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A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)*P[n-1]+x*P[n-2].

%I A128966 #19 Feb 04 2025 11:35:44
%S A128966 0,1,1,1,2,1,1,4,4,1,1,6,10,6,1,1,8,20,20,8,1,1,10,34,50,34,10,1,1,12,
%T A128966 52,104,104,52,12,1,1,14,74,190,258,190,74,14,1,1,16,100,316,552,552,
%U A128966 316,100,16,1,1,18,130,490,1058,1362,1058,490,130,18,1,1,20,164
%N A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)*P[n-1]+x*P[n-2].
%C A128966 A variant of A008288 (they satisfy the same recurrence).
%H A128966 Reinhard Zumkeller, <a href="/A128966/b128966.txt">Rows n = 0..125 of triangle, flattened</a>
%F A128966 P[n](x) = (x+1) * ( ((x+1+sqrt(x^2+6x+1))/2)^n - ((x+1-sqrt(x^2+6x+1))/2)^n ) / sqrt(x^2+6x+1) - _Max Alekseyev_, Mar 10 2008
%F A128966 P[n](x) = (x+1) * (sqrt(x)*I)^(n-1) * U[n-1](-I*(x+1)/sqrt(x)/2), where U[n](t) is Chebyshev polynomial of the 2nd kind. - _Max Alekseyev_, Mar 10 2008
%e A128966 Triangle begins:
%e A128966 0
%e A128966 1, 1
%e A128966 1, 2, 1
%e A128966 1, 4, 4, 1
%e A128966 1, 6, 10, 6, 1
%e A128966 1, 8, 20, 20, 8, 1
%e A128966 1, 10, 34, 50, 34, 10, 1
%e A128966 1, 12, 52, 104, 104, 52, 12, 1
%e A128966 1, 14, 74, 190, 258, 190, 74, 14, 1
%e A128966 1, 16, 100, 316, 552, 552, 316, 100, 16, 1
%p A128966 P[0]:=0;
%p A128966 P[1]:=x+1;
%p A128966 for n from 2 to 14 do
%p A128966 P[n]:=expand((x+1)*P[n-1]+x*P[n-2]);
%p A128966 lprint(P[n]);
%p A128966 lprint(seriestolist(series(P[n],x,200)));
%p A128966 od:
%t A128966 t[n_, k_] := 2^(1-n)*Binomial[n, k]*Sum[Binomial[n, 2*m+1]*HypergeometricPFQ[{-k, -m, k-n}, {1/2-n/2, -n/2}, -1], {m, 0, (n-1)/2}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2014, after _Max Alekseyev_ *)
%o A128966 (PARI) { T(n,k) = sum(m=0,(n-1)\2, binomial(n,2*m+1) * sum(j=0,m, binomial(m,j) * binomial(n-2*j,k-j) * 2^(2*j+1-n) ) ) } \\ _Max Alekseyev_, Mar 10 2008
%o A128966 (Haskell)
%o A128966 a128966 n k = a128966_tabl !! n !! k
%o A128966 a128966_row n = a128966_tabl !! n
%o A128966 a128966_tabl = map fst $ iterate
%o A128966    (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
%o A128966                       zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([0], [1, 1])
%o A128966 -- _Reinhard Zumkeller_, Jul 20 2013
%Y A128966 Cf. A163271 (row sums), A110170 (central terms).
%Y A128966 Cf. A102413.
%K A128966 nonn,tabl
%O A128966 0,5
%A A128966 _N. J. A. Sloane_, May 10 2007