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A136247 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial h(n,x) with h(0,x)=1, h(1,x)=1-x and recursively h(n,x) = 1 + n -(1-x)*(1-h(n-1,x)) - n*h(n-2,x).

%I A136247 #10 Jul 22 2015 02:52:57
%S A136247 1,1,-1,1,-1,1,1,2,2,-1,1,6,-4,-3,1,1,-4,-20,6,4,-1,1,-40,8,44,-8,-5,
%T A136247 1,1,-12,188,-6,-80,10,6,-1,1,308,136,-546,-10,130,-12,-7,1,1,416,
%U A136247 -1864,-628,1256,50,-196,14,8,-1,1,-2664,-3640,6696,1984,-2506,-126,280,-16,-9,1
%N A136247 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial h(n,x) with h(0,x)=1, h(1,x)=1-x and recursively h(n,x) = 1 + n -(1-x)*(1-h(n-1,x)) - n*h(n-2,x).
%C A136247 Row sums are 1, 0, 1, 4, 1, -14, 1, 106, 1, -944, 1, 10396, 1, -135134, 1, 2027026, 1, -34459424, 1, 654729076, 1...
%C A136247 [Row sums s(n) appear to obey s(n) -2*s(n-1) +(n+1)*s(n-2) +2*(1-n)*s(n-3) +(n-2)* s(n-4)=0. - _R. J. Mathar_, Dec 04 2011]
%D A136247 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43.
%e A136247 1;
%e A136247 1, -1;
%e A136247 1, -1, 1;
%e A136247 1, 2, 2, -1;
%e A136247 1, 6, -4, -3, 1;
%e A136247 1, -4, -20,6, 4, -1;
%e A136247 1, -40, 8, 44, -8, -5, 1;
%e A136247 1, -12, 188, -6, -80,10, 6, -1;
%e A136247 1, 308, 136, -546, -10, 130, -12, -7, 1;
%e A136247 1, 416, -1864, -628, 1256, 50, -196, 14, 8, -1;
%e A136247 1, -2664, -3640, 6696, 1984, -2506, -126,280, -16, -9, 1;
%p A136247 h := proc(n,x)
%p A136247     if n = 0 then
%p A136247         1 ;
%p A136247     elif n = 1 then
%p A136247         1-x ;
%p A136247     else
%p A136247         1+n-(1-x)*(1-procname(n-1,x)) -n*procname(n-2,x) ;
%p A136247         expand(%) ;
%p A136247     end if;
%p A136247 end proc:
%p A136247 A136247 := proc(n,k)
%p A136247     coeftayl(h(n,x),x=0,k) ;
%p A136247 end proc:
%p A136247 seq(seq(A136247(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Dec 04 2011
%t A136247 Clear[h, a, n, x, y, c, d] (*Solve linear Shabat transform for Hermite type recursion*) Solve[c*x0 + d - x*(c*x1 + d) + n*(c*x2 + d) == 0, x0] c = -1; d = 1; Solve[y = c*x + d == 0, x] h[x, 0] = 1; h[x, 1] = 1 - x; h[x_, n_] := h[x, n] = -(-1 - n + (1 - x) - (1 - x)* h[ x, n - 1] + n *h[x, n - 2]); Table[ExpandAll[h[x, n]], {n, 0, 10}]; a = Table[CoefficientList[h[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[h[x, n], x]], {n, 0, 10}];
%Y A136247 Cf. A137286.
%K A136247 easy,tabl,sign
%O A136247 0,8
%A A136247 _Roger L. Bagula_, Mar 17 2008