A208341 - OEIS

%I #44 Aug 21 2021 02:45:12

%S 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 64,1,8,34,96,192,272,256,128,1,9,43,138,321,552,688,576,256,1,10,53,

%U 190,501,1002,1520,1696,1280,512,1,11,64,253,743,1683,2972,4048

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

%C Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section.

%C Row sums: (1,3,8,...), even-indexed Fibonacci numbers.

%C Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers.

%C v(n,2) = A107839(n), v(n,n) = 2^(n-1), v(n+1,n) = A001792(n),

%C v(n+2,n) = A049611, v(n+3,n) = A049612.

%C 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 Essentially triangle in A049600. - _Philippe Deléham_, Mar 23 2012

%H Reinhard Zumkeller, <a href="/A208341/b208341.txt">Rows n = 0..124 of triangle, flattened</a>

%F 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 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 G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - _Philippe Deléham_, Mar 12 2012

%F T(n,k) = A106195(n-1,n-k), k = 1..n. - _Reinhard Zumkeller_, Dec 16 2013

%F From _Peter Bala_, Aug 11 2015: (Start)

%F The following remarks assume the row and column indexing start at 0.

%F 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 Riordan array (1/(1 - x), x*(2 - x)/(1 - x)).

%F 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 Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End)

%F 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 First five rows:

%e   1;

%e   1, 2;

%e   1, 3,  4;

%e   1, 4,  8,  8;

%e   1, 5, 13, 20, 16;

%e First five polynomials v(n,x):

%e   1

%e   1 + 2x

%e   1 + 3x +  4x^2

%e   1 + 4x +  8x^2 +  8x^3

%e   1 + 5x + 13x^2 + 20x^3 + 16x^4

%e (1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:

%e   1;

%e   1, 0;

%e   1, 2,  0;

%e   1, 3,  4,  0;

%e   1, 4,  8,  8,  0;

%e   1, 5, 13, 20, 16,  0;

%e   1, 6, 19, 38, 48, 32, 0;

%e Triangle in A049600 begins:

%e   0;

%e   0, 1;

%e   0, 1, 2;

%e   0, 1, 3,  4;

%e   0, 1, 4,  8,  8;

%e   0, 1, 5, 13, 20, 16;

%e   0, 1, 6, 19, 38, 48, 32;

%e   ... - _Philippe Deléham_, Mar 23 2012

%p T := (n,k) -> hypergeom([n-k+1, -k],[1],-1):

%p seq(lprint(seq(simplify(T(n,k)),k=0..n)),n=0..7); # _Peter Luschny_, May 20 2015

%t u[1, x_] := 1; v[1, x_] := 1; z = 13;

%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]   (* A160232 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]   (* A208341 *)

%o (Haskell)

%o a208341 n k = a208341_tabl !! (n-1) !! (k-1)

%o a208341_row n = a208341_tabl !! (n-1)

%o a208341_tabl = map reverse a106195_tabl

%o -- _Reinhard Zumkeller_, Dec 16 2013

%o (PARI) T(n,k) = sum(i = 0, k, 2^(k-i)*binomial(n-k,i)*binomial(k,i));

%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ _Michel Marcus_, Aug 14 2015

%Y Cf. A160232, A000045, A049600, A106195.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Feb 25 2012

%E New name from _Peter Luschny_, May 20 2015

%E Offset corrected by _Joerg Arndt_, Aug 12 2015