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A193844 Triangular array: the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.

%I A193844 #52 Mar 26 2020 04:26:16
%S A193844 1,1,3,1,5,7,1,7,17,15,1,9,31,49,31,1,11,49,111,129,63,1,13,71,209,
%T A193844 351,321,127,1,15,97,351,769,1023,769,255,1,17,127,545,1471,2561,2815,
%U A193844 1793,511,1,19,161,799,2561,5503,7937,7423,4097,1023
%N A193844 Triangular array:  the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.
%C A193844 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%C A193844 A193844 is also the fission of (p1(n,x)) by (q1(n,x)), where p1(n,x)=x^n+x^(n-1)+...+x+1 and q1(n,x)=(x+2)^n.
%C A193844 Essentially A119258 but without the main diagonal. - _Peter Bala_, Jul 16 2013
%C A193844 From _Robert Coquereaux_, Oct 02 2014: (Start)
%C A193844 This is also a rectangular array A(n,p) read down the antidiagonals:
%C A193844 1 1 1 1 1 1 1 1 1
%C A193844 3 5 7 9 11 13 15 17 19
%C A193844 7 17 31 49 71 97 127 161 199
%C A193844 15 49 111 209 351 545 799 1121 1519
%C A193844 31 129 351 769 1471 2561 4159 6401 9439
%C A193844 ...
%C A193844 Calling Gr(n) the Grassmann algebra with n generators, A(n,p) is the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence. If p is odd A(n,p) is the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n). If p is even, the dimension of this cohomology group is A(n,p)+1. A(n,p) = 2^n*A059260(p,n-1)-(-1)^p.
%C A193844 (End)
%C A193844 The n-th row are also the coefficients of the polynomial P=sum_{k=0..n} (X+2)^k (in falling order, i.e., that of X^n first). - _M. F. Hasler_, Oct 15 2014
%H A193844 Jean-François Chamayou, <a href="http://arxiv.org/abs/1410.1708">A Random Difference Equation with Dufresne Variables revisited</a>, arXiv:1410.1708 [math.PR], 2014.
%H A193844 R. Coquereaux and E. Ragoucy, <a href="http://dx.doi.org/10.1016/0393-0440(94)00014-U">Currents on Grassmann algebras</a>, J. of Geometry and Physics, 1995, Vol 15, pp 333-352.
%H A193844 R. Coquereaux and E. Ragoucy, <a href="http://arxiv.org/abs/hep-th/9310147">Currents on Grassmann algebras</a>, arXiv:hep-th/9310147, 1993.
%H A193844 C. Kassel, <a href="http://dx.doi.org/10.1007/BF01459145">A Künneth formula for the cyclic cohomology of Z2-graded algebras</a>, Math.  Ann. 275 (1986) 683.
%F A193844 From _Peter Bala_, Jul 16 2013: (Start)
%F A193844 T(n,k) = sum {i = 0..k} (-1)^i*binomial(n+1,k-i)*2^(k-i).
%F A193844 O.g.f.: 1/( (1 - x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 3*x)*t + (1 + 5*x + 7*x^2)*t^2 + ....
%F A193844 The n-th row polynomial R(n,x) = 1/(x+1)*( (2*x+1)^(n+1) - x^(n+1) ). (End)
%F A193844 T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 17 2014
%F A193844 T(n,k) = 2^k*binomial(n+1,k)*hyper2F1(1,-k,-k+n+2, 1/2). - _Peter Luschny_, Jul 23 2014
%e A193844 First six rows:
%e A193844 1
%e A193844 1....3
%e A193844 1....5....7
%e A193844 1....7....17....15
%e A193844 1....9....31....49....31
%e A193844 1....11...49....111...129...63
%p A193844 A193844 := (n,k) -> 2^k*binomial(n+1,k)*hypergeom([1,-k],[-k+n+2],1/2);
%p A193844 for n from 0 to 5 do seq(round(evalf(A193844(n,k))),k=0..n) od; # _Peter Luschny_, Jul 23 2014
%p A193844 # Alternatively
%p A193844 p := (n,x) -> add(x^k*(1+2*x)^(n-k), k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od; # _Peter Luschny_, Jun 18 2017
%t A193844 z = 10;
%t A193844 p[n_, x_] := (x + 1)^n;
%t A193844 q[n_, x_] := (x + 1)^n
%t A193844 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193844 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193844 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193844 h[n_] := CoefficientList[d[n, x], {x}]
%t A193844 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193844 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193844 *)
%t A193844 TableForm[Table[h[n], {n, 0, z}]]
%t A193844 Flatten[Table[h[n], {n, -1, z}]]  (* A193845 *)
%o A193844 (Sage) # uses[fission from A193842]
%o A193844 p = lambda n,x: (x+1)^n
%o A193844 A193844_row = lambda n: fission(p, p, n)
%o A193844 for n in range(7): print(A193844_row(n)) # _Peter Luschny_, Jul 23 2014
%Y A193844 Cf. A193842, A193845, A119258.
%Y A193844 Columns, diagonals: A000225, A000337, A055580, A027608, A211386, A211388, A000012, A005408, A056220, A199899.
%Y A193844 A145661 is an essentially identical triangle.
%K A193844 nonn,tabl
%O A193844 0,3
%A A193844 _Clark Kimberling_, Aug 07 2011