# [seqfan] How is A116672 defined

Richard J. Mathar mathar at mpia-hd.mpg.de
Tue Jul 18 15:22:34 CEST 2017

```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

```