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.

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, 52, 40, 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, 345, 188, 88, 41, 19, 9, 4, 2, 1, 1, 600, 694, 400, 189, 88, 41, 19, 9, 4, 2, 1, 1, 1104, 1386, 846, 406, 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

Offset: 0

Author

Juri-Stepan Gerasimov, Jul 08 2008

Keywords

• nonn
• tabl

Comments

From Petros Hadjicostas, Jun 12 2019: (Start)

This is a mirror image of the 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 on 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.)

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, which has only 2 infinite sets of initial conditions, namely, G(n, 0) = G(n+1, n+1) = 1 for n >= 0).

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).

(End)

Examples

			Triangle begins:
  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  52  40  19   9  4  2  1
  1  96  83  41  19  9  4  2 1
  1 177 170  88  41 19  9  4 2 1
  1 326 345 188  88 41 19  9 4 2 1
  1 600 694 400 189 88 41 19 9 4 2 1
  ...
E.g., G(14, 2) = G(11, 1) + G(11, 2) + G(12, 2) + G(13, 2) = 600 + 694 + 1386 + 2751 = 5431.
		

Links

• Robert Price, Table of n, a(n) for n = 0..1325
• Juri-Stepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...

Crossrefs

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

Programs

Mathematica

nlim = 50;
Do[G[n, 0] = 1, {n, 0, nlim}];
Do[G[n + 1, n + 1] = 1, {n, 0, nlim}];
Do[G[n + 2, n + 1] = 2, {n, 0, nlim}];
Do[G[n + 3, n + 1] = 4, {n, 0, nlim}];
Do[G[n + 4, m] =
   G[n + 1, m - 1] + G[n + 1, m] + G[n + 2, m] + G[n + 3, m], {n, 0,
   nlim}, {m, 1, n + 1}];
A140997 = {}; For[n = 0, n <= nlim, n++,
 For[k = 0, k <= n, k++, AppendTo[A140997, G[n, k]]]];
A140997 (* Robert Price, Aug 25 2019 *)

Formula

From Petros Hadjicostas, Jun 12 2019: (Start)

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

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)).

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.

(End)

Extensions

Typo in definition corrected by R. J. Mathar, Sep 19 2008

Name edited by and more terms from Petros Hadjicostas, Jun 12 2019

Deleted extraneous term at a(29) by Robert Price, Aug 25 2019

Added 13 missing terms at a(79) by Robert Price, Aug 25 2019