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A172971 Triangle T(n, k) = c(n) - c(k) - c(n-k), where c(n) = Product_{j=0..n} Partitions(j), read by rows.

%I A172971 #8 Dec 05 2022 04:41:49
%S A172971 -1,-1,-1,-1,-1,-1,-1,0,0,-1,-1,1,2,1,-1,-1,7,9,9,7,-1,-1,35,43,44,43,
%T A172971 35,-1,-1,191,227,234,234,227,191,-1,-1,1199,1391,1426,1432,1426,1391,
%U A172971 1199,-1,-1,10079,11279,11470,11504,11504,11470,11279,10079,-1
%N A172971 Triangle T(n, k) = c(n) - c(k) - c(n-k), where c(n) = Product_{j=0..n} Partitions(j), read by rows.
%H A172971 G. C. Greubel, <a href="/A172971/b172971.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172971 T(n, k) = c(n) - c(k) - c(n-k), where c(n) = Product_{j=0..n} Partitions(j).
%F A172971 T(n, n-k) = T(n, k).
%e A172971 Triangle begins as:
%e A172971   -1;
%e A172971   -1,    -1;
%e A172971   -1,    -1,    -1;
%e A172971   -1,     0,     0,    -1;
%e A172971   -1,     1,     2,     1,    -1;
%e A172971   -1,     7,     9,     9,     7,    -1;
%e A172971   -1,    35,    43,    44,    43,    35,    -1;
%e A172971   -1,   191,   227,   234,   234,   227,   191,    -1;
%e A172971   -1,  1199,  1391,  1426,  1432,  1426,  1391,  1199,    -1;
%e A172971   -1, 10079, 11279, 11470, 11504, 11504, 11470, 11279, 10079,  -1;
%t A172971 c[n_]:= Product[PartitionsQ[j], {j,n}];
%t A172971 T[n_, k_]:= c[n] - (c[k] + c[n-k]);
%t A172971 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A172971 (Magma)
%o A172971 A000009:= Coefficients(&*[1+x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Sergei Haller's code
%o A172971 c:= func< n | (&*[A000009[j+1]: j in [0..n]]) >;
%o A172971 A172971:= func< n,k | c(n) - c(k) - c(n-k) >;
%o A172971 [A172971(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 04 2022
%o A172971 (SageMath)
%o A172971 def EulerTransform(a):
%o A172971     @cached_function
%o A172971     def b(n):
%o A172971         if n == 0: return 1
%o A172971         s = sum(sum(d * a(d) for d in divisors(j)) * b(n-j) for j in (1..n))
%o A172971         return s//n
%o A172971     return b
%o A172971 a = BinaryRecurrenceSequence(0, 1)
%o A172971 b = EulerTransform(a) # Peter Luschny's code for A000009
%o A172971 @CachedFunction
%o A172971 def c(n): return product(b(j) for j in range(n+1))
%o A172971 def A172971(n,k): return c(n) - c(k) - c(n-k)
%o A172971 flatten([[A172971(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Dec 04 2022
%Y A172971 Cf. A000009.
%K A172971 sign,tabl,less
%O A172971 0,13
%A A172971 _Roger L. Bagula_, Feb 06 2010
%E A172971 Edited by _G. C. Greubel_, Dec 04 2022