A122445 - OEIS

%I #17 Feb 10 2023 11:53:43

%S 1,1,0,0,1,1,1,0,0,1,2,3,2,1,0,0,1,3,6,10,8,3,1,0,0,1,4,10,22,36,28,

%T 12,4,1,0,0,1,5,15,39,83,135,107,47,17,5,1,0,0,1,6,21,62,155,324,525,

%U 418,189,72,23,6,1,0,0,1,7,28,92,259,629,1298,2094,1676,773,305,104,30,7,1,0,0

%N Pendular trinomial triangle, read by rows of 2n+1 terms (n>=0), defined by the recurrence: if 0 < k < n, T(n,k) = T(n-1,k) + 2*T(n,2n-1-k); otherwise, if n-1 < k < 2n-1, T(n,k) = T(n-1,k) + T(n,2n-2-k); with T(n,0) = T(n+1,2n) = 1 and T(n+1,2n+1) = T(n+1,2n+2) = 0.

%C The diagonals may be generated by iterated convolutions of a base sequence B with the sequence C of central terms. The g.f. B(x) of the base sequence satisfies: B = 1 + x*B^2 + 2x^2*(B^2 - B); the g.f. C(x) of the central terms satisfies: C(x) = 1/(1+x - xB(x)).

%H G. C. Greubel, <a href="/A122445/b122445.txt">Rows n = 0..50 of the triangle, flattened</a>

%e To obtain row 4, pendular sums of row 3 are carried out as follows.

%e   [1, 2, 3,  2, 1, 0, 0]: given row 3;

%e   [1, _, _, __, _, _, _]: start with T(4,0) = T(3,0) = 1;

%e   [1, _, _, __, _, _, 1]: T(4,6) = T(4,0) + 2*T(3,6) = 1 + 2*0 = 1;

%e   [1, 3, _, __, _, _, 1]: T(4,1) = T(4,6) + 1*T(3,1) = 1 + 1*2 = 3;

%e   [1, 3, _, __, _, 3, 1]: T(4,5) = T(4,1) + 2*T(3,5) = 3 + 2*0 = 3;

%e   [1, 3, 6, __, _, 3, 1]: T(4,2) = T(4,5) + 1*T(3,2) = 3 + 1*3 = 6;

%e   [1, 3, 6, __, 8, 3, 1]: T(4,4) = T(4,2) + 2*T(3,4) = 6 + 2*1 = 8;

%e   [1, 3, 6, 10, 8, 3, 1]: T(4,3) = T(4,4) + 1*T(3,3) = 8 + 1*2 = 10;

%e   [1, 3, 6, 10, 8, 3, 1,0,0]: complete row 4 by appending two zeros.

%e Triangle begins:

%e   1;

%e   1, 0,  0;

%e   1, 1,  1,  0,   0;

%e   1, 2,  3,  2,   1,   0,   0;

%e   1, 3,  6, 10,   8,   3,   1,   0,   0;

%e   1, 4, 10, 22,  36,  28,  12,   4,   1,  0,  0;

%e   1, 5, 15, 39,  83, 135, 107,  47,  17,  5,  1, 0, 0;

%e   1, 6, 21, 62, 155, 324, 525, 418, 189, 72, 23, 6, 1, 0, 0;

%e Central terms are:

%e   C = A122447 = [1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, ...].

%e Lower diagonals start:

%e   D1 = A122448 = [1, 1, 3, 10, 36, 135, 525, 2094, 8524, ...];

%e   D2 = A122449 = [1, 2, 6, 22, 83, 324, 1298, 5302, 22002, ...].

%e Diagonals above central terms (ignoring leading zeros) start:

%e   U1 = A122450 = [1, 3, 12, 47, 189, 773, 3208, 13478, 57222, ...];

%e   U2 = A122451 = [1, 4, 17, 72, 305, 1300, 5576, 24068, 104510, ...].

%e There exists the base sequence:

%e   B = A122446 = [1, 1, 2, 7, 24, 88, 336, 1321, 5316, 21788, ...]

%e which generates all diagonals by convolutions with central terms:

%e   D2 = B * D1 = B^2 * C

%e   U2 = B * U1 = B^2 * C"

%e where C" = [1, 2, 8, 28, 107, 418, 1676, 6848, 28418, ...]

%e are central terms not including the initial [1,0].

%p T:= proc(n, k) option remember;

%p       if k=0 and n=0 then 1

%p     elif k<0 or k>2*(n-1) then 0

%p     elif n=2 and k<3 then 1

%p     else T(n-1, k) + `if`(k<n, T(n, 2*n-k-1), T(n-1, k) + T(n, 2*n-k-2))

%p       fi

%p     end:

%p seq(seq(T(n, k), k=0..2*n), n=0..12); # _G. C. Greubel_, Mar 16 2021

%t T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, T[n-1, k] + If[k<n, T[n, 2*n-k-1], T[n-1, k] + T[n, 2*n-k-2]]]]];

%t Table[T[n, k], {n,0,12}, {k,0,2*n}] // Flatten (* _G. C. Greubel_, Mar 16 2021 *)

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

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (n==0 and k==0): return 1

%o     elif (k<0 or k>2*(n-1)): return 0

%o     elif (n==2 and k<3): return 1

%o     else: return T(n-1, k) + ( T(n, 2*n-k-1) if k<n else T(n-1, k) +T(n, 2*n-k-2) )

%o flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 16 2021

%Y Cf. A122446, A122447 (central terms), A122452 (row sums).

%Y Diagonals: A122448, A122449, A122450, A122451.

%Y Variants: A118340, A118345, A118350, A119369.

%K nonn,tabf

%O 0,11

%A _Paul D. Hanna_, Sep 07 2006