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A140997 Triangle G(n,k) read by rows, for 0 <= k <= n, where G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, and G(n+4, m) = G(n+1, m-1) + G(n+1, m) + G(n+2, m) + G(n+3, m) for n >= 0 and m = 1..n+1. 23

%I

%S 1,1,1,1,2,1,1,4,2,1,1,8,4,2,1,1,15,9,4,2,1,1,28,19,9,4,2,1,1,2,52,40,

%T 19,9,4,2,1,1,96,83,41,19,9,4,2,1,1,177,170,88,41,19,9,4,2,1,1,326,

%U 345,188,88,41,19,9,4,2,1,1,600,694,400,189,88,41,19,9,4,2,1,1,2031,2751,1779,871,406,189,88,41,19,9,4,2,1,1,3736,5431,3719,1866,872,406,189,88,41,19,9,4,2,1

%N Triangle G(n,k) read by rows, for 0 <= k <= n, where G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, and G(n+4, m) = G(n+1, m-1) + G(n+1, m) + G(n+2, m) + G(n+3, m) for n >= 0 and m = 1..n+1.

%C From _Petros Hadjicostas_, Jun 12 2019: (Start)

%C This is a mirror of image of triangular array A140994. The current array has index of asymmetry s = 2 and index of obliqueness (obliquity) e = 0. Array A140994 has the same index of asymmetry, but has index of obliqueness e = 1. (In other related sequences, the author uses the letter y for the index of asymmetry and the letter z for the index of obliqueness, but the stone slab that appears over a tomb in a picture that he posted in those sequences, the letters s and e are used instead. See, for example, the documentation for sequences A140998, A141065, A141066, and A141067.)

%C In general, if the index of asymmetry (from the Pascal triangle A007318) is s, then the order of the recurrence is s + 2 (because the recurrence of the Pascal triangle has order 2). There are also s + 2 infinite sets of initial conditions (as opposed to the Pascal triangle that has only 2 infinite sets of initial conditions, namely, G(n, 0) = G(n+1, n+1) = 1 for n >= 0).

%C Pascal's triangle A007318 has s = 0 and is symmetric, arrays A140998 and A140993 have s = 1 (with e = 0 and e = 1, respectively), and arrays A140996 and A140995 have s = 3 (with e = 0 and e = 1, respectively).

%C (End)

%H Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...</a>

%F From _Petros Hadjicostas_, Jun 12 2019: (Start)

%F G(n, k) = A140994(n, n-k) for 0 <= k <= n.

%F Bivariate g.f.: Sum_{n,k >= 0} G(n,k)*x^n*y^k = (1 - x - x^2 - x^3 + x^2*y + x^4*y)/((1 - x) * (1 - x*y) * (1 - x - x^2 - x^3 - x^3*y)).

%F Differentiating once w.r.t. y and setting y = 0, we get the g.f. of column k = 1: x/((1 - x) * (1 - x - x^2 - x^3)). This is the g.f. of sequence A008937.

%F (End)

%e Triangle begins:

%e 1

%e 1 1

%e 1 2 1

%e 1 4 2 1

%e 1 8 4 2 1

%e 1 15 9 4 2 1

%e 1 28 19 9 4 2 1

%e 1 52 40 19 9 4 2 1

%e 1 96 83 41 19 9 4 2 1

%e 1 177 170 88 41 19 9 4 2 1

%e 1 326 345 188 88 41 19 9 4 2 1

%e 1 600 694 400 189 88 41 19 9 4 2 1

%e ...

%e E.g., G(14, 2) = G(11, 1) + G(11, 2) + G(12, 2) + G(13, 2) = 600 + 694 + 1386 + 2751 = 5431.

%Y Cf. A007318, A008937, A140993, A140994, A140995, A140996, A140998, A141015, A141018, A141020, A141021, A141065, A141066, A141067.

%K nonn,tabl

%O 0,5

%A _Juri-Stepan Gerasimov_, Jul 08 2008

%E Typo in definition corrected by _R. J. Mathar_, Sep 19 2008

%E Name edited by and more terms from _Petros Hadjicostas_, Jun 12 2019

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Last modified August 21 23:28 EDT 2019. Contains 326169 sequences. (Running on oeis4.)