A301305 - OEIS

%I #16 Mar 26 2018 13:14:30

%S 1,-1,1,3,-5,2,-14,33,-25,6,85,-261,292,-140,24,-621,2363,-3516,2546,

%T -892,120,5236,-23872,44537,-43405,23228,-6444,720,-49680,264860,

%U -596396,733983,-532095,226644,-52356,5040,521721,-3193029,8448004,-12605668,11586756,-6707208,2383248,-474144,40320,-5994155,41506739,-126480376,222424796,-248535142,182793154,-88379152,27046632,-4745376,362880,74701055,-578419961,1998774636,-4056699966,5351696394,-4791391134,2945757656,-1226765624,330797184,-52079040,3628800

%N G.f. L(x,y) satisfies: L(x,y) = x * (1 + y*x*L'(x,y)) / (1 + x*L'(x,y)) where L'(x,y) = d/dx L(x,y), as a triangle read by rows.

%C Main diagonal equals the factorials.

%C Column 0 forms A088716, signed.

%C Rows sums are zeros after the initial row.

%C Absolute row sums equal A301388.

%H Paul D. Hanna, <a href="/A301305/b301305.txt">Table of n, a(n) for n = 1..1275 giving rows 1..50 as a flattened triangle read by rows.</a>

%F G.f. L(x,y) satisfies: [x^n] exp( -n * L(x,y) ) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp( -n * L(x,y) ) for n>=1.

%e G.f. L(x,y) = Sum_{n>=1, k=0..n-1} T(n,k) * x^n*y^k begins:

%e L(x,y) = x + (-1 + y)*x^2 + (3 - 5*y + 2*y^2)*x^3 + (-14 + 33*y - 25*y^2 + 6*y^3)*x^4 + (85 - 261*y + 292*y^2 - 140*y^3 + 24*y^4)*x^5 + (-621 + 2363*y - 3516*y^2 + 2546*y^3 - 892*y^4 + 120*y^5)*x^6 + (5236 - 23872*y + 44537*y^2 - 43405*y^3 + 23228*y^4 - 6444*y^5 + 720*y^6)*x^7 + (-49680 + 264860*y - 596396*y^2 + 733983*y^3 - 532095*y^4 + 226644*y^5 - 52356*y^6 + 5040*y^7)*x^8 + ...

%e where L = L(x,y) satisfies:

%e L = x*(1 + y*x*L') / (1 + x*L').

%e TRIANGLE.

%e This triangle of coefficients T(n,k) in L(x,y) begins:

%e 1;

%e -1, 1;

%e 3, -5, 2;

%e -14, 33, -25, 6;

%e 85, -261, 292, -140, 24;

%e -621, 2363, -3516, 2546, -892, 120;

%e 5236, -23872, 44537, -43405, 23228, -6444, 720;

%e -49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040;

%e 521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320;

%e -5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880; ...

%e LIMITS.

%e In this triangle, the largest real root of the n-th row polynomial in y converges to the constant t = 2.845344903202547217277843362090557097661... (A301389).

%e RELATED SERIES.

%e exp(L(x,y)) = 1 + x + (-1 + 2*y)*x^2/2! + (13 - 24*y + 12*y^2)*x^3/3!  +  (-263 + 660*y - 540*y^2 + 144*y^3)*x^4/4!  +  (8381 - 26800*y + 31380*y^2 - 15840*y^3 + 2880*y^4)*x^5/5!  +  (-379409 + 1485870*y - 2280180*y^2 + 1706520*y^3 - 619200*y^4 + 86400*y^5)*x^6/6!  +  (22915369 - 106759128*y + 203726880*y^2 - 203269920*y^3 + 111449520*y^4 - 31691520*y^5 + 3628800*y^6)*x^7/7! + ...

%e satisfies: [x^n] exp(-n*L(x,y)) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp(-n*L(x,y)) for n>=1.

%o (PARI) {T(n,k) = my(L=x); for(i=0,n, L = x*(1 + y*x*L')/(1 + x*L' +x*O(x^n)) ); polcoeff(polcoeff(L,n,x),k,y)}

%o for(n=1,12, for(k=0,n-1, print1(T(n,k),", "));print(""))

%Y Cf. A088716, A301388, A301389.

%K sign,tabl

%O 1,4

%A _Paul D. Hanna_, Mar 20 2018