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A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.

%I A208341 #44 Aug 21 2021 02:45:12
%S A208341 1,1,2,1,3,4,1,4,8,8,1,5,13,20,16,1,6,19,38,48,32,1,7,26,63,104,112,
%T A208341 64,1,8,34,96,192,272,256,128,1,9,43,138,321,552,688,576,256,1,10,53,
%U A208341 190,501,1002,1520,1696,1280,512,1,11,64,253,743,1683,2972,4048
%N A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.
%C A208341 Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section.
%C A208341 Row sums: (1,3,8,...), even-indexed Fibonacci numbers.
%C A208341 Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers.
%C A208341 v(n,2) = A107839(n), v(n,n) = 2^(n-1), v(n+1,n) = A001792(n),
%C A208341 v(n+2,n) = A049611, v(n+3,n) = A049612.
%C A208341 Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 12 2012
%C A208341 Essentially triangle in A049600. - _Philippe Deléham_, Mar 23 2012
%H A208341 Reinhard Zumkeller, <a href="/A208341/b208341.txt">Rows n = 0..124 of triangle, flattened</a>
%F A208341 u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1.
%F A208341 As DELTA-triangle with 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 12 2012
%F A208341 G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - _Philippe Deléham_, Mar 12 2012
%F A208341 T(n,k) = A106195(n-1,n-k), k = 1..n. - _Reinhard Zumkeller_, Dec 16 2013
%F A208341 From _Peter Bala_, Aug 11 2015: (Start)
%F A208341 The following remarks assume the row and column indexing start at 0.
%F A208341 T(n,k) = Sum_{i = 0..k} 2^(k-i)*binomial(n-k,i)*binomial(k,i) = Sum_{i = 0..k} binomial(n-k+i,i)*binomial(k,i).
%F A208341 Riordan array (1/(1 - x), x*(2 - x)/(1 - x)).
%F A208341 O.g.f. 1/(1 - (2*t + 1)*x + t*x^2) = 1 + (1 + 2*t)*x + (1 + 3*t + 4*t^2)*x^2 + ....
%F A208341 Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End)
%F A208341 For k<n, T(n,k) = T(n-1,k) + Sum_{i=1..k} T(n-i,k-i). - _Glen Whitney_, Aug 17 2021
%e A208341 First five rows:
%e A208341   1;
%e A208341   1, 2;
%e A208341   1, 3,  4;
%e A208341   1, 4,  8,  8;
%e A208341   1, 5, 13, 20, 16;
%e A208341 First five polynomials v(n,x):
%e A208341   1
%e A208341   1 + 2x
%e A208341   1 + 3x +  4x^2
%e A208341   1 + 4x +  8x^2 +  8x^3
%e A208341   1 + 5x + 13x^2 + 20x^3 + 16x^4
%e A208341 (1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
%e A208341   1;
%e A208341   1, 0;
%e A208341   1, 2,  0;
%e A208341   1, 3,  4,  0;
%e A208341   1, 4,  8,  8,  0;
%e A208341   1, 5, 13, 20, 16,  0;
%e A208341   1, 6, 19, 38, 48, 32, 0;
%e A208341 Triangle in A049600 begins:
%e A208341   0;
%e A208341   0, 1;
%e A208341   0, 1, 2;
%e A208341   0, 1, 3,  4;
%e A208341   0, 1, 4,  8,  8;
%e A208341   0, 1, 5, 13, 20, 16;
%e A208341   0, 1, 6, 19, 38, 48, 32;
%e A208341   ... - _Philippe Deléham_, Mar 23 2012
%p A208341 T := (n,k) -> hypergeom([n-k+1, -k],[1],-1):
%p A208341 seq(lprint(seq(simplify(T(n,k)),k=0..n)),n=0..7); # _Peter Luschny_, May 20 2015
%t A208341 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208341 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208341 v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];
%t A208341 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208341 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208341 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208341 TableForm[cu]
%t A208341 Flatten[%]   (* A160232 *)
%t A208341 Table[Expand[v[n, x]], {n, 1, z}]
%t A208341 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208341 TableForm[cv]
%t A208341 Flatten[%]   (* A208341 *)
%o A208341 (Haskell)
%o A208341 a208341 n k = a208341_tabl !! (n-1) !! (k-1)
%o A208341 a208341_row n = a208341_tabl !! (n-1)
%o A208341 a208341_tabl = map reverse a106195_tabl
%o A208341 -- _Reinhard Zumkeller_, Dec 16 2013
%o A208341 (PARI) T(n,k) = sum(i = 0, k, 2^(k-i)*binomial(n-k,i)*binomial(k,i));
%o A208341 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ _Michel Marcus_, Aug 14 2015
%Y A208341 Cf. A160232, A000045, A049600, A106195.
%K A208341 nonn,tabl
%O A208341 0,3
%A A208341 _Clark Kimberling_, Feb 25 2012
%E A208341 New name from _Peter Luschny_, May 20 2015
%E A208341 Offset corrected by _Joerg Arndt_, Aug 12 2015