A201385 - OEIS

A201385

Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: T(n,1) = 2^n - 1; for k>1, T(n,k) = 0 for n <= 2*(k-1); otherwise T(n+1,k) = T(n,k-1) + T(n,k).

%I #38 Nov 08 2023 09:40:02

%S 1,3,7,3,15,10,31,25,10,63,56,35,127,119,91,35,255,246,210,126,511,

%T 501,456,336,126,1023,1012,957,792,462,2047,2035,1969,1749,1254,462,

%U 4095,4082,4004,3718,3003,1716,8191,8177,8086,7722,6721,4719,1716

%N Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: T(n,1) = 2^n - 1; for k>1, T(n,k) = 0 for n <= 2*(k-1); otherwise T(n+1,k) = T(n,k-1) + T(n,k).

%C A "Pascal Staircase".

%C The zero entries simplify the definition, but are not part of the official triangle.

%H Alois P. Heinz, <a href="/A201385/b201385.txt">Rows n = 1..200, flattened</a>

%H Ozer Ozturk and Piotr Pragacz, <a href="http://arxiv.org/abs/1111.6612">On Schur function expansions of Thom polynomials</a>, arXiv:1111.6612 [math.AG], 2011-2012. See (59), p. 22.

%e Triangle begins:

%e 1

%e 3

%e 7 3

%e 15 10

%e 31 25 10

%e 63 56 35

%e 127 119 91 35

%e ...

%t With[{rowmax=20},DeleteCases[Transpose[PadLeft[NestWhileList[Accumulate[#[[2;;-2]]]&,2^Range[rowmax]-1,Length[#]>2&]]],0,2]] (* _Paolo Xausa_, Nov 07 2023 *)

%Y Columns k = 1, 2, 3 give A000225, A000247, A272352(n+1).

%Y Row sums give A130783.

%K nonn,tabf,easy

%O 1,2

%A _Jonathan Vos Post_, Nov 30 2011

%E Entry revised by _N. J. A. Sloane_, Nov 07 2023