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%I #5 Dec 31 2021 19:36:08
%S 1,1,1,1,4,1,3,10,10,3,3,18,22,18,3,5,28,58,58,28,5,7,46,103,158,103,
%T 46,7,9,68,187,313,313,187,68,9,11,94,306,614,698,614,306,94,11,15,
%U 133,502,1174,1636,1636,1174,502,133,15,19,188,763,2038,3358,4030,3358,2038,763,188,19
%N Triangle T(n, k) = binomial(n, k)*(A000009(n) + A000009(n-k) + A000009(k)) - 2, read by rows.
%H G. C. Greubel, <a href="/A156224/b156224.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = binomial(n, k)*(A000009(n) + A000009(n-k) + A000009(k)) - 2.
%F T(n, n-k) = T(n, k).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 3, 10, 10, 3;
%e 3, 18, 22, 18, 3;
%e 5, 28, 58, 58, 28, 5;
%e 7, 46, 103, 158, 103, 46, 7;
%e 9, 68, 187, 313, 313, 187, 68, 9;
%e 11, 94, 306, 614, 698, 614, 306, 94, 11;
%e 15, 133, 502, 1174, 1636, 1636, 1174, 502, 133, 15;
%e 19, 188, 763, 2038, 3358, 4030, 3358, 2038, 763, 188, 19;
%t T[n_, k_]:= Binomial[n, k]*(PartitionsQ[n] +PartitionsQ[n-k] +PartitionsQ[k]) -2;
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
%o (Sage)
%o # Uses Peter Luschny's program for A000009
%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 P = EulerTransform(a)
%o def T(n,k): return binomial(n,k)*(P(n) + P(n-k) + P(k)) - 2
%o flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Dec 31 2021
%Y Cf. A000009.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 06 2009
%E Edited by _G. C. Greubel_, Dec 31 2021
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