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A141020 Pascal-like triangle with index of asymmetry y = 4 and index of obliqueness z = 0. 19
1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, 32, 16, 8, 4, 2, 1, 1, 63, 33, 16, 8, 4, 2, 1, 1, 124, 67, 33, 16, 8, 4, 2, 1, 1, 244, 136, 67, 33, 16, 8, 4, 2, 1, 1, 480, 276, 136, 67, 33, 16, 8, 4, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The left column is set to 1. The four rightmost columns start with powers of 2:

T(n, 0) = T(n, n)=1; T(n, n-1)=2; T(n, n-2)=4; T(n, n-3)=8; T(n, n-4)=16.

Recurrence: T(n, k) = T(n-1, k) + T(n-2, k) + T(n-3, k) + T(n-4, k) + T(n-5, k) + T(n-5,k-1), k = 1..n-5.

From Petros Hadjicostas, Jun 14 2019: (Start)

In the attached photograph we see that the index of asymmetry is denoted by s (rather than y) and the index of obliqueness by e (rather than z).

The general recurrence is G(n+s+2, k) = G(n+1, k-e*s+e-1) + Sum_{1 <= m <= s+1} G(n+m, k-e*s+m*e-2*e) for n >= 0 with k = 1..(n+1) when e = 0 and k = (s+1)..(n+s+1) when e = 1. The initial conditions are G(n+x+1, n-e*n+e*x-e+1) = 2^x for x=0..s and n >= 0. There is one more initial condition, namely, G(n, e*n) = 1 for n >= 0.

For s = 0, we get Pascal's triangle A007318. For s = 1, we get A140998 (e = 0) and A140993 (e = 1). For s = 2, we get A140997 (e = 0) and A140994 (e = 1). For s = 3, we get A140996 (e = 0) and A140995 (e = 1). For s = 4, we have the current array (with e = 0) and array A141021 (with e = 1). In some of these arrays, the indices n and k are sometimes shifted.

(End)

LINKS

Table of n, a(n) for n=0..66.

Juri-Stepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...

FORMULA

From Petros Hadjicostas, Jun 14 2019: (Start)

T(n, k) = A141021(n, n-k) for 0 <= k <= n.

Bivariate g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = (1 - x - x^2 - x^3 - x^4 - x^5 + y*x^2*(1 + x + x^2 + x^4)) / ((1 - x) * (1 - x*y) * (1 - x - x^2 - x^3 - x^4 - x^5 - x^5*y)).

Differentiating the bivariate w.r.t. y and setting y = 0, we get the g.f. of the column k = 1: x/((-1 + x)*(x^5 + x^4 + x^3 + x^2 + x - 1)). This is the g.f. of a shifted version of sequence A001949.

(End)

EXAMPLE

Pascal-like triangle with y = 4 and z = 0 begins as follows:

  1

  1   1

  1   2   1

  1   4   2   1

  1   8   4   2   1

  1  16   8   4   2  1

  1  32  16   8   4  2  1

  1  63  33  16   8  4  2  1

  1 124  67  33  16  8  4  2 1

  1 244 136  67  33 16  8  4 2 1

  1 480 276 136  67 33 16  8 4 2 1

  1 944 560 276 136 67 33 16 8 4 2 1

  ...

MAPLE

A141020 := proc(n, k) option remember ; if k<0 or k>n then 0 ; elif k=0 or k=n then 1 ; elif k=n-1 then 2 ; elif k=n-2 then 4 ; elif k=n-3 then 8 ; elif k=n-4 then 16 ; else procname(n-1, k) +procname(n-2, k)+procname(n-3, k)+procname(n-4, k) +procname(n-5, k)+procname(n-5, k-1) ; fi; end:

for n from 0 to 20 do for k from 0 to n do printf("%d, ", A141020(n, k)) ; od: od: # R. J. Mathar, Sep 19 2008

CROSSREFS

Cf. A001949, A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141021, A141031, A141064, A141065, A141066, A141067, A141069, A141070, A141072, A141073.

Sequence in context: A059623 A140997 A140996 * A152568 A155038 A057728

Adjacent sequences:  A141017 A141018 A141019 * A141021 A141022 A141023

KEYWORD

nonn,tabl

AUTHOR

Juri-Stepan Gerasimov, Jul 11 2008

EXTENSIONS

Partially edited by N. J. A. Sloane, Jul 18 2008

Recurrence rewritten by R. J. Mathar, Sep 19 2008

STATUS

approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)