# [seqfan] Re: How is A116672 defined

Neil Sloane njasloane at gmail.com
Wed Jul 19 16:31:37 CEST 2017

```I agree that A116672 is mysterious.  I made a few grammatical improvements,
and I created a new triangle A289656 to serve as a possible missing link
between A116672 and Pascal's A007318.  However, the mystery remains.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Tue, Jul 18, 2017 at 9:22 AM, Richard J. Mathar <mathar at mpia-hd.mpg.de>
wrote:

> Does someone know how the terms of A116672 (related to the inverse
> binomial transform of Euler transforms of the diagonals of Pascal's
> triangle)
> are defined?
> A glimpse of what might be intended is given in A008778, A008779 and
> A008780, but the NAME which relates this to A007318 appears to be
> obfuscating, and I cannot figure out how the comments (which are just an
> example) are related to the NAME.
>
> We observe that the Euler transform of the diagonal of Pascal's
>   1, 1, 1, 1, 1, 1, 1, 1, 1, 1
> is
>   1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77 (A000041).
>
> The Euler transform of the first sub-diagonal
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
> is
> 1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479 (A000219)
>
> The Euler transform of the 3rd sub-diagonal
> 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120
> is
> 1, 1, 4, 10, 26, 59, 141, 310, 692, 1483, 3162, 6583 (A000294)
>
> The Euler transform of the 4th sub-diagonal
> 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455...
> is
> 1, 1, 5, 15, 45, 120, 331, 855, 2214, 5545, 13741, 33362 (A000335)
>
> The Euler transform of the 5th sub-diagonal
> 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820
> is
> 1, 1, 6, 21, 71, 216, 672, 1982, 5817, 16582, 46633, 128704 (A000391)
>
> How might this be related to Arnold's triangle?
> Is this just an overall (failed) attempt to rephrase some approximate
> formulas of higher dimensional partitions?
>
> Richard Mathar
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```