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A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*(2*k+1)*x + x^2).

%I A335333 #47 Aug 30 2025 10:12:52
%S A335333 1,1,1,1,3,1,1,5,13,1,1,7,37,63,1,1,9,73,305,321,1,1,11,121,847,2641,
%T A335333 1683,1,1,13,181,1809,10321,23525,8989,1,1,15,253,3311,28401,129367,
%U A335333 213445,48639,1,1,17,337,5473,63601,458649,1651609,1961825,265729,1
%N A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*(2*k+1)*x + x^2).
%H A335333 Seiichi Manyama, <a href="/A335333/b335333.txt">Antidiagonals n = 0..139, flattened</a>
%H A335333 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>.
%F A335333 T(n,k) is the coefficient of x^n in the expansion of (1 + (2*k+1)*x + k*(k+1)*x^2)^n.
%F A335333 T(n,k) = Sum_{j=0..n} k^j * (k+1)^(n-j) * binomial(n,j)^2.
%F A335333 T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(n+j,j).
%F A335333 n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (n-1) * T(n-2,k).
%F A335333 T(n,k) = P_n(2*k+1), where P_n is n-th Legendre polynomial.
%F A335333 From _Seiichi Manyama_, Aug 30 2025: (Start)
%F A335333 T(n,k) = (-1)^n * Sum_{j=0..n} (1/(2*(2*k+1)))^(n-2*j) * binomial(-1/2,j) * binomial(j,n-j).
%F A335333 T(n,k) = Sum_{j=0..floor(n/2)} (k*(k+1))^j * (2*k+1)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
%F A335333 E.g.f. of column k: exp((2*k+1)*x) * BesselI(0, 2*sqrt(k*(k+1))*x). (End)
%e A335333 Square array begins:
%e A335333   1,    1,     1,      1,      1,       1, ...
%e A335333   1,    3,     5,      7,      9,      11, ...
%e A335333   1,   13,    37,     73,    121,     181, ...
%e A335333   1,   63,   305,    847,   1809,    3311, ...
%e A335333   1,  321,  2641,  10321,  28401,   63601, ...
%e A335333   1, 1683, 23525, 129367, 458649, 1256651, ...
%t A335333 T[n_, k_] := LegendreP[n, 2*k + 1]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 03 2021 *)
%o A335333 (PARI) T(n, k) = pollegendre(n, 2*k+1);
%Y A335333 Columns k=0..4 give A000012, A001850, A006442, A084768, A084769.
%Y A335333 Rows n=0..6 give A000012, A005408, A003154(n+1), A160674, A144124, A335338, A144126.
%Y A335333 Main diagonal gives A331656.
%Y A335333 T(n,n-1) gives A331657.
%Y A335333 Cf. A307883, A307884.
%K A335333 nonn,tabl,changed
%O A335333 0,5
%A A335333 _Seiichi Manyama_, Jun 02 2020