Original entry on oeis.org
1, -3, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452
Offset: 0
G.f. = 1 - 3*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...
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m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 +Sqrt(1-4*x))/2 -2*x)); // G. C. Greubel, Aug 04 2018
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CoefficientList[Series[(1 +Sqrt[1-4*x])/2 -2*x, {x, 0, 50}], x] (* G. C. Greubel, Aug 04 2018 *)
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{a(n) = if( n<2, (n==0) - 3*(n==1), - binomial(2*n - 2, n-1) / n)};
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{a(n) = if( n<0, 0, polcoeff( (1 + sqrt(1 - 4*x + x * O(x^n))) / 2 - 2*x, n))};
A309867
Expansion of Product_{k>0} (1+sqrt(1-4*x^k))/2.
Original entry on oeis.org
1, -1, -2, -2, -5, -9, -36, -104, -365, -1219, -4213, -14617, -51570, -183084, -656536, -2370066, -8613590, -31478538, -115632718, -426676244, -1580878746, -5878933054, -21936060630, -82100980070, -308146839623, -1159545407027, -4373730398473, -16533813947503
Offset: 0
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nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1-4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2021 *)
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N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1-4*x^k))/2))
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N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, binomial(2*j-2, j-1)*x^(i*j)/j)))
A237621
Riordan array (1+x, x*(1-x)); inverse of Riordan array A237619.
Original entry on oeis.org
1, 1, 1, 0, 0, 1, 0, -1, -1, 1, 0, 0, -1, -2, 1, 0, 0, 1, 0, -3, 1, 0, 0, 0, 2, 2, -4, 1, 0, 0, 0, -1, 2, 5, -5, 1, 0, 0, 0, 0, -3, 0, 9, -6, 1, 0, 0, 0, 0, 1, -5, -5, 14, -7, 1, 0, 0, 0, 0, 0, 4, -5, -14, 20, -8, 1, 0, 0, 0, 0, 0, -1, 9, 0, -28, 27, -9, 1
Offset: 0
Triangles begins:
1;
1, 1;
0, 0, 1;
0, -1, -1, 1;
0, 0, -1, -2, 1;
0, 0, 1, 0, -3, 1;
0, 0, 0, 2, 2, -4, 1;
0, 0, 0, -1, 2, 5, -5, 1;
0, 0, 0, 0, -3, 0, 9, -6, 1;
0, 0, 0, 0, 1, -5, -5, 14, -7, 1;
...
Production matrix is:
1, 1;
-1, -1, 1;
0, -1, -1, 1;
-1, -2, -1, -1, 1;
-2, -5, -2, -1, -1, 1;
-6, -14, -5, -2, -1, -1, 1;
-18, -42, -14, -5, -2, -1, -1, 1;
-57, -132, -42, -14, -5, -2, -1, -1, 1;
-186, -429, -132, -42, -14, -5, -2, -1, -1, 1;
... (columns are A126983 and A115140)
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T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[n<2, 1, T[n-1,k-1] - T[n-2,k-1] ]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2022 *)
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def T(n,k): # T = A237621
if (k<0 or k>n): return 0
elif (n<2): return 1
else: return T(n-1, k-1) - T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2022