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A026584 Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even. 26

%I #27 Dec 12 2021 09:30:46

%S 1,1,0,1,1,1,2,1,1,1,1,4,2,4,1,1,1,2,5,7,8,7,5,2,1,1,2,8,9,20,14,20,9,

%T 8,2,1,1,3,9,19,28,43,40,43,28,19,9,3,1,1,3,13,22,56,62,111,86,111,62,

%U 56,22,13,3,1,1,4,14,38,69,140,167,259,222,259,167,140,69,38,14,4,1

%N Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.

%C Row sums are in A026597. - _Philippe Deléham_, Oct 16 2006

%C T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)| <= 1 if s(i-1) odd, |s(i)-s(i-1)| = 1 if s(i-1) is even, for i = 1..n.

%H Clark Kimberling, <a href="/A026584/b026584.txt">Rows 0..100, flattened</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n, k) = T(n-1, k-2) + T(n-1, k) if ( (n+k) mod 2 ) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), where T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2).

%e First 5 rows:

%e 1

%e 1 0 1

%e 1 1 2 1 1

%e 1 1 4 2 4 1 1

%e 1 2 5 7 8 7 5 2 1

%t z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2]; t[n_, k_] := Floor[n/2] /; k == 2 n - 1; t[n_, k_] := t[n, k] = If[EvenQ[n + k], t[n - 1, k - 2] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];

%t TableForm[u] (* A026584 array *)

%t v = Flatten[u] (* A026584 sequence *)

%o (Sage)

%o @CachedFunction

%o def T(n,k):

%o if (k==0 or k==2*n): return 1

%o elif (k==1 or k==2*n-1): return (n//2)

%o else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)

%o flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Dec 11 2021

%Y Cf. A026519, A026536, A026552, A026568, A027926.

%Y Cf. A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599.

%K nonn,tabf

%O 1,7

%A _Clark Kimberling_

%E Updated by _Clark Kimberling_, Aug 29 2014

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Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)