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%I #22 Feb 18 2021 00:29:15
%S 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,7,4,1,0,1,5,12,12,5,1,0,1,6,18,30,
%T 18,6,1,0,1,7,25,55,55,25,7,1,0,1,8,33,88,143,88,33,8,1,0,1,9,42,130,
%U 273,273,130,42,9,1,0,1,10,52,182,455,728,455,182,52,10,1,0
%N Pendular triangle (p=0), 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), otherwise T(n,k) = T(n,n-1-k) + p*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.
%C The last five rows in the example section appear in the table on p. 8 of Getzler. Cf. also A173075. - _Tom Copeland_, Jan 22 2020
%H G. C. Greubel, <a href="/A167763/b167763.txt">Rows n = 0..100 of the triangle, flattened</a>
%H E. Getzler, <a href="http://arxiv.org/abs/alg-geom/9612005">The semi-classical approximation for modular operads</a>, arXiv:alg-geom/9612005, 1996.
%F T(2n+m) = [A001764^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A001764.
%F If n > 2k, T(n,k) = binomial(n+k+1,k)*(n-2k+1)/(n+k+1), else T(n,k) = T(n,n-1-k), with conditions: T(n,0)=1 for n>=0 and T(n,n)=0 for n > 0. - _Paul D. Hanna_, Nov 12 2009
%F Sum_{k=0..n} T(n, k, p=0) = A093951(n). - _G. C. Greubel_, Feb 17 2021
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 1, 0;
%e 1, 3, 3, 1, 0;
%e 1, 4, 7, 4, 1, 0;
%e 1, 5, 12, 12, 5, 1, 0; ...
%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,0], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 17 2021 *)
%o (PARI) {T(n,k)=if(k==0,1,if(n==k,0,if(n>2*k,binomial(n+k+1,k)*(n-2*k+1)/(n+k+1),T(n,n-1-k))))} \\ _Paul D. Hanna_, Nov 12 2009
%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, 0) 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,0): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2021
%Y Cf. this sequence (p=0), A118340 (p=1), A118345 (p=2), A118350 (p=3).
%Y Cf. A001764, A006013, A006629, A093951, A102893, A173075.
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Nov 11 2009
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