This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A157077 #42 Feb 16 2025 08:33:09
%S A157077 1,0,2,-2,0,6,0,-12,0,20,6,0,-60,0,70,0,60,0,-280,0,252,-20,0,420,0,
%T A157077 -1260,0,924,0,-280,0,2520,0,-5544,0,3432,70,0,-2520,0,13860,0,-24024,
%U A157077 0,12870,0,1260,0,-18480,0,72072,0,-102960,0,48620,-252,0,13860,0,-120120,0,360360,0,-437580,0,184756
%N A157077 Triangle read by rows, coefficients of the Legendre polynomials P(n, x) times 2^n: T(n, k) = 2^n * [x^k] P(n, x), n >= 0, 0 <= k <= n.
%H A157077 Paul W. Haggard, <a href="https://doi.org/10.1155/S0161171288000481">Some applications of Legendre numbers</a>, International Journal of Mathematics and Mathematical Sciences, vol. 11, Article ID 538097, 8 pages, 1988. See Table 3 p. 412.
%H A157077 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>
%F A157077 Row sums are 2^n.
%F A157077 From _Peter Luschny_, Dec 19 2014: (Start)
%F A157077 T(n,0) = A126869(n).
%F A157077 T(n,n) = A000984(n).
%F A157077 T(n,1) = (-1)^floor(n/2)*A005430(floor(n/2)+1) if n is odd else 0.
%F A157077 Let Q(n, x) = 2^n*P(n, x).
%F A157077 Q(n,0) = (-1)^floor(n/2)*A126869(floor(n/2)) if n is even else 0.
%F A157077 Q(n,1) = A000079(n).
%F A157077 Q(n,2) = A069835(n).
%F A157077 Q(n,3) = A084773(n).
%F A157077 Q(n,4) = A098269(n).
%F A157077 Q(n,5) = A098270(n). (End)
%F A157077 From _Fabián Pereyra_, Jun 30 2022: (Start)
%F A157077 n*T(n,k) = 2*(2*n-1)*T(n-1,k-1) - 4*(n-1)*T(n-2,k).
%F A157077 T(n,k) = (-1)^floor((n-k)/2)*binomial(n+k,k)*binomial(n,floor((n-k)/2))*(1+(-1)^(n-k))/2.
%F A157077 O.g.f.: A(x,t) = 1/sqrt(1-4*x*t+4*x^2) = 1 + (2*t)*x + (-2+6*t^2)*x^2 + (-12*t+20*t^3)*x^3 + (6-60*t^2+70*t^4)*x^4 + .... (End)
%e A157077 The term order is Q(x) = a_0 + a_1*x + ... + a_n*x^n. The coefficients of the first few polynomials in this order are:
%e A157077 {1},
%e A157077 {0, 2},
%e A157077 {-2, 0, 6},
%e A157077 {0, -12, 0, 20},
%e A157077 {6, 0, -60, 0, 70},
%e A157077 {0, 60, 0, -280, 0, 252},
%e A157077 {-20, 0, 420, 0, -1260, 0, 924},
%e A157077 {0, -280, 0, 2520, 0, -5544, 0, 3432},
%e A157077 {70, 0, -2520, 0, 13860, 0, -24024, 0, 12870},
%e A157077 {0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620},
%e A157077 {-252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756}.
%e A157077 .
%e A157077 From _Jon E. Schoenfield_, Jul 04 2022: (Start)
%e A157077 As a right-aligned triangle:
%e A157077 1;
%e A157077 0, 2;
%e A157077 -2, 0, 6;
%e A157077 0, -12, 0, 20;
%e A157077 6, 0, -60, 0, 70;
%e A157077 0, 60, 0, -280, 0, 252;
%e A157077 -20, 0, 420, 0, -1260, 0, 924;
%e A157077 0, -280, 0, 2520, 0, -5544, 0, 3432;
%e A157077 70, 0, -2520, 0, 13860, 0, -24024, 0, 12870;
%e A157077 0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620;
%e A157077 -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756. (End)
%p A157077 with(orthopoly):with(PolynomialTools): seq(print(CoefficientList (2^n*P(n, x), x,termorder=forward)),n=0..10); # _Peter Luschny_, Dec 18 2014
%t A157077 Table[CoefficientList[2^n*LegendreP[n, x], x], {n, 0, 10}]; Flatten[%]
%o A157077 (PARI) tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ _Michel Marcus_, Dec 18 2014
%o A157077 (Sage)
%o A157077 def A157077_row(n):
%o A157077 if n==0: return [1]
%o A157077 T = [c[0] for c in (2^n*gen_legendre_P(n, 0, x)).coefficients()]
%o A157077 return [0 if is_odd(n+k) else T[k//2] for k in (0..n)]
%o A157077 for n in range(9): print(A157077_row(n)) # _Peter Luschny_, Dec 19 2014
%Y A157077 Cf. A100258, A126869, A000984, A005430, A000079, A069835, A084773, A098269, A098270.
%K A157077 tabl,sign
%O A157077 0,3
%A A157077 _Roger L. Bagula_, Feb 22 2009
%E A157077 Name clarified and edited by _Peter Luschny_, Dec 18 2014