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%I #9 Feb 18 2021 00:28:55
%S 1,1,0,1,1,0,1,2,1,0,1,3,5,1,0,1,4,11,6,1,0,1,5,18,30,7,1,0,1,6,26,70,
%T 40,8,1,0,1,7,35,121,201,51,9,1,0,1,8,45,184,487,286,63,10,1,0,1,9,56,
%U 260,873,1445,386,76,11,1,0,1,10,68,350,1375,3592,2147,502,90,12,1,0
%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) + 2*T(n-1,k), for n>=k>=0, with T(n,0) = 1 and T(n,n) = 0^n.
%C See A118340 for definition of pendular triangles and pendular sums.
%H G. C. Greubel, <a href="/A118345/b118345.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(2*n+m,n) = [A118346^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A118346.
%e Row 6 equals the pendular sums of row 5:
%e [1, 4, 11, 6, 1, 0], where the pendular sums proceed as follows:
%e [1, __, __, __, __, __]: T(6,0) = T(5,0) = 1;
%e [1, __, __, __, __, 1]: T(6,5) = T(6,0) + 2*T(5,5) = 1 + 2*0 = 1;
%e [1, 5, __, __, __, 1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;
%e [1, 5, __, __, 7, 1]: T(6,4) = T(6,1) + 2*T(5,4) = 5 + 2*1 = 7;
%e [1, 5, 18, __, 7, 1]: T(6,2) = T(6,4) + T(5,2) = 7 + 11 = 18;
%e [1, 5, 18, 30, 7, 1]: T(6,3) = T(6,2) + 2*T(5,3) = 18 + 2*6 = 30;
%e [1, 5, 18, 30, 7, 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, 5, 1, 0;
%e 1, 4, 11, 6, 1, 0;
%e 1, 5, 18, 30, 7, 1, 0;
%e 1, 6, 26, 70, 40, 8, 1, 0;
%e 1, 7, 35, 121, 201, 51, 9, 1, 0;
%e 1, 8, 45, 184, 487, 286, 63, 10, 1, 0;
%e 1, 9, 56, 260, 873, 1445, 386, 76, 11, 1, 0;
%e 1, 10, 68, 350, 1375, 3592, 2147, 502, 90, 12, 1, 0; ...
%e Central terms are T(2*n,n) = A118346(n);
%e semi-diagonals form successive self-convolutions of the central terms:
%e T(2*n+1,n) = A118347(n) = [A118346^2](n),
%e T(2*n+2,n) = A118348(n) = [A118346^3](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,2], {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)+2*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,2) 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,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2021
%Y Cf. A118346, A118347, A118348, A118349, A118340.
%Y Cf. A167763 (p=0), A118340 (p=1), this sequence (p=2), A118350 (p=3).
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Apr 26 2006
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