A118350 - OEIS

A118350

Pendular triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) + 3*T(n-1,k), for n>=k>=0, with T(n,0)=1 and T(n,n)=0^n.

%I #9 Feb 18 2021 00:29:06

%S 1,1,0,1,1,0,1,2,1,0,1,3,6,1,0,1,4,13,7,1,0,1,5,21,42,8,1,0,1,6,30,96,

%T 54,9,1,0,1,7,40,163,325,67,10,1,0,1,8,51,244,770,445,81,11,1,0,1,9,

%U 63,340,1353,2688,583,96,12,1,0,1,10,76,452,2093,6530,3842,740,112,13,1,0

%N Pendular triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) + 3*T(n-1,k), for n>=k>=0, with T(n,0)=1 and T(n,n)=0^n.

%C See definition of pendular triangle and pendular sums at A118340.

%H G. C. Greubel, <a href="/A118350/b118350.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(2*n+m,n) = [A118351^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of the central terms A118351.

%e Row 6 equals the pendular sums of row 5,

%e [1, 4, 13, 7, 1, 0], where the sums proceed as follows:

%e [1, __, __, __, __, __]: T(6,0) = T(5,0) = 1;

%e [1, __, __, __, __, 1]: T(6,5) = T(6,0) + 3*T(5,5) = 1 + 3*0 = 1;

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

%e [1, 5, __, __, 8, 1]: T(6,4) = T(6,1) + 3*T(5,4) = 5 + 3*1 = 8;

%e [1, 5, 21, __, 8, 1]: T(6,2) = T(6,4) + T(5,2) = 8 + 13 = 21;

%e [1, 5, 21, 42, 8, 1]: T(6,3) = T(6,2) + 3*T(5,3) = 21 + 3*7 = 42;

%e [1, 5, 21, 42, 8, 1, 0] finally, append a zero to obtain row 6.

%e Triangle begins:

%e 1;

%e 1, 0;

%e 1, 1, 0;

%e 1, 2, 1, 0;

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

%e 1, 4, 13, 7, 1, 0;

%e 1, 5, 21, 42, 8, 1, 0;

%e 1, 6, 30, 96, 54, 9, 1, 0;

%e 1, 7, 40, 163, 325, 67, 10, 1, 0;

%e 1, 8, 51, 244, 770, 445, 81, 11, 1, 0;

%e 1, 9, 63, 340, 1353, 2688, 583, 96, 12, 1, 0;

%e 1, 10, 76, 452, 2093, 6530, 3842, 740, 112, 13, 1, 0;

%e 1, 11, 90, 581, 3010, 11760, 23286, 5230, 917, 129, 14, 1, 0; ...

%e Central terms are T(2*n,n) = A118351(n);

%e semi-diagonals form successive self-convolutions of the central terms:

%e T(2*n+1,n) = A118352(n) = [A118351^2](n),

%e T(2*n+2,n) = A118353(n) = [A118351^3](n).

%t T[n_, k_, p_]:= T[n,k,p] = If[n<k || k<0, 0, If[k==0, 1, If[k==n, 0, If[n<=2*k, T[n,n-k-1,p] + p*T[n-1,k,p], T[n,n-k,p] + T[n-1,k,p] ]]]];

%t Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 17 2021 *)

%o (PARI) T(n,k)=if(n<k || k<0,0,if(k==0,1,if(n==k,0, if(n>2*k,T(n,n-k)+T(n-1,k),T(n,n-1-k)+3*T(n-1,k)))))

%o (Sage)

%o @CachedFunction

%o def T(n, k, p):

%o if (k<0 or n<k): return 0

%o elif (k==0): return 1

%o elif (k==n): return 0

%o elif (n>2*k): return T(n,n-k,p) + T(n-1,k,p)

%o else: return T(n, n-k-1, p) + p*T(n-1, k, p)

%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 17 2021

%o (Magma)

%o function T(n,k,p)

%o if k lt 0 or n lt k then return 0;

%o elif k eq 0 then return 1;

%o elif k eq n then return 0;

%o elif n gt 2*k then return T(n,n-k,p) + T(n-1,k,p);

%o else return T(n,n-k-1,p) + p*T(n-1,k,p);

%o end if;

%o return T;

%o end function;

%o [T(n,k,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2021

%Y Cf. A118351, A118352, A118353, A118354.

%Y Cf. A167763 (p=0), A118340 (p=1), A118345 (p=2), this sequence (p=3).

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Apr 26 2006