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A247644 Triangle formed from the odd-numbered rows of A088855.

%I A247644 #20 Sep 29 2021 12:44:28
%S A247644 1,1,1,1,1,2,4,2,1,1,3,9,9,9,3,1,1,4,16,24,36,24,16,4,1,1,5,25,50,100,
%T A247644 100,100,50,25,5,1,1,6,36,90,225,300,400,300,225,90,36,6,1,1,7,49,147,
%U A247644 441,735,1225,1225,1225,735,441,147,49,7,1,1,8,64,224,784,1568,3136,3920,4900,3920,3136,1568,784,224,64,8,1
%N A247644 Triangle formed from the odd-numbered rows of A088855.
%C A247644 The rows give the coefficients in the numerator polynomials of the o.g.f.s for the columns of triangle A055898. - _Georg Fischer_, Aug 16 2021
%C A247644 They also occur (with a factor 2*x) in the numerator polynomials of the difference A157052-A157074. - _Georg Fischer_, Sep 27 2021
%H A247644 Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%e A247644 Triangle begins:
%e A247644 1,
%e A247644 1,1,1,
%e A247644 1,2,4,2,1,
%e A247644 1,3,9,9,9,3,1,
%e A247644 1,4,16,24,36,24,16,4,1,
%e A247644 1,5,25,50,100,100,100,50,25,5,1,
%e A247644 1,6,36,90,225,300,400,300,225,90,36,6,1,
%e A247644 1,7,49,147,441,735,1225,1225,1225,735,441,147,49,7,1,
%e A247644 1,8,64,224,784,1568,3136,3920,4900,3920,3136,1568,784,224,64,8,1,
%e A247644 ...
%t A247644 row[n_] := CoefficientList[Sum[Binomial[n, k]^2 *x^(2*k), {k, 0, n}] + Sum[Binomial[n, k]*Binomial[n, k - 1]* x^(2*k - 1), {k, 0, n}], x];
%t A247644 Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jun 07 2018 *)
%o A247644 (PARI) T(n, k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ A088855
%o A247644 row(n) = vector(2*n-1, k, T(2*n-1, k)); \\ _Michel Marcus_, Sep 27 2021
%Y A247644 Cf. A055899, A055900, A055901, A055902, A055903, A055904, A088855.
%Y A247644 Cf. A088459 (even numbered rows of A088855).
%K A247644 nonn,tabf
%O A247644 1,6
%A A247644 _N. J. A. Sloane_, Sep 23 2014
%E A247644 Row n=8 corrected by _Jean-François Alcover_, Jun 07 2018
%E A247644 Offset changed to 1 by _Georg Fischer_, Sep 27 2021