%I #8 Dec 05 2022 04:41:49

%S -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 35,-1,-1,191,227,234,234,227,191,-1,-1,1199,1391,1426,1432,1426,1391,

%U 1199,-1,-1,10079,11279,11470,11504,11504,11470,11279,10079,-1

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

%H G. C. Greubel, <a href="/A172971/b172971.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = c(n) - c(k) - c(n-k), where c(n) = Product_{j=0..n} Partitions(j).

%F T(n, n-k) = T(n, k).

%e Triangle begins as:

%e -1;

%e -1, -1;

%e -1, -1, -1;

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

%e -1, 1, 2, 1, -1;

%e -1, 7, 9, 9, 7, -1;

%e -1, 35, 43, 44, 43, 35, -1;

%e -1, 191, 227, 234, 234, 227, 191, -1;

%e -1, 1199, 1391, 1426, 1432, 1426, 1391, 1199, -1;

%e -1, 10079, 11279, 11470, 11504, 11504, 11470, 11279, 10079, -1;

%t c[n_]:= Product[PartitionsQ[j], {j,n}];

%t T[n_, k_]:= c[n] - (c[k] + c[n-k]);

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

%o (Magma)

%o A000009:= Coefficients(&*[1+x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Sergei Haller's code

%o c:= func< n | (&*[A000009[j+1]: j in [0..n]]) >;

%o A172971:= func< n,k | c(n) - c(k) - c(n-k) >;

%o [A172971(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 04 2022

%o (SageMath)

%o def EulerTransform(a):

%o @cached_function

%o def b(n):

%o if n == 0: return 1

%o s = sum(sum(d * a(d) for d in divisors(j)) * b(n-j) for j in (1..n))

%o return s//n

%o return b

%o a = BinaryRecurrenceSequence(0, 1)

%o b = EulerTransform(a) # Peter Luschny's code for A000009

%o @CachedFunction

%o def c(n): return product(b(j) for j in range(n+1))

%o def A172971(n,k): return c(n) - c(k) - c(n-k)

%o flatten([[A172971(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Dec 04 2022

%Y Cf. A000009.

%K sign,tabl,less

%O 0,13

%A _Roger L. Bagula_, Feb 06 2010

%E Edited by _G. C. Greubel_, Dec 04 2022