A103978
Expansion of (sqrt(1-12*x^2)+12*x^2+2*x-1)/(2*x*sqrt(1-12*x^2)).
Original entry on oeis.org
1, 3, 6, 9, 54, 54, 540, 405, 5670, 3402, 61236, 30618, 673596, 288684, 7505784, 2814669, 84440070, 28146690, 956987460, 287096238, 10909657044, 2975361012, 124965162504, 31241290626, 1437099368796, 331638315876, 16581915793800
Offset: 0
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rec:= -(n+1)*a(n)+2*(n-1)*a(n-1)+12*(2*n-3)*a(n-2)+24*(2-n)*a(n-3)+144*(4-n)*a(n-4):
f:= gfun:-rectoproc({rec=0,a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 9},a(n),remember):
map(f, [$0..30]); # Robert Israel, Sep 13 2020
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CoefficientList[Series[(Sqrt[1-12x^2]+12x^2+2x-1)/(2x Sqrt[1-12x^2]),{x,0,30}],x] (* Harvey P. Dale, Aug 06 2022 *)
Original entry on oeis.org
2, 3, 8, 46, 252, 1558, 9800, 64115, 428546, 2921527, 20220128, 141746372, 1004278856, 7180301580, 51739691584, 375370204876, 2739615168344, 20100885190508, 148179065429664, 1096966770610372, 8151826588836472, 60787793832205004, 454719634089674432
Offset: 1
Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
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with(numtheory):
b:= proc(n) option remember;
`if`(n=0, 1, add(add((2^d)*binomial(2*d-2,d-1),
d=divisors(j)) *b(n-j), j=1..n)/n)
end:
a:= proc(n) option remember;
`if`(n<1, -1, -add(a(n-i) *b(i), i=1..n))
end:
seq(a(n), n=1..30); # Alois P. Heinz, Jan 27 2012
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b[n_] := b[n] = If[n==0, 1, Sum[Sum[2^d*Binomial[2*d-2, d-1], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; a[n_] := a[n] = If[n<1, -1, -Sum[a[n-i]* b[i], {i, 1, n}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
A336524
Triangular array read by rows. T(n,k) is the number of unlabeled binary trees with n internal nodes and exactly k distinguished external nodes (leaves) for 0 <= k <= n+1 and n >= 0.
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 6, 6, 2, 5, 20, 30, 20, 5, 14, 70, 140, 140, 70, 14, 42, 252, 630, 840, 630, 252, 42, 132, 924, 2772, 4620, 4620, 2772, 924, 132, 429, 3432, 12012, 24024, 30030, 24024, 12012, 3432, 429
Offset: 0
Taylor series starts: (y + 1) + x*(y + 1)^2 + 2*x^2*(y + 1)^3 + 5*x^3*(y + 1)^4 + 14*x^4*(y + 1)^5 + ...
Triangle T(n, k) begins:
1, 1;
1, 2, 1;
2, 6, 6, 2;
5, 20, 30, 20, 5;
14, 70, 140, 140, 70, 14;
42, 252, 630, 840, 630, 252, 42;
...
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nn = 5; b[z_] := (1 - Sqrt[1 - 4 z])/(2 z);Map[Select[#, # > 0 &] &,Transpose[Table[CoefficientList[Series[D[v b[v z], {v, k}]/k! /. v -> 1, {z, 0, nn}], z], {k, 0, nn + 1}]]] // Grid
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T(n,m):=(binomial(n+m,n)*binomial(2*n+1,n+m))/(2*n+1); /* Vladimir Kruchinin, Oct 16 2020 */
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for(n=1,8,for(k=0,n,print1(binomial(n,k)*binomial(2*n-2,n-1)/n,", "));print()) \\ Hugo Pfoertner, Oct 16 2020
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