cp's OEIS Frontend

A156050 Triangle T(n,m) = binomial(n,m)+2*P(n,m) read by rows, where P(n,m) = 1+A000041(n)-A000041(m)-A000041(n-m).

%I A156050 #7 Jan 24 2013 03:05:16
%S A156050 1,1,1,1,4,1,1,5,5,1,1,8,10,8,1,1,9,16,16,9,1,1,14,25,32,25,14,1,1,15,
%T A156050 35,51,51,35,15,1,1,22,48,82,96,82,48,22,1,1,25,64,118,164,164,118,64,
%U A156050 25,1,1,34,83,170,264,310,264,170,83,34,1
%N A156050 Triangle T(n,m) = binomial(n,m)+2*P(n,m) read by rows, where P(n,m) = 1+A000041(n)-A000041(m)-A000041(n-m).
%C A156050 The element-by-element sum of: the Pascal triangle A007318 plus two times the elements of P(n,m).
%C A156050 Row sums are 2^n+2*( (n+1)*(1+A000041(n)) -2*A000070(n) ), starting 1, 2, 6, 12, 28, 52, 112, 204, 402, 744, 1414,..., always below factorial(n+1).
%C A156050 The remarkable thing about this sub-Eulerian numbers triangle is that it a Sierpinski gasket modulo 2.
%e A156050 P(n,m) starts in row n= 0 as
%e A156050 0
%e A156050 0, 0
%e A156050 0, 1, 0
%e A156050 0, 1, 1, 0
%e A156050 0, 2, 2, 2, 0
%e A156050 0, 2, 3, 3, 2, 0
%e A156050 0, 4, 5, 6, 5, 4, 0
%e A156050 0, 4, 7, 8, 8, 7, 4, 0
%e A156050 0, 7, 10, 13, 13, 13, 10, 7, 0
%e A156050 0, 8, 14, 17, 19, 19, 17, 14, 8, 0
%e A156050 0, 12, 19, 25, 27, 29, 27, 25, 19, 12, 0
%e A156050 to yield T(n,m) from row n=0 on:
%e A156050 1,
%e A156050 1, 1,
%e A156050 1, 4, 1,
%e A156050 1, 5, 5, 1,
%e A156050 1, 8, 10, 8, 1,
%e A156050 1, 9, 16, 16, 9, 1,
%e A156050 1, 14, 25, 32, 25, 14, 1,
%e A156050 1, 15, 35, 51, 51, 35, 15, 1,
%e A156050 1, 22, 48, 82, 96, 82, 48, 22, 1,
%e A156050 1, 25, 64, 118, 164, 164, 118, 64, 25, 1,
%e A156050 1, 34, 83, 170, 264, 310, 264, 170, 83, 34, 1
%t A156050 Clear[f];
%t A156050 t[n_, m_] = 1 + PartitionsP[n] - PartitionsP[m] - PartitionsP[n - m]; \! Table[(Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]])/2, {n, 0, 10}];
%t A156050 Table[Table[Binomial[n, m], {m, 0, n}] + (Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]]), {n, 0, 10}];
%t A156050 Flatten[%]
%K A156050 nonn,tabl,easy
%O A156050 0,5
%A A156050 _Roger L. Bagula_, Feb 02 2009
%E A156050 Row sum formula and P(n,m) examples added - The Assoc. Eds. of the OEIS, Aug 29 2010