal (M=4) of triangle A115154 (called Y(3,1)).
Original entry on oeis.org
1, 121, 1762, 20989, 235543, 2588641, 28259704, 308277688, 3369314986, 36937817182, 406378312834, 4487124168115, 49722977591155, 552884158181785, 6167643983348920, 69012419067802540, 774413615153430670
Offset: 0
a(0)= 1 = 190 - 9*(25-4); a(2) = 1762 = 14506 - 9*(1606-190).
A115191
Fifth diagonal (M=5) of triangle A115154 (called Y(3,1)).
Original entry on oeis.org
1, 364, 6379, 85384, 1039873, 12146074, 139010356, 1574773276, 17751401206, 199685662642, 2245313830945, 25260355535020, 284496154963705, 3208676674986850, 36246188218732000, 410128577783927380
Offset: 0
a(0)= 1 = 1606 - 3*(4*190-9*25); a(1) = 364 = 14506 - 3*(4*1606-9*190).