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Triangular array, read by rows: pairwise sums of trinomial array A027907.
15

%I #24 Sep 08 2019 02:07:54

%S 1,1,2,1,1,3,4,3,1,1,4,8,10,8,4,1,1,5,13,22,26,22,13,5,1,1,6,19,40,61,

%T 70,61,40,19,6,1,1,7,26,65,120,171,192,171,120,65,26,7,1,1,8,34,98,

%U 211,356,483,534,483,356,211,98,34,8,1,1,9,43,140,343,665,1050,1373

%N Triangular array, read by rows: pairwise sums of trinomial array A027907.

%C Counting the top row as row 0, T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k) = n and the string does not contain the substring "00". E.g., T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112. - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007

%F T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 2, 1], [1, 3, 4, 3, 1].

%F G.f.: (1+yz)/[1-z(1+y+y^2)].

%e 1

%e 1 2 1

%e 1 3 4 3 1

%e 1 4 8 10 8 4 1

%e 1 5 13 22 26 22 13 5 1

%t T[_, 0] = 1; T[1, 1] = 2; T[n_, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[_, _] = 0;

%t Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* _Jean-François Alcover_, Jul 22 2018 *)

%o (PARI) {T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,2,1][k+1], if( n==2, [1,3,4,3,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

%o (PARI) T(n,k)=polcoeff(Ser(polcoeff(Ser((1+y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

%o (PARI) {T(n, k) = if( k<0 || k>2*n, 0, if(n==0, 1, polcoeff( (1 + x + x^2)^n, k)+ polcoeff( (1 + x + x^2)^(n-1), k-1)))};

%Y Columns include A025565, A025566, A025567, A025568.

%Y Cf. A025177.

%K nonn,tabf,easy

%O 0,3

%A _Clark Kimberling_

%E Edited by _Ralf Stephan_, Jan 09 2005

%E Edited by _Clark Kimberling_, Jun 20 2012