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A156224 Triangle T(n, k) = binomial(n, k)*(A000009(n) + A000009(n-k) + A000009(k)) - 2, read by rows.

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