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 A277930 Array of coefficients a(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = ((2*k+1)*x+sqrt(1+4*k*(k+1)*x^2))/(1-x^2), k>=0. 0
 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 3, 1, 1, 9, 25, 5, -3, 1, 1, 11, 41, 7, -59, 3, 1, 1, 13, 61, 9, -263, 5, 29, 1, 1, 15, 85, 11, -759, 7, 805, 3, 1, 1, 17, 113, 13, -1739, 9, 6649, 5, -131, 1, 1, 19, 145, 15, -3443, 11, 31241, 7, -12155, 3, 1, 1, 21, 181, 17, -6159, 13, 106261, 9, -200711, 5, 765, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The A(k,x) satisfy A(k,x)^2 = 1+(4*k+2)*x*A(k,x)+x^2*A(k,x)^2 for k>=0. The terms of odd-numbered columns a(k,2*n+1) are simple with (2*k+1)*x/(1-x^2), analogous the even-numbered columns a(k,2*n) with the o.g.f. of A000108. LINKS FORMULA a(k,0) = 1 and a(k,2*n+2) = 1-2*(Sum_{i=0..n} A000108(i)*(-k*(k+1))^(i+1)) and a(k,2*n+1) = 2*k+1 for k >= 0 and n >= 0. A(k,x) = (1+(2*k+1)*x+2*k*(k+1)*x^2*C(-k*(k+1)*x^2))/(1-x^2) for k >= 0, where C is the o.g.f. of A000108. A(k,x)*A(k,-x) = 1/(1-x^2) for k >= 0. Conjecture: a(k,2*n+2) = 1+2*k+2*(-k)^(n+2)*(Sum_{i=0..n} A234950(n,i)*k^i) for k>=0 and n>=0. - Werner Schulte, Aug 03 2017 EXAMPLE The terms define the array a(k,n) for k >= 0 and n >= 0, i.e., k\n  0   1    2   3       4   5        6   7           8   9         10  11  ... 0:   1   1    1   1       1   1        1   1           1   1          1   1  ... 1:   1   3    5   3      -3   3       29   3        -131   3        765   3  ... 2:   1   5   13   5     -59   5      805   5      -12155   5     205573   5  ... 3:   1   7   25   7    -263   7     6649   7     -200711   7    6766585   7  ... 4:   1   9   41   9    -759   9    31241   9    -1568759   9   88031241   9  ... 5:   1  11   61  11   -1739  11   106261  11    -7993739  11  672406261  11  ... 6:   1  13   85  13   -3443  13   292909  13   -30824051  13  ... 7:   1  15  113  15   -6159  15   696305  15   -97648655  15  ... 8:   1  17  145  17  -10223  17  1482769  17  -267255791  17  ... 9:   1  19  181  19  -16019  19  2899981  19  ... 10:  1  21  221  21  -23979  21  5300021  21  ... etc. The formal power series corresponding to row 2 is A(2,x) = 1+5*x+13*x^2+5*x^3 .. The terms define the triangle T(k,n) = a(k-n,n) for 0 <= n <=k, i.e., k\n  0  1   2  3   4  5  ... 0:   1 1:   1  1 2:   1  3   1 3:   1  5   5  1 4:   1  7  13  3   1 5:   1  9  25  5  -3  1 etc. MATHEMATICA A[k_, n_]:=If[n==0, 1, If[EvenQ[n], 1 - 2 Sum[CatalanNumber[i] (-k(k + 1))^(i + 1), {i, 0, (n - 2)/2}], 2k + 1]]; Table[A[n - k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Indranil Ghosh, Aug 03 2017 *) PROG (Python) from sympy import catalan def A(k, n): return 1 if n==0 else 1 - 2*sum([catalan(i)*(-k*(k + 1))**(i + 1) for i in xrange(n/2)]) if n%2==0 else 2*k + 1 for n in xrange(13): print [A(n - k, k) for k in xrange(n + 1)] # Indranil Ghosh, Aug 03 2017 CROSSREFS Cf. A000108, A234950. Sequence in context: A113245 A103450 A128254 * A026714 A008288 A238339 Adjacent sequences:  A277927 A277928 A277929 * A277931 A277932 A277933 KEYWORD sign,easy,tabl AUTHOR Werner Schulte, Nov 04 2016 STATUS approved

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Last modified January 20 21:36 EST 2019. Contains 319336 sequences. (Running on oeis4.)